Econ Homework

Chapter 2: Market forces Supply & Demand

This chapter includes four important elements:

1. A “change in quantity” demanded or supplied as a result of a change in the current price.

This is a movement along the demand or supply curve. This helps us understand the slope of demand or supply with respect to the price and then estimate their own price elasticities.

2. A “shift or change in the demand or supply” as a result of a change in a relevant “factor other” than the current price. This change represents a change in the entire demand or supply or a shift. Understanding the factors that shift the demand or supply help us specify and estimate a demand or supply equation and estimate the other factors’ elasticities

How do we distinguish a “change in quantity demanded or supplied” from a “change in demand or supply”? If the factor that changes is on any of the axes (such as the current price is on the vertical axis), then there is a “change in quantity demanded or supplied”. But if the change is in a factor that is not on any of the axes such as income or cost of production, then there is a “shift or change in demand or supply”.

The student should define the slope of direct demand or supply with respect to current price as “change in quantity over change in price”. Not the other way!

Example:

(Direct) demand: Qdx = 6,060 – 3Px. Slope of demand = ∆Q/∆P = -3

Inverse demand: Px = 2020 -1/3Qdx. Slope of inverse demand = ∆P/∆Q= -1/3

3. Consumer and producer surplus

What is the usefulness of calculating the consumer surplus for the manager? The manager can use it in price discrimination and in valuing full economic prices. What’s the usefulness of knowing the producer surplus? The producer can use it to bargain with the distributor over the surplus above minimum cost of producing the good accruing to the distributor.

4. Market equilibrium and disequilibrium (or price restrictions)

“Market equilibrium” means supply equals demand and there is no surplus or shortage. This helps determine equilibrium price and quantity.

“Market disequilibrium” means that supply and demand do not intersect or are not equal at any price in the market. In this case, we have either a surplus (quantity supplied exceeds quantity demanded) or a shortage (quantity demanded exceeds quantity supplied). This helps us determine the size of shortage or surplus.

When a government intervenes in the market and buys the surplus to set a price above the equilibrium price, then there is a “price floor” as is the case with agricultural products.

If the government issues a decree and sets the price below the equilibrium price then there is a “price ceiling or control” which leads to shortages. Some governments set a rent control for apartments.

THE SUPPLY FUNCTION

Supply function and Shifts in Market Supply

Supply Specification: The simple supply equation is defined as:

Qs = a + bP

and the slope with the respect to the price ∆Q/∆P is positive. That is the supply curve is upward sloping.

The general (direct) market supply equation can be expressed as a function.

QS = f (P; Production cost, Taxes, Expected Price).

where “P” is the current period price for this good and is different from the expected future price. Change in current price causes a change in quantity supplied or a movement along the curve. The other factors (after the semi colon) are the shifters which cause a change or a shift in the entire supply. Production cost includes the cost of labor, represented by the wage rate “ PW ”, capital cost represented by “PR”, the rental price of capital (equipment), and the price of raw materials “PM”. “ T” will represent taxes and Pex represents the expected future price for this good. Then the general (direct) supply function can be rewritten as QS= f (P; PW, PR, PM, T, Pex)

For example, Qs = 2000 + 3P –PW - 4PR – 2PM –T + - Pex.

where (direct) slope of supply with respect to (w. r. t.) current price P, ∆Q/∆P, is +3, and the slope w.r.t., PW, ∆Q/∆PW, is -1, and w. r. t. PR , ∆Q/∆PR , is -4, ∆Q/∆Pex > 0 or <0.

If PW = $20, PR = $40, PM = $10, T = $100 and Pex = $15, then after substitution for the constant shifters, the general (direct) supply equation collapses to the simple (direct) supply equation:

Qs = 1715 + 3P, which is generally written as Qs = a + bP.

All the “other variables” have been lumped with the intercept and the simple direct slope is ∆Q/∆P = +3. In the simple (direct) supply equation, all the variables after the “; ” are the factors that are held constant and are usually lumped together to form the intercept a . They are the “Shifters”. As indicated above, simple direct supply equation is given by:

Qs = a + bP. (Here, a is the horizontal intercept and b is the direct slope).

The simple “inverse” supply function is P = -a/b + (1/b)Qs

where +1/b is the inverse supply slope and -a/b is the vertical intercept.

A graph of the simple supply function is given by S1 below (P is placed on the vertical axis). Using the above example, P = -1715 / 3 + 1/3*Qs where 1/3 is the inverse slope.

Examples of shifts in Supply: Suppose labor production cost decreases and also assume no changes in the other variables including the current price “P”. A reduction in production cost implies an increase in profit (the difference between total revenues and costs), which should increase quantity supplied. The increase in the quantity supplied while the current price is assumed constant implies a right shift (an increase) in the supply curve from S1 to S2 in the graph below. The sign for wage rate should be -.

In conclusion, a decrease in the wage rate (W) implies an increase in the quantity supplied QS, assuming P is constant, which means a rightward shift in supply curve, and vice versa for an increase in (W), which implies a shift in supply to the left.

The same logic applies to decreases or increases in PR and PM.

Changes in Expected Future Price (Pex): These changes are applied to the price expected to prevail in the next period. Their effect on quantity supplied in this period depends on the storability of the good in question.

( Storable Good; e.g., oil )

In the special case when the good is storable (e.g., oil, gold, … etc) then an increase in the expected price implies storing the good instead of producing more of it. Then at the current price, current quantity supplied decreases, representing a shift to the left in the supply curve.

In general, an increase in expected price should shift the supply curve to the right, which is the normal case. This is particularly true for non-storable goods. If the good is non-storable such as milk, then an increase in Pex leads to an increase in current QS because the production of non-storable goods does not take much time to bring them on stream and thus, the firms worry about maintaining their market shares. Therefore, the supply curve shifts to the right, assuming the other variables are constant.

Taxes: There are basically two types of taxes: Specific and ad valorem. The specific tax is a fixed amount of money per unit sold (e.g., 10 cents per pound), while ad valorem is proportional to the value or the price (e.g., 10% of the price) which may not be constant. An example of a specific tax is the excise tax, which is a constant $ tax on each unit sold and the tax revenue is collected from the supplier. In this case the (inverse) supply curve shifts up in a parallel fashion by the amount of the tax.

Fig. 2.7 A per Unit (Excise) Specific Tax

If the tax is ad valorem, then if the price increases, the amount of the proportional tax increases with the price. Suppose the tax rate =20%. If P =$10 then the tax amount is $2. If P=$20, then the tax is $4. In this case the shift in supply is really an upward rotation .

Note that after tax, P1 = S1 = (1+t%)*S0 where supply S0 =P0 is expressed as an inverse supply equation:

P0 = -a/b+ (1/b)Qs0 which is S0, where P0 is price before tax.

Then S1 is

P1= (1+ t%)*P0 = (1+ t%)* [-a/b + (1/b)Qs0], where P1 is price after tax.

Solve for Qs1 as a function of P1.

Direct supply after tax: Qs1 = a + (b/(1+t%))P1= a + bP1 + t*bP1

(note that (1+ t%) = 1+ 20%) = 1.20 in the above example)

Fig. 2-8 An Ad Valorem Tax (t=20%)

That is, inverse S0 rotates upward to S1.

Producer Surplus

The points on the supply curve measure the minimum amounts (or prices) the producers are willing to “accept” for producing the good because the supply curve is a cost curve. Those amounts are tantamount to minimum costs necessary to produce different levels of the good, and these costs are usually lower than the market price. Supply price is a minimum price.

Suppose the (direct) supply equation for TVs is given by:

Qs = 2000 + 3P –PW - 4PR = 2000 + 3P –2000 - 4*100 = -400+ 3P.

where PR is the rental price of monitors (a complement) per unit representing capital cost and PW is the price of an input like labor or the wage rate.

Suppose PR = $100 and PW = $2000.

Then the simple (direct) supply equation is: Qs = -400 + 3P (where -400 is horizontal intercept).

The inverse supply equation is: P = 400/3 + (1/3) Qs (400/3 is vertical intercept).

Fig. 2.9: Producer surplus

In Fig 2.9 (Producers Surplus), the cost per unit to produce the first unit of the good is $400/3 (point C) and to produce the 800 units per unit is $400 (point B).In this figure, suppose the market price is $400 and this market price applies to all units. Upon substitution, the quantity is 800 units Then the sales revenues received by the producers are P*Q = $400 * 800 = $320,000. This is the area of rectangle [0 A B 800].

The area under the supply curve up to the point where the price line intersects the supply curve is the minimum cost associated with producing 800 units (an integral). Then

Producer Surplus = Revenues received – minimum amount necessary to produce the good or PS = TR –VC where VC is variable cost.

= Area of the triangle ABC

= ½ * H * B

= ½ *($400 – 400/3)*800

= $106,668. (for the wholesaler A)

Graphically, PS is the area below the price line and above the supply curve. It is a powerful tool for managers. In the above figure, suppose that the 800 units are 800 pounds of meat supplied by the meat wholesaler to the retailer which is the producer of steak (the restaurant). In this case, the restaurant manager (the retailer) can bargain with the meat wholesaler over the producer surplus (a maximum of $106,668) to capture some of it in the form of a lower price. Thus, the retailer can use the PS against the wholesaler.

MARKET DEMAND FUNCTION:

Specification as a general (direct) function:

QDX = f(PX ; Income, Prices of related goods, Advertising, other variables) or

QDX = f(PX; M, PY, PZ, A,H)

where goods “X” and “Y” are substitutes, and “X” and “Z” are complements. The variables after the semi colon are the shifters of the demand curve.

The simple (direct) demand depends on current own price and assumes all the other variables are constant. QD = c - dP (add or subtract the shifters to or from horizontal intercept). Direct price slope ∆QD /∆P = - d.

When it comes to income (M), there are two types of goods: Normal and Inferior. In case of normal goods, an increase in income (holding the other variables constant), would lead to an increase in purchasing power, which manifests itself in an increase in demand. Demand curve shifts to the right, assuming no change in current price. ∆QDX /∆M > 0

In case of inferior goods, an increase in income leads to a reduction in quantity demanded at the same price. Thus, the demand curve shifts to the left. ∆QDX /∆M < 0

Related goods can be substitutes or complements.

· If goods X and Y (Tea and Coffee) are substitutes , then an increase in PY (of coffee) would lead to a decrease in quantity demanded of Y(coffee). On the other hand, it is assumed that there is no change in PX (tea) then people switch from coffee to tea, which means quantity demanded of tea (QX) increases at the same price of tea PX. This implies a shift in demand for tea (X). Relation between PY and QX is ∆QDX /∆PY > 0.

· If the two goods X and Z (Printers and Computers) are complementary , then an increase in price of computers (PZ) would lead to a leftward shift in demand for printers (DX). Relation between PZ and QX is negative, ∆QDX /∆PZ < 0 .

Advertising (A) also shifts the demand curve. An increase in advertising shifts the demand curve to the right. There are two types of advertising: informative advertising which provides information about the existence or quality of a product, and persuasive advertising which alters the underlying taste of the consumer “You must buy it” or “The only thing you should buy”. ∆QDX /∆A > 0 .

Consumer Expectations (changes in expected prices, expected income etc): Demand for durable goods (e.g., cars) is affected by changes in expected prices. However, demand for perishable products (e.g. milk, eggs) is not affected much by expectation of higher prices.

Other factors (H) are special factors related to certain products such as “Health Scares” related to cigarettes” or the birth of a baby related to diapers.

The general linear (direct) demand equations can be written as

QDX = b0 - b1PX + b2PY - b3PZ + b4M + b5A.

The own direct slope with respect to PX is: ∆QDX /∆PX = -b1 < 0 (simple demand has a negative slope). (What’s the “indirect” demand slope for the simple demand?) Is it -1/b1?

The sign for income (M) depends on whether good X is normal or inferior.

For example, if the slope with respect to M is ∆ QDX /∆ M = +b4 (then the good is normal). If ∆ QDX /∆ M is negative, the good is inferior.

The slope with respect to PY is: ∆QDX /∆PY = +b2 >0 (positive means X and Y are substitutes ). If ∆QDX /∆PZ = -b3 < 0 then X and Z are complements. ∆QDX /∆Z = b5

Demonstration Problem 2-1

A firm’s manager was given the estimate of the direct demand function or equation for his/her firm’s product X:

Qdx = 12,000 – 3Px + 4Py – 1M + 2Ax

Please answer the following questions:

1. What type of goods are X and Y (with respect to the price of Y)?

2. What type of good is X (with respect to income)? Normal? Inferior? Why?

3. How does advertising affect this firm’ product?

4. Let Py = $400, M =$1,000 and A = $100. Derive the simple inverse demand and calculate the inverse slope (hint: plug the numbers in the equation and solve for Px ).

Is it: PX =12,800/3 -1/3QD?

Consumer Surplus (area below the demand curve above the price line)

Points on the demand curve signals the maximum amount a consumer is willing to pay per unit for a certain amount of a product. This maximum amount falls as more of a product is consumed and it is also different from the market price. Demand price is a maximum price.

Lets us look at the demand for water. Suppose at zero the consumer is willing to pay $5 to have the first liter of water (see Fig. 2-5a).

Fig. 2-5: Consumer Surplus

In discrete terms , after this consumer consumes the first liter he/she is willing to pay $4 for the second liter. Once this consumer has enjoyed 2 liters, it is willing to pay $3 per liter and so on.

For the continuous, case, what is the total value (benefit) of 2 liters of water? (Area under the demand curve to the horizontal axis = area of rectangle + area of triangle).

Max total benefits = $3 x (2 liters) + ½*($5 - $3)*(2 liters) = $6 + $2 = $8.

In the market, the consumer does not pay different prices for different units. Here, the market price after buying the second liter is $3. Total consumer expenses are $3x2 units = $6.

Consumer Surplus = Total Maximum Willingness to pay - Total expenses = $8 - $6=$2.

This concept is useful in disciplines that emphasize price discrimination where producers try to capture CS from consumers. You can also calculate CS directly by calculating the area of the shaded triangle above. CS = ½*H*B = ½*($5 - $3)*(2 liters) = $2.

Market Equilibrium

Market Equilibrium: Supply intersects demand. It includes the equilibrium quantity “Qe” (or Q*) and equilibrium price Pe (or P*). After the equilibrium, there is no shortage or surplus. Quantity supplied equals quantity demanded as shown in the graph below.

(Fig. 2-10: Market Equilibrium)

Demonstration Problem 2-4:

Simple Direct Market Demand: QD = 6 - 0.5*P.

Simple Direct Market Supply: QS = 4 + 2*P.

Market Equilibrium: QD = QS .

6 – 0.5Pe = 4 + 2Pe

0.5Pe + 2Pe = 6 - 4

2.5Pe = 2

Solve for Pe. Then

Pe = 2 / 2.5 = $ 0.8 (equilibrium price also called P* ).

Plug Pe in either supply or demand equation to determine the equilibrium quantity:

Qe = 4 + 2Pe = 4 + 2(0.8) = 5.6 units (equilibrium quantity)

Thus, market equilibrium = (Pe; Qe) = ($0.8; 5.6 units).

The graph of this market equilibrium is given by

(Fig. 2-10: Market Equilibrium)

Free Market Mechanism: The tendency of the market price to change as a result of market forces in order to clear the market (i.e., to equate QS and QD).

· If P1 > Pe then QS > QD (Surplus).

Then there would be a downward pressure on “P”, and once “, until QS = QD at Pe.

· If P2 < Pe, then QD > QS (shortage) and there would be an upward pressure on the price, shrinking the shortage, until QS = QD at Pe.

PRICE RESTRICTIONS AND MARKET (DIS)EQUILIBRIUM

There are two types of market disequilibrium: Price controls (or ceiling) and Price floor (or support). Disequilibrium means supply does not equal demand.

Price Control or Ceiling ( PC) : Government’s intervention (e.g., rent control) prevents market price from moving up to clear the market and achieve equilibrium. Thus PC < Pe ; where PC is the ceiling price. That is, price ceiling is below equilibrium price.

http://daphne.palomar.edu/llee/101Chapter08.pdf

(Figure 2-11a: Price ceiling)

Price controls such as rent controls lead to shortages because the controlled price is too low.

Total shortages = QCD – QCS.

If these are apartments, then the total shortage can be divided relative to equilibrium into two parts:

Qe - QCS = # of existing apartments that are taken off the market relative to Qe.

QCD – Qe = # of new apartments which are sought by new renters relative to Qe

Demonstration Problem 2-5 (apartments)

Demand: QD = 100 – 5P (where Q is in 10,000 units and P is in $100, and the zeros can be ignored).

Supply: QS = 50 + 5P

a. Calculate market equilibrium

QD = QS

100 – 5Pe = 50 + 5Pe → Pe = $5 (one hundred) and Qe = 75 (0,000) units

b. Assume ceiling PC = $1 (one hundred). Calculate the total shortage .

QCD = 100 - 5 PC = 95(0,000 ) units (total quantity demanded at price ceiling).

QCS = 50 + 5 PC = 55 (0,000) units.

Total shortage = QCD – QCS = 95 – 55 = 40 (0,000) apartments.

c. If the average apartment has three persons, then

# of displaced residents = (3)*( Qe - QCS ) =(3)*(75-55) = 60 (0,000) persons

# of new residents = (3)*( QCD – Qe ) = (3)*(95-75) = 60 (0,000) persons

How Do Businesses Deal with Losses Created by Price Ceilings?

Price ceilings provide a gain for buyers and a loss for sellers. Sellers would like to avoid the loss if they can.

1. One way to do so is called a black market . In this case, the sellers illegally raise the price and hope to get away with it. So, for example, tickets to popular events are sold by scalpers at high prices. (In California, ticket scalping is not illegal if it is not conducted at the place the event takes place.) While there are many other examples, black markets are not smart; it is just too easy to be caught. It is also not smart because of the existence of gray markets.

2. A gray market is a way of getting around the price ceiling without actually doing anything illegal. There are two forms of gray market. (a) One form of gray market involves charging for goods or services that were formerly provided free . If the rent cannot be raised on the apartment, there is nothing preventing the landlord from charging for the parking space, charging for use of the elevator, charging for gardening and cleaning services, forcing the tenants to pay for electricity and water, and so forth. In New York, a rent-controlled apartment near Central Park might rent for $300 to $400 per month; in a free market, the rent would probably be $2,000 per month. To get in, one needs the key. This has been known to cost $1,000. This is not a refundable deposit; this is a charge to have the key. It is obviously worth it to be able to rent the apartment for $300 to $400 per month. A Berkeley apartment owner converted his apartment into a church. To be able to live there, one had to pay church dues of $1,200 per year in addition to the rent. Gasoline stations would commonly charge for washing the windows, checking the tires, and so forth. The price of oil used in oil changes would be raised. (Those having oil changes at the station were favored in access to gasoline during the years of the price ceiling. In these years, Americans had the cleanest engines in history.) Some gas station owners ran the line to the gasoline pump through the car wash. One San Diego station forced people to have a $7 car wash to get to the gasoline pump. ($7 in these years is the equivalent of about $20 today.). This practice was later declared illegal. (b) The second form of gray market is to provide less service for the same price .

Welfare Impact of Price Ceiling

Since there is a shortage, there should be an allocation mechanism to allocate the good among the consumers. The most common mechanism is (first come, first served). In times of severe shortage, consumers must spend some time to wait in line or search for the good or apartment. Suppose the demand is for gasoline and the consumer wants to buy 10 gallons. Moreover, assume this consumer must wait for two hours in line to get the gasoline and that this consumer’s time is worth $5 an hour. This means the consumer is spending $1 per gallon in terms of waiting time to purchase gasoline (non-pecuniary price), in addition to the price ceiling per gallon (pecuniary price). This is called the full economic price.

Example:

Full economic price can be depicted graphically as:

( Figure 2-11b: Full Economic Price and Welfare Impact of Price Ceiling)

Example : Suppose the maximum price the consumers are “willing to pay” per unit is PF = $11 (called full economic price and is assumed) and the pecuniary price ceiling per unit is PC = $5.

Thus ( PF – PC ) = (11-6) = $6 is called the non pecuniary price per unit the consumers are willing to pay by waiting in line (the implicit price per unit for waiting in line). Full econ price per unit is PF = PC + (PF - PC)

Full economic price = pecuniary dollar price + non pecuniary price. Note that PF is greater than the equilibrium price Pe.

Example: Calculating Full Economic Price Using Equations. In the apartment example above: The supply equation under the ceiling is

QSC = 50 + 5 PC = 55 units (by plugging PC = $1 in this supply equation) (STEP 1)

Next, set the demand equation under ceiling equal to 55 units and change PC to PF in this equation:

QDC = 100 – 5PF = 55 units (STEP 2)

Then solve this equation for full economic price, PF = (100 - 55)/5 = $9. (STEP 3)

Compare this full economic price to:

Equilibrium Pe = $5 and to ceiling price PC = $1.

The non-pecuniary price of the good is: (STEP 4)

PF – PC = $9 - $1 = $8. This is the value of your time waiting in line or searching per unit.

Non busy consumers with very low opportunity cost of waiting time may benefit from the price ceiling, while those with high value for opportunity cost of waiting time may be hurt by the relatively low price ceiling. If a politician’s constituents have a relatively low opportunity cost of time, that politician naturally will attempt to invoke a price ceiling.

Another mechanism to allocate the good that is in short supply is to sell the good to the regular customers (e.g., gas stations during crises sell to their regular customers).

How to calculate the cost of welfare (CS + PS) lost due to price ceiling? It is the area of the shaded triangle in Fig. 2-11b.

= 1/2*(PF - Pc)*(Qe – Q Sc) = ½($9 - $1)*(75- 55) = $80

=1/2 *nonpecuniary price* supply shortage relative to equilibrium

Price Floor or Support: The government sets the price floor (Pf ) above the equilibrium price to support farmers’ income. Price support leads to surpluses, which are usually purchased by the government. Thus,

Pf > P* above equilibrium price.

Because the intervention price (Pf) is set too high, there is a surplus of this agricultural commodity.

http://daphne.palomar.edu/llee/101Chapter08.pdf

Total Surplus = QfS - QfD . For the price to stay at Pf , the government must purchase the surplus.

Cost of purchasing the surplus is illustrated in Figure 2-12 (A Price Floor).

The cost of purchasing the surplus = amount of surplus * price floor.

How Do Businesses Solve the Surplus Problem?

There were many ways to solve the problem of surpluses.

1. Occasionally, a store simply broke the manufacturer's policy . The store lowered the price to get rid of the surplus. The manufacturer had threatened that the store would be prohibited from selling the manufacturer's product; the store either believed that the manufacturer would not carry-out the threat or did not care. For example, Crown Books began lowering the prices of its books and a company called Discount Records began lowering the prices of phonograph records.

2. More likely, stores would try to get around the price floor without actually violating. (a) One common solution was to provide more service for the same money. Stereo stores could add free CDs or other free accessories. Washing machine stores used to virtually give away the dryer. Gas stations gave away glasses, knives, and Blue Chip Stamps. (b) A second solution was to simply absorb the surplus . Your textbook producers would have a surplus of textbooks. At the end of each edition, the books would be returned to the publisher and the paper was recycled. (c) A third solution was to change the name of the product in order to reduce the price. Surplus gasoline was sold to independent dealers who would sell it as Thrifty, 7-11, or Discount Gas at a lower price. Surplus liquor was bottled with a different label and sold as Slim Price, or Yellow Wrap at a lower price. Surplus washing machines and refrigerators were sold, for example, to Sears and marketed as Kenmore at a lower price. When automobiles were fair-traded, the dealers could not lower the price; however, they would give a trade-in value that was much greater than the trade-in car was actually worth. The main point here is that, even if someone interferes with the market process, there are powerful forces to return to equilibrium

COMPARATIVE STATICS (within supply /demand framework)

Changes in Demand

Suppose there is an increase in income (the case of normal good), or in the price of the substitute or in the expected price (the case of a durable good). These variables are determinants of demand. Then the demand curve will shift up. In the supply/demand framework, both equilibrium price and quantity change when there is a shift in demand. Both will increase in this case.

The opposite shift in demand will happen if there is an increase in price of a complement.

or increase of income and the good is inferior

Changes in Supply

In reality, both P and QS change when supply shifts. For example, if there is an increase in PR ( rental price of capital) or Pw (wage rate) or the production cost then the supply curve will shift to the left, creating new market equilibrium with a higher equilibrium price ( P*2) and lower equilibrium quantity ( Q*2 ).

(Increase in R)

Chapter 3: Quantitative Demand Analysis

Chapter 3: Qualitative Demand Analysis

Assignment: The regression spreadsheet at the end of chapter.

This chapter includes three important elements:

1. In contrast to the previous chapter which examines the direction of change

(positive or negative slope), this chapter examines the magnitude of change or percentage of change (i.e., elasticities)

2. Elasticities. Any elasticity is defined as a percentage change over a percentage change. The slope, which is a part of the elasticities, is a change over a change. There are three elasticities for demand. The own price elasticity helps marketing mangers in deciding whether to increase the price or decrease it in order to increase sales revenues. The cross price elasticities help mangers determine the effect of a change in the price of a substitute or complementary product on the demand of their product. The income elasticity measures the responsiveness to changes in income.

THE ELASTCITY CONCEPT

(Elasticity = % ∆ / % ∆ )

A price elasticity of demand, for example, measures how much quantity will change in percentage terms when a price changes by a certain percentage. (Direct price Elasticity = % ∆ Q /% ∆P )

Example: suppose:

%∆P = + 5%; Price elasticity = – 2; then %∆Q = (elasticity)* % ∆P = -2 *5% = -10%.

OWN (direct) PRICE ELASTICITY OF DEMAND

“Own” means we use the % change in the quantity and the % change in price for the same good, say x. “Direct” means % ∆QD / % ∆P and not the inverse.

First, I will present the two definitions of the point elasticities then I will provide the definition of the midpoint or arc elasticity which is more relevant for the “total revenue test”.

First definition: point direct elasticity (moving from say point A to point B).

EPD = % ∆QD / % ∆P.

This definition can be rewritten for a direct demand schedule as

EPD = ∆Q / Q = (Q2 –Q1) / Q1

∆P/P (P2 –P1) / P1

Example:

P	QD
$9 (P1 )	15 Units Q1
7 (P2 )	25 Q2

EPD = ( Q1 - Q2 )/ Q1

(P1 - P2) / P1

= (25 - 15) / 15 = - 3

(7 - 9) / 9

Second Definition: point elasticity (moving from point B to point A)

EPD = ( Q1 - Q2 )/ Q2

(P1 - P2) / P2

= -1.4 (the same example above but with different elasticities)

First Def.: Moving From Point A to Point B EPD = ( Q2-Q1) / Q1 = (25 - 15)/15 (P2-P1) / P1 (7 - 9) / 9 = -3

Second Def.: Moving From Point B to Point A EPD = (Q1 - Q2 )/ Q2 = (15 -25)/25 (P1 - P2) / P2 (9 - 7) / 7 = -1.4

The third definition: Mid point elasticity

In the graph above, we move from point A or B to the midpoint .

EPD = (Q2 - Q1) / ½(Q1 + Q2) = ( Q2 - Q1) / average Q =

(P2 - P1) / ½ (P1 + P2) (P2 - P1) / average P

where P and Q with bars in the graph are averages for quantities 1/2*(Q1 +Q2) and for the prices ½*(P1 + P2), respectively. Those averages are equal to 8 and 20 for quantities and prices in the above graph, respectively.

Applying the midpoint (arc) elasticity formula to the above example, we have

(25 - 15)/ ½ (15 + 25)

(7 - 9) / ½ (7 + 9)

= -2

The movement is from point A to the midpoint (not to point B as is the case in the point elasticity in the graph above). See INSIDE BUSINESS 31 P. 80 for an example on calculating the midpoint (Arc ) elasticity for the housing market over one month change.

Own Price elasticity for a direct demand equation :

Let the direct demand equation be: QD = a – bP where ∆QD / ∆P = -b is the direct price slope. Then the direct elasticity = ∆QD/Q / ∆P/P = (∆QD / ∆P)*(average P / average Q) where ∆QD / ∆P is the slope of direct demand and (average P / average Q) is the location point on the demand curve. To calculate the “Averages”: Sum up all the values and then divide the sum by the number of observations.

Example : If QD = 6 - 1.5P

and average P = $ 2

average Q = 5 units.

Recall, direct slope = ∆Q/∆P = -1.5 in the equation above.

Then EPD = (∆Q/∆P)*average P/average Q = (-1.5)*2/5 = - 3/5

Co-efficient of EPD = │EPD│= absolute value of EPD .

This is used in order to avoid comparing two negative numbers for the elasticity but the price elasticity of demand is still negative.

Types of Elasticities : (see p. 81, Table 3-2 for real world estimates of elasticity)

If 0 < │ EPD │ < 1 (-.3, -.75, -.9 etc); then demand is price inelastic [see INSIDE BUSINESS 3-2 on demand for prescription drugs on Page 84]

If │ EPD │ > 1 (e.g., -1.3, -2, -5.6 etc); then demand is price elastic.

If │ EPD │= 1; then demand is unitary price elastic.

If │ EPD │ = ∞; then demand is perfectly price elastic. Here demand is a horizontal line.

If price drops then the quantity can go to infinity. Similarly, if price increases, quantity can drop to zero by an infinite amount. Thus, %∆Q = ∞ or (%∆Q / %∆P = ∞/ % ∆P.

If │EPD│ = 0; (%∆Q / % ∆P = 0/ % ∆P) then demand is perfectly price inelastic. Here demand is a vertical line. The quantity demanded does not change when price changes.

The quantity is not sensitive to changes in the price at al.

Examples: Demand for a heart transplant, demand for insulin.

Demand for illegal drugs is almost vertical. Putting drug pushers in jail is not enough.

Demand for cigarettes by youth smokers? Answer: Inelastic. Is there a difference in price elasticity of smoking between black and white Youth? Between youth with less educated and more educated parents? Answer: Black youth and youth with less educated parents have greater price elasticity of demand. The following uses coefficient of elasticity

(these elasticities are coefficients of elasticity but this elasticity is always negative)

Derivation of a Linear Demand Equation (without using regression)

Given two points on the demand curve, we can estimate the direct slope (-b) and the intercept (a) and have a derived or estimated simple demand equation.

P Q

$9 15 Units

$7 25 Units

The simple form of a linear demand equation:

QD = a – b P

Direct slope = ∆ QD /∆ P = -b = (25 – 15)/ (7 – 9) = -5

Therefore, –b = -5 and QD = a – 5 P

Then plug this into the general form for demand and solve for the intercept (a) at any one point, say ($9, 15), we have

15 = a - 5*9

Therefore, a = 60

Thus, the derived linear demand equation is QD = 60 –5P.

One can get the same answer by using the second point ($7, 25) to solve for (a).

Estimation of Price Elasticity of Demand along a Linear Demand Curve:

Recall EPD = ( ∆Q / ∆P)*(P/Q)

where ∆Q / ∆P is the slope of demand.

Example of a linear demand:

Q = 12 – 3 P

Then the direct slope = ∆Q / ∆P = -3 (constant). Find the endpoints on both axes.

Then estimate the price elasticities along the straight line demand curve as follows:

At point A : EPD = (∆Q/∆P)*P/Q= (-3) * (4/0) = - ∞ ( perfectly price elastic ).

At point B : EPD = (-3) * (2/6) = - 1 ( unitary price elastic ).

At point C : EPD = (-3) * (0/12) = 0 ( perfectly price inelastic ).

Total revenue Test :

In the following table, compare the change in the price and total revenue. Then relate this relationship to the type of price elasticity

P	Q	TR= P*D	Mid Point │EPD│	Conclusion
$9	15	$135	-
7	25	175 increases	2	P-elastic
5	35	175 no change	1	Unitary elastic
3	45	135 decreases	0.5	P-inelastic

1. If │Mid-point EPD│ > 1 (elastic), P and TR move in Opposite direction.

2. If │Mid-point EPD │ < 1 (inelastic), P and TR move in Same direction.

This test can be explained in the figure below which is different from the table above. In the range where demand is inelastic, an increase in the price corresponds with an increase in total revenues. In the elastic range, total revenue will decrease if price increases.

Determinants of the Own Price Elasticity of Demand:

1. Availability of substitutes: The greater the number of viable substitutes for a certain product, the greater the demand elasticity of that product. (Consumers move to the substitutes as a result of higher price and Q drops)

2. Time : For non-capital products (e.g., gasoline), short-term elasticity is less than long term elasticity in absolute value. Demand elasticities for these products grow over time. The opposite is true for capital goods.

3. Importance of a product in total budget: (or share of expenses on a certain product in the total budget). The lower the share of the product, the lower the elasticity (less elastic). Example: expenses on salt.

Compare price elasticity of food with that for transportation. Hint: In 2000 US consumers spent 14% of their incomes on food and 4% on transportation.

Time

Non-Capital Products (gasoline ):

Short run EPD < Long run EPD in absolute value

In the short run, people would merely drive less. In the long run, in addition to driving less, people replace their large cars with smaller and more fuel efficient cars. Thus,

‌ LR %∆QD ‌ > ‌ SR %∆QD ‌ in absolute value (more elastic in the L/R)

which means for a given % increase in the price, the long-run price elasticity in absolute value is greater than the short-run price elasticity.

Capital Goods : (Cars) :

In the short run , there will be a deferment of buying new cars by both first-time buyers and repeat buyers after the increase in the price of cars. But in the long run, the deferment will be by the first-time buyers only. Thus,

‌ ‌ SR % ∆ QD > LR % ∆ QD ‌ in absolute value‌.

Short-run price elasticity is greater than the long run price elasticity (i.e., more elastic in the short run).

Examples : (Table 3-3 on Page 82 for estimates of short-term and long-term elasticities).

Other example: Estimates of short- and long-run elasticities for non-capital and capital goods (gasoline and automobiles).

Non-Capital Goods (Gasoline):

The following are estimates of price elasticities of demand for gasoline after the oil price increased in 1974 and in 1979-80. Those estimates show that the elasticities change in the long run. The long-run price elasticities grew over time.

Years Following the Gasoline Price Increase

Elasticity	1	2	3….	5………..	10………….	15
EPD	-0.11	-0.22	-0.32	-0.49	-0.82	-1.17

The conclusion is for non capital goods: │EPD SR│ < │EPD LR│

Capital Goods (Automobiles)

Years Following the Price Increase

Elasticity	1	2	3….	5………..	10………….	15
EPD	-1.20	-0.93	-0.73	-0.55	-0.42	-0.40

The conclusion is for capital goods: │EPD SR│ > │EPD LR│

Marginal Revenue and Own Price Elasticity of Demand

Marginal revenue (MR) is the change of total revenue over the change in quantity. That is, MR = ∆R / ∆Q. MR is linked to the own price elasticity of demand. Notice first that if demand curve is linear (straight line), MR revenue bisects the distance on the horizontal axis between zero and where the demand curve hits the horizontal axis, and thus divides this distance into two equal parts. In the case MR is twice the slope of the inverse demand curve.

Example. Suppose direct demand is Q = 30 - 3P. Slope of direct demand ∆Q/∆P = -3.

Then the inverse demand is given by

P = 10 - 1/3*Q (inverse demand)

Slope of the inverse demand (∆P/∆Q ) is -1/3.

The slope of MR = 2*(-1/3) = -2/3 (twice slope of inverse demand). Then the equation for MR which has the same intercept as the inverse demand is:

MR = 10- 2/3*Q

Notice in the graph below, when MR = 0 the own price elasticity of demand is unitary. If MR is positive the demand is elastic, and if MR is negative the demand is inelastic

Example 2: Direct demand Q = 6 – P . Then P = 6 - Q and MR = 6 – 2Q .

Figure 3-3 Demand and Marginal Revenue.

The relationship between MR and the own (direct) price elasticity [E = %∆Q / % ∆P =

= (∆Q / ∆P)*(P/Q)] is given by

MR = P*[(1+E)/E], where E is the direct price elasticity of demand.

· Calculate MR if E = -2 (elastic). Then MR = P*(1-2)/-2 = 1/2P which is positive.

· Calculate MR if E = -1/2 (inelastic) Then MR = P*[(1-1/2)/-1/2] = -P ( which is negative?)

· Calculate MR if E = -1 (unitary elastic). Then MR = 0 (i.e., TR is at its maximum)

CROSS PRICE ELASTICITY OF DEMAND:

Two related goods: X and Y. Our good is X and the price of related good Y changed. Then the price elasticity of demand for X with respect to a change in price of Y is:

__

%∆QX ∆ QX PY

EDXPY = ______ = ____ * ___

%∆ PY ∆ PY QX

__ __

where PY and QX are average values, or values at a particular point.

If X and Y are substitutes then

EDXPY > 0. That is, the cross price elasticity is positive.

If X and Y are complements then,

EDXPY < 0. That is, the cross price elasticity is negative.

Example: Direct demand is given by QXD = 31 – 2PX + 0.5 PY

Note: The own direct slope with respect to X, ∆QX / ∆PX, is (-2) and the cross slope with respect to the price of Y, ∆QX / ∆PY, is (+0.5) and positive. In color:

QXD = 31 – 2PX + 0.5 PY

Suppose the averages are given by:

__ PX	__ PY	__ QX
$8	$10	20 Units

Then own price elasticity of demand for X with respect to own price X is:

___

∆ QX PX

EDXPX = * ___ = (-2)*(8/20) = -0.8 (inelastic)

∆ PX QX

Then price elasticity of demand for X with respect to the cross price of Y is:

___

∆ QX PY

EDXPY = * ___ = (+0.5)*(10/20) = +0.25 (Substitutes)

∆ PY QX

INCOME ELASTICITY OF DEMAND:

EMD = % ∆Q/ %∆M

EMD = %∆QD = ∆Q / average Q

%∆M ∆M/ average M

Or EMD = ∆Q * average M

∆ M average Q

where ∆Q is the slope of the demand with respect to income.

∆ M

If EMD > 0 (i.e., income slope is positive), then the good is normal (e.g., EMD for food =+ 0.80 which implies that food is a normal good).

If EMD < 0 , then the good is inferior (e.g., EMD for corned beef = -1.94).

If 0 < EMD < 1 , then the normal good is a necessity (e.g., food)

If EM0 > 1 , then the normal good is a luxury (e.g., recreation)

OBTAINING ELASTICITIES FROM DEMAND FUNCTIONS

First we will consider elasticities from linear demand functions which use linear regression, and the elasticities should be calculated. Then we proceed to elasticities of nonlinear demand functions which use log linear regression and elasticities are constants.

Linear demand equation ( without lagged dependent variable):

Qt = a – bPt + cMt + dAt + e PY

where t refers to time period, M denotes income and A denotes advertising.

The coefficients b, c, d and e are direct slopes with respect to P, M, A and PY, respectively, and these slopes can be used in deriving the elasticities by multiplying them by the averages (or locations). For example, ∆Q/∆P= -b. Then to form the price elasticity we have:

EDP = (∆QX /∆PX)*(average PX / average QX) = (-b)*(average PX / average QX) < 0.

Then to form the income elasticity, we have

EDM = (∆Q /∆M)*(average M / average Q) = (+c)*(average M / average Q) > <0

Cross price elasticity with respect to PY

EDPy = (∆QX/∆Py)*(average Py / average QX) = (e)*(average Py / average QX) > < 0.

Linear demand equation ( with lagged dependent variable):

Qt = a – bPt + cMt + dAt + e PY + fQt-1

The lagged Q, Qt-1 , quantifies habit forming behavior. If there is a habit of having X in the last period than we expect last period’s quantity to influence the current period’s quantity. Here, we can distinguish between the short-run elasticities and long-run elasticities. Note the estimate of the slope for Qt-1, which is in the equation is + f.

LR Price elasticity= SR EDP / (1- f) < 0, where f is estimated slope for lagged Q, Qt-1, and the slope should be positive.

LR Income elasticity= SR EDM / (1- f) > or < 0.

LR Cross price elasticit y = SR EDPY / (1- f) > or < 0 and so on.

Log linear demand equations (without a lagged dependent variable):

lnQ = a1 – b1lnP + c1lnM + d1lnA

where a1 = ln(A) and A is the intercept in the linear case. You can derive the original ” a” without the “log” from a1 by calculating the exponential of a1. That is, a= ea1. In this log-linear case, the parameters b1, c1 and d1 are constant elasticities of price, income and advertising, respectively. Specifically, - b1 = %ΔQ/%ΔP, c1 = %ΔQ/%ΔM and … so on. “ln” is the natural log symbol. Nothing should be done to these parameters because they are already estimated elasticities and they are not slopes. In excel, = ln(cell).

Note that the above functions can include the lagged dependent variable as one of the regressors to capture habit forming and in order to calculate both the short- and long-run elasticities. (see HW assignment for chapter 3)

Log linear Demand Equation (with a lagged Q) :

lnQ = a1 – b1lnP + c1lnM + e1lnQt-1

Note that the estimates of slope of lagged Q is the estimate of e1.

First, prepare the Excel spreadsheet (Copy the Table). Example for linear and log linear:

Spreadsheet for linear and log linear demand functions with Qt-1 (Three Independent Variables: P, M and lagged Q)
	Linear equation: Qt = a – bPt + cMt + fQt-1	Log Linear equation lnQ = a1 – b1lnP + c1lnM + e1lnQt-1
Year	Q	P	M	Lagged Q	lnQt	lnPt	lnMt	lnQt-1
1988	6	28	10		1.791759	3.332205	2.302585
1989	10	25	10	6	2.302585	3.218876	2.302585	1.791759
1990	13	18	10	10	2.564949	2.890372	2.302585	2.302585
1991	18	17	15	13	2.890372	2.833213	2.70805	2.564949
1992	22	15	15	18	3.091042	2.70805	2.70805	2.890372
1993	24	13	17	22	3.178054	2.564949	2.833213	3.091042
1994	27	12	20	24	3.295837	2.484907	2.995732	3.178054
1995	32	10	22	27	3.465736	2.302585	3.091042	3.295837
1996	36	10	25	32	3.583519	2.302585	3.218876	3.465736
Average	22.75	15	16.75		No averages

Skip the first row because Excel cannot run regressions with empty cells. To find the averages divide the sum by 8 (skip first row) in the example above and you may exclude the first row in doing the summation.

Estimation of a Linear Demand Function with Qt-1 (no price of Y in this example)
Regression Statistics
Multiple R	0.9971618
R Square	0.99433165
Adjusted R Square	0.99008039
Standard Error	0.89201694
Observations	8
ANOVA
	df	SS	MS	F	Significance F
Regression	3	558.3172231	186.1057	233.891	6.01301E-05
Residual	4	3.182776866	0.795694
Total	7	561.5
	Slopes	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	3.29129981	5.224732673	0.629946	0.562922	-11.21491369	17.79751	-11.2149	17.79751
Price	-0.132603	0.220193368	-0.60221	0.579505	-0.743959095	0.478753	-0.74396	0.478753
Income	0.68458864	0.295716861	2.315014**	0.081582	-0.136454693	1.505632	-0.13645	1.505632
Lagged Q	0.52530978	0.246056519	2.134915**	0.099656	-0.157854049	1.208474	-0.15785	1.208474
Write the estimates as an equation below (no price of a substitute is included in this equation):
Qt = 3.291 - 0.133 Pt + 0.685 Mt + 0.525 Qt-1
(0.63) (-0.60) (2.32) (2.13)
where ∆Q/∆P = --0.132603	and ∆Q/∆M =0.68458864	and so on
In this linear case, the estimated coefficients are the slopes. All the variables Price, Income and Lagged Q have the correct signs for a demand equation.
The price is not statistically significant at any level. Use the standard (large sample) ranges for statistical significance (%) and not the table P-values given with this regression output (see Question 1 in the HW for t-statistics ranges).
Short-run own price elasticity of demand = (The slope of the price) * (Average price / Average quantity)
= - 0.133 *(15/22.75) = - 0.088
See the text for more definitions of elasticities
Long run price elasticity of demand = SR P elasticity/(1-slope of lagged Q) = -0.088/(1-0.525) = - 0.185 or = [(Slope of price) / (1 - slope of lagged Q)]*(Average price/Average quantity)
= [(-0.133)/(1 - 0.525)]*(15/22.75) = -0.088/(1-0.525)= - 0.185 where 0.525 is the estimated slope for lagged Q.
The income elasticities can be estimated the same way by using ∆ income and average income instead of ∆price and average price in the above short run and long run formulas. (see P. 31 for more information on the formula for M -elasticity) Try it!! For the cross price elasticity use ∆PY and average PY to write the cross price elasticity. See P. 28 and P. 30).

Estimation of Log Linear Demand Function with Qt-1 ( no price of Y) lnQ = a1 – b1lnP + c1lnM + e1lnQt-1
Regression Statistics
Multiple R	0.998255
R Square	0.996513
Adjusted R Square	0.993897
Standard Error	0.034373
Observations	8
ANOVA
	df	SS	MS	F	Significance F
Regression	3	1.350558	0.450186	381.0214	2.28E-05
Residual	4	0.004726	0.001182
Total	7	1.355284
	Elasticities	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	0.689069	0.981391	0.702134	0.521302	-2.03572	3.413853	-2.03572	3.413853
Ln Price	-0.08368	0.224922	-0.37203	0.728742	-0.70816	0.540807	-0.70816	0.540807
Ln Income	0.456731	0.123854	3.687659	0.021062	0.112857	0.800605	0.112857	0.800605
Ln Lagged Q	0.465942	0.13362	3.487067	0.02519	0.094952	0.836931	0.094952	0.836931
ln Qt = ln a - a1 lnPt + a2 lnMt + a3 lnQt-1
where: Q is the Quantity. The coefficients a1, a2 and a3 are ELASTICITES.
P is the Price
M is the Income
t is the time period
ln is the natural log
The coefficients are elasticities.
a1 = %ΔQ/%ΔP = -0.084 = Short- run Price elasticity of demand (Do not make any changes)
a1/(1- slope of lagged Q) = a1 / (1 - a3) = -0.084/(1- 0.465942) = long-run price elasticity of demand
a2 = %ΔQ / %ΔI = 0.457 = Short- run Income elasticity of demand
a2//(1- slope of lagged Q) = a2/(1 - a3) = 0.457 /(1-0.466) long-run Income elasticity of demand
If income elasticity a2 > 0, then the good is normal

REGRESSION ANALYSIS

Please refer to pages 95 to 107 in the textbook for more information on regression analysis.

Also, see the link

http://www2.chass.ncsu.edu/garson/PA765/regress.htm

We will estimate a demand function using linear and log-linear regressions with lagged Q.

· Linear Regression (three independent variables) : The following demand function has three regressors P, M and Qt-1 .

Qt = a + bPt + cMt + dQt-1

where: Q is the Quantity (dependent variable)

P is the Price

M is the Income

Qt-1 is the lagged Q

t is the time period

· Input or copy the data on an EXCEL sheet, clearly specifying the dependent Y variable to be the quantity (Qt) (highlight its column), and the independent X variables to be the price (Pt), income (Mt) and the lagged Qt-1 or as the situation warrants.. Here we have three regressors: (Pt), income (Mt) and the lagged Qt-1 (highlight all of them at the same time).

· To enter values for the lagged Qt-1, you may copy the whole data under Qt and paste it in a new column added to the given sheet under the lagged Qt-1. Pasting should start such that the first observation under Qt will be the first observation under the lagged Qt-1 starting with the second row.

· Click on Excel icon on top left, Excel Options at the bottom of pop up menu,  Add-ins in the left hand column, then Analysis Toolpak, then hit ok.

·  

· if it does not come up, then hit go and make sure that Analysis Toolpak is checked.

· then under Data, Data analysis, Regression, ok.

·  

· If you have Analysis Toopak in your computer, then the road to regression is shorter. Click on Excel icon, Data, Data Analysis in the up far right  then Regression.

· Go to TOOLS menu and click DATA ANALYSIS. Pick up REGRESSION from the ANALYSIS TOOLS presented in the pop up menu and click OK.

· First highlight the dependent variable (Qt) cell range from the spreadsheet starting from the second row (skip the row with the empty cell), and click OK on the REGRESSION pop up menu to insert the selected data range in the Input Y range box. Similarly select the relevant data range for all the independent variables together including lagged Q and insert the selected data range in the Input X range box. Double check your cell ranges.

· Click on “LABEL” to include the symbols or names of variables in the regression output.

· In the OUTPUT OPTIONS, click New Worksheet Ply and say OK. The Regression output will be available to you on a newly created worksheet.

How to add DATA ANALYSIS to your TOOLS menu?

· If the TOOLS menu in your computer does not have DATA ANALYSIS, you can add it by doing the following.

· Open TOOLS

· Click on ADD-INS

· Include ANALYSIS TOOLPACK from the pop up menu dialog box and click OK.

· Go back to TOOLS and you will find DATA ANALYSIS at the bottom of the menu.

The Questions required for the homework assignment are listed

Below:

Homework assignment: Questions

QUESTION 1:

Copy the database below into an excel sheet.

Run QX on the four regressors: PX, M, PY and lagged Qx.

Write down the estimated linear demand equation with t-statistics under the estimated coefficients as done above. In addition, write down the R-square and explain what it means. Explain the statistical significance of the t-statistics for each regressor. Significance of T-statistics is usually given by the P-values in the regression output. We will not use it in here because we have a small sample which will bias the P-values. There are three levels of significance: 1%, 5% and 10% represented by ***, ** and *, respectively. Do not use the computed P-values of this small sample regression. Instead, use the following conventional t-statistics significance ranges used for large data:1.63 <t < 1.96 (10%); 1.96 < t < 2.54 (5%); and t > 2.54 (1%). This means in your regression output, look at the t-statistics column for each regressor. Then place the value of that computed t-statistic in one of the above ranges. The P-values given in the regression output are sensitive to sample size and are not accurate.

QUESTION 2

Check the signs of the estimated coefficients. Do the signs follow the theory as expected? Examine the sign for each regressor and point out what they mean.

QUESTION 3:

Calculate the short-run and long-run price and income elasticities of demand for good X using the averages for the quantity, price and income? Based on the income elasticity, what type is good X?

Short Run P elasticity for a linear Eq. = [slope of price]*(Average Price/Average quantity)

Long Run P elasticity for a linear Eq. = (SR P elasticity) /(1- slope of lagged Q)

or = [slope of price / (1- slope of the lagged variable)]*(Average Price/Average quantity).

They are the same.

Average = sum/n, skipping first row.

The short-run and long run income elasticities are calculated the same way. Here the slope is for income and the average for income (see page 31 or the solved regression on pp 32-33). What type of good is X with respect to income elasticities?

Short Run Income elasticity for a linear Eq. = [slope of Income]*(Average Income/Average quantity)

Long Run Income elasticity for a linear Eq. = (SR Income elasticity)/(1- slope of lagged Q)

QUESTION 4:

Calculate the short-run and long-run cross price elasticities with respect to Py (see p. 28 and p. 30 in the notes). What type of goods are X and Y with respect to these elasticities?

QUESTION 5

Can you think of another independent variable that you may add to the above equation? What will the sign of this variable be? Specify the name of this variable. Do not include Weather in this equation.

QUESTION 6

Is this a supply or demand equation? Why? Forget about signs. Look for other clues in the equation.

SEE DATA BELOW:

Copy the data from Word to excel.

After transferring the data set from Word to excel, make sure you follow these steps;

Highlight all the cells in excel.

Right click on any cell in the data sheet in excel.

Click on FORMAT CELLS.

Under CATEGORY, click on NUMBER.

Then click OK.

Spring 2010: Regression Assignment Data Sheet (linear case only))

When you copy in Excel 2007: COPY, PASTE SPECIAL then TEXT.

Year	Qx	Px	M	Py	Lagged QX
1984	9	29	14	11
1985	10	28	15	12	9
1986	12	25	18	14	10
1987	14	23	20	15	12
1988	16	20	23	17	14
1989	17	19	26	19.5	16
1990	18	17	29	21	17
1991	21	16	34	22	18
1992	26	14	37	23	21
1993	28	12.5	35	23.5	26
1994	29	12	38	25	28
1995	30	10	41	23	29
1996	33	14	44	20	30
1997	35	15	47	19	33
1998	38	18	51	20	35
1999	39	19	55	21	38
2000	40	21	58	22	39
2001	42	18	61	23	40
2002	45	18	63	25	42
2003	46	17	65	26	45
2004	50	15	66	21	46
2005	55	14	68	25	50
2006	57	12	70	27	55
2007	58	10	73	28	57
2008	61	9	74	28.5	58
2009	65	8.5	79	30	61
2010	66	7	80	31	65

Chapter 4: The Theory of Individual Behavior

CONSUMER BEHAVIOR

In any economy, there are many goods and services, and the consumers buy baskets (or bundles) of these goods and services. Consumers compare goods and services before they buy them. Comparison of possible baskets is based on tastes or preferences and not in terms of costs and prices. If there is a comparison, we say that there is a preference ordering.

Example: Alternative Baskets for Food and Clothing

Basket	Units of Food	Units of Clothing
A	20 units	30 units
B	10	50
D	40	20
E	30	40
G	10	20
H	10	40

Basic Properties of Preference Ordering :

The theory of consumer behavior begins with three basic assumptions about peoples’ preferences.

1. Property 4-1: Completeness. Preferences are assumed to be complete in the sense that consumers can compare and rank all possible baskets. This means that for any two baskets say A and B a consumer will prefer A to B, B to A or is indifferent between them. In the above example, take baskets A and E. The consumer prefers E to A (more of both). If you take baskets A and B, the consumer cannot rank these baskets. Thus, completeness does not hold for all possible baskets in the above example. The consumer is needed to express her or his preference or indifference among baskets. This assumption is needed for the manager to predict the consumer’s consumption patterns with reasonable accuracy.

2. Property 4-2: All goods are good not bad (i.e., desirable). Consumers prefer more of any good to less. Graphically, this assumption means the direction of increase in satisfaction is the Northeast. What will be the direction if one of the two goods is “bad”? Northwest?

3. Property 4-3: Transitivity. For any three bundles: S, T and U, if S is preferred to T and T is preferred to U, and then S is preferred to U if transitivity holds. This assumption rules out the possibility that the consumer will be caught in a perpetual preference ordering cycle in which it will never be able to make a choice at the end.

Since not all the baskets can be compared and ranked (that is completeness is not satisfied) we need additional information on preferences to rank all bundles. This additional information is the “indifference curve.”

Indifference Curves :

An indifference curve includes all the baskets (points) that generate the same level of satisfaction. If we graph the above example we can produce an indifference curve that compares the baskets which we could not compare before. This indifference curve (µ1) can include the baskets (A, B and D) without violating any of the above assumptions. This means that A is indifferent to B and D, and vice versa. We cannot include H in here.

In this case we can compare and rank any two baskets using the three basic assumptions and the indifference curve.

For Example:

E is preferred to A

A is Preferred to G

E is preferred to G (transitive preference).

Characteristics of Indifference curves:

i. Indifference curves are person-specific and time-specific, changing time period may

change the curvature of the curves for the same person. A set of indifference curves

curves may be steeper than another set. Steeper curves signal that stronger preference

is given to the good on the horizontal axis than to the one on the vertical axis and vice

versa. Curve is relatively flat when more preference is given to good on vertical axis

ii. An indifference curve between two goods such as food and clothing slopes downward

(has a negative slope) ∆C/∆F < 0. This is because all goods are good (desirable) and

thus, if one good is increased the other should be decreased to maintain the same

satisfaction or moving along the same indifference curve.

iii. Any point that lies above and to the right of a given indifference curve, say µ1, is

preferred to any point on the curve µ1, and vice versa for any point below this curve.

This should define the direction of increase of satisfaction for an indifferent map. This

is a result of Property 4-2 of preferences.

iv. Indifference curves can not intersect for the same person, the same time period. This is a result of Properties 4-2 and 4-3 of preferences.

A R B (by assumption; they lie on the same curve) (R indicates” indifferent to”)

The Marginal Rate of Substitution:

Marginal Rate of Substitution

Example: Preferences between food and clothing are given in the following table.

Basket	Satisfaction	F	C	Slope=∆C/∆F	MRS
A	µ1	1	14	-	-
B	µ1	2	10	-4/1	+4
D	µ1	3	7	-3/1	+3
E	µ1	4	6	-1/1	+1

Starting at point A and moving to point B, the individual consumer is willing to give up 4 units of good C to obtain one unit of good F, while keeping satisfaction the same (moving along the same indifference curve) and so on. Giving up a certain amount of one good to obtain more of another good while keeping satisfaction the same is called the marginal rate of substitution (MRS).

MRSF,C = maximum amount of good C that will be given up for one additional unit of good F, keeping satisfaction the same (i.e., moving along the same indifference curve).

MRSF,C = - ∆C / ∆F = - slope of indifference curve > 0 (absolute value of slope).

In the above diagram, marginal rate of substitution is diminishing.

The value and the change in this rate reveal information about the shape of the indifference curves, which in turn has to do with locating the consumer equilibrium or choice. Some indifference curves are straight lines, convex, right-angled, vertical lines or horizontal lines. Straight line curves give corner solutions.

4. Property 4-4 Diminishing MRS. This 4th assumption implies that the indifference

curves are convex. In the above example, moving from point A to point B, MRS is 4.

Then moving from B to D, MRS is 1 (MRS is diminishing). This means that the

preference ordering in this example most likely gives rise to convex indifference

curves, and in this case the solution or the equilibrium includes positive amounts of

both goods (internal solution). This assumption if imposed rules out other shapes of

indifference curves. (What will be the solution if indifference curves are straight lines?)

CONSTRAINTS: The Budget Constraint:

The Budget Constraint includes all baskets (points) where total expenditures on the goods included in any given basket equal to income.

Let:

Pf be the price per unit of food.

F be the quantity of food

Pc be the price of clothing per unit.

C be the quantity of clothing

M be the income

Then the budget constrain equation is Pf *F + PC*C = M

Total expenditure on F and C = Income.

For the budget or opportunity set, the equation is written as an inequality:

Pf *F + Pc*C M

Graphs of Budget Constraint and Set:

Since the budget constraint equation is linear it suffices just to determine the end points (horizontal and vertical intercepts) and then connect them with a straight line.

Pf *F + Pc*C = M

· If C = 0 then Pf *F = M and

_

F = M/Pf (horizontal intercept),

_

where F is the maximum amount of food that can be purchased with the whole income.

· If F = 0 then Pc*C = M and

_

C = M/Pc (vertical intercept,

__

where C is the maximum amount of clothing that can be purchased with whole income.

The budget set includes all the baskets inside the whole triangle.

Slope of the Budget Constraint:

As shown above, the budget constraint’s standard equation is

Pf*F + Pc*C = M

this equation can be rewritten in the format of the intercept and the slope as

C = M/Pc – (Pf / Pc)*F (where M/Pc is the vertical intercept)

Then the slope is ∆C/∆F = - Pf / Pc is the slope of the budget constraint.

That is, slope of the budget constraint is the price ratio and vertical intercept is M/Pc.

This intercept-slope format of the budget constraint follows the graph where the variable on the vertical axis is the variable on the left-hand side of the equation:

C = M/Pc – (Pf /Pc)*F

This expression of the budget constraint is more in line with the graph and it clearly shows its slope and its vertical intercept.

Shifts in the Budget Constraint:

Changes in one of the Prices: Outward Rotation of budget constraint:

Suppose Pf decreases from P1f to P2f while PC and M remain the same.

On the other hand, if Pf increases there will be an inward rotation.

Parallel shift

· If income (M) increases while the two prices (Px and Py) stay the same, there will be an upward parallel shift in the budget constraint and no change in the slope.

Fig. 4-5 Changes in Income Shrink or Expand Opportunities.

Demonstration Problem 4-1

Let P1f = $1 /unit, P1c = $2/unit and M = $80

Then the slope of B. C. = - Pf / Pc = -1/2 = slope of the solid line in the graph below:

If Pf increases from $1 to $2 while Pc and I stay the same, the budget constraint rotates inward (the dotted line). New slope= -2/2 = -1.

CONSUMER EQUILIBRIUM

The consumer maximizes utility or satisfaction by choosing the most desirable basket out of all the affordable baskets defined by the budget constraint.

Thus consumer choice, equilibrium or the optimal basket must satisfy two conditions:

I. Be affordable or lie on the budget constraint.

II. Give the most preferred combination of goods or services (optimal).

Graphically, this means that the consumer equilibrium is the tangency point between the budget constraint and the indifference curve that gives the highest satisfaction.

Point D is the most desirable but is not affordable.

Point B is affordable but is not the most desirable (it lies on indifference curve μ1)

Point B′ is affordable but is not the most desirable.

Point A is both the most desirable and affordable.

Then Point A is the consumer’s optimal choice or equilibrium and it is a tangency between the budget constraint and indifference curve (µ2)

Characterization of Consumer Choice or Equilibrium for Interior Solution:

Slope of the indifference curve = Slope of budget constraint.

∆C / ∆F = - Pf / PC. Multiply both sides by a minus we will have:

- ∆C / ∆F = Pf / PC

or MRSF,C = Pf / PC (for well-behaved (or convex) indifference curves and for interior solutions. For other shapes of indifference curves (such as straight lines) this equality may not hold (and we may have a corner solution).

COMPARATIVE STATITCS

In this section, we change either a price or income at a time and examine the change in consumer equilibrium. In the case of changes in income we must distinguish between normal and inferior goods

Fig.4-9: Price Changes and Consumer Equilibrium

The budget constraint was rotated twice: once rotated inward when P1f increased to P2f and the second rotated outward when it decreased to P0f. There are three tangency points or consumer choices (or equilibria): A, B and D. If you connect these three equilibrium points, you will get price consumption curve for food (PCCF)

In this section, we change income but keep both prices constant. This implies parallel shifts in the budget constraints. Assume that the good is normal.

Fig. 4-11: Income Changes and Consumer Equilibrium

There is a tangency point between an indifference curve and each one of the budget constraints, forming three consumer equilibria. If you connect these three equilibrium points you will get the income consumption curve. Both goods are normal goods because their consumption at equilibrium increases when income increases, and vice versa.

We can examine consumer equilibrium when income changes for the inferior good case. In Fig. 4-12 below the initial consumer equilibrium is point A. When income increases from M0 to M1, the consumer moves back from point A to point B, implying a decrease in the choice of good X. In this case, good X is an inferior good. Examples of inferior goods include bus trips, used clothes, generic jeans, used books…etc. Fig. 4-12 also shows that good Y is a normal good because after the increase in income the consumer chose more of good Y.

Fig 4-12 inferior good

(An Increase in Income Decreases the consumption of Good X)

INCOME AND SUBSTITUTION EFFECTS:

Suppose the absolute Pf drops while PC stays the same then the lower relative price of food Pf / PC has two effects. First, is the Substitution Effect where the relatively cheaper good (food) is substituted for the more expensive good clothing, keeping satisfaction the same. Graphically, this effect means moving along the original indifference curve using the new budget constraint which is defined by the new relative price.

The second is the real (not nominal) income effect, which resulted because of the change in the relative price. Real income changes with the change in relative price but nominal income is constant. Graphically, in Fig 4-13a for a normal good, this effect is shown by a parallel shift in the new budget constraint from the substitution effect point B to point D. The whole movement is the price effect from A to D. Thus,

P.E. = S.E. + I.E

Fig 4-13a. Substitution and Income effects for Normal Goods (S.E. and I.E)

The movement from A to B along the original indifference curve μ1 is the substitution effect while the movement from B to D (jumping from the new budget constraint) to its parallel at point D is the real income effect. In the case of normal goods, I. E. reinforces S. E. The whole movement from A to D is the price effect for a normal good. Food increases from F*A to F*D.

Inferior Goods (S.E. and I.E)

The substitution effect is the same for both the normal and inferior goods. The difference is in the income effect which is negative for inferior goods. In the normal good case, income effect is positive while for inferior good this effect is negative or an increase in real income reduces the quantity as shown by the movement from B to D in Fig 4-13b below. Income effect partially offsets substitution effect.

Footnote: The original budgets constraint which is tangent to indifference curve is missing in Fig -13b. I cannot add it because I do not have the software.

Fig 4-13b. Substitution and Income effects for Inferior Goods (S.E. and I.E)

APPLICATIONS OF INDIFFERENCE CURVE ANALYSIS

APPLICATION 1. The Bonus case:

Suppose there are two goods: X and Y.

PX = $4/Unit

PY = $2/Unit

M = $80

· Draw the Budget constraint; placing X on the Horizontal axis:

PX*X + PY*Y = M

$4X + $2Y = $80

· Suppose there is a promotional plan, which pays Six units of X for the first Ten units of X purchased. There are no bonuses after this. Draw the budget constraint.

a) Assume X = 0 (no bonus in this case); then B.C. is 0 + PY*Y = $80.

_

Y = 80/2 = 40 units.

b) Assume X < 10 units (right before bonus). The budget constraint is:

PX*X + PY*Y = $80

If X = 10 (eligible for bonus), then the budget constraint becomes:

$4*10 + $2*Yb = 80 or Yb = ($80 -$4*10)/$2 = 20 units.

Yb = 20 Units (subscript b is a notation that refers to the bonus case).

Fig.4-14: Buy Certain Units; Get other Units Free (the bonus case)

c) The Bonus Case. Add the SIX bonuses to X without any change in Yb = 20. This means there is a horizontal portion to the budget constraint from X = 10 to X = 16 units, while Yb = 20 units.

d) Assume Y = 0 (with the X bonus); then the budget constraint equation becomes

PX*X + 0 = 80 +$Bonus on X

where $Bonus on X = PX*Bonus X = $4*6 = $24

Substitute:

4 X + 0 = $80 + $24 = $104

__

Maximum X =104 / 4 = 26 units.

APPLICATION 2. The in-kind Gift Certificate case: Valid at Store X only

The original black budget line is the budget constraint before the consumer receives the $10-gift certificate for store of good X only. The consumer equilibrium is point A. Once the consumer receives the gift certificate for X (only), this budget line shifts out in a parallel way to the lighter line (see Text, page 138 ?) because if it spends all income on Y it still can use the X-certificate. On the other hand, if Y = 0 then the consumer will spend all income on X and as well use the X-certificate (new intercept on the horizontal axis). Consumer equilibrium is now point C as in Fig 4-16

Fig. 4-16: A Gift certificate Valid for Store X

How would you draw the budget constraint if the gift is cash and is not constrained to store X or Y?

RELATIONSHIP BETWEEN INDIFFERENCE CURVES AND DEMAND CURVES

The budget constraint was rotated twice: once rotated inward when P1f increased to P2f and the second rotated outward when it decreased to P0f, given income and price of clothing. There are three tangency points or consumer choices (equilibria) A, B and D.

Fig 4-20: Derivation of Individual demand for Food from indifference curves.

The points on this individual demand curve are associated with the consumer choices or equilibriums.

The individual demand curve has two properties:

1. Each point on this curve is part of a consumer equilibrium which satisfies the equilibrium condition (MRSF,C = Pf / PC).

2. Utility changes as we move along this curve. The lower the price, the higher the level of utility.

Note: All points on the demand curve are associated with the same income. If income changes then the demand curve will shift.

Deriving the Market Demand Curve from Individual demand Curves

Suppose there are two individual consumers whose individual demand curves are given by D1 and D2. The market demand is DM.

The market demand is the horizontal sum of quantities demanded by all the individual consumers in a given market for each possible price. For example, at price equals $3, consumer 1’s quantity is 2 units and consumer 2’s quantity is 1. The market quantity at the price of $3 is 4 units on the demand DM. This process is repeated and the locus of the point on DM is the market demand curve.

Fig. 4-21: Deriving Market Demand

CHAPTER 5: The Production Process and Costs

In this chapter, we will present tools that help managers in deciding which inputs and how much of each input to use to produce the output efficiently or optimally.

THE PRODUCTION FUNCTION.

The production function summarizes the technology that is used in converting inputs such as labor, steel and machinery into output such as an automobile. In this chapter, we will use two inputs: Capital (K) which involves machines, and labor (L) to produce the output (Q). The output should be produced efficiently if it is part of a production function.

This function is an engineering relation that defines the maximum amount of output that can be produced with a given set of inputs. Mathematically, this function is denoted as

Q = F(K, L),

That is, the maximum amount of output Q that can be produced with given K units of capital and L units of labor. Production functions assume efficiency.

Specific example (exponential function):

Q = A K1/3 L2/3

This is called Cobb-Douglas type production function. The parameter A is the efficiency or multi-factor productivity parameter that converts inputs into output.

The Short–Run Vs. the Long –Run:

· The Short–Run is the time period during at least one of the input is kept fixed and cannot be changed. This fixed input is usually capital (K*) such as equipment. In this short run period output can change by varying the intensity of operation, not the size of the firm. In this case, the S/R production function can be rewritten as

Q = F(K*, L) = f(L) (that is, output is a function of labor L and K* is a constant)

· The Long–Run is the amount of time it takes to make all inputs variable. Here, the firm contemplates different sizes of its plants. If it chooses a specific size (capital or K*) then the firm is in the short run. The long run is just a planning period.

Example :

Long run: Q = 10 K0.5 L0.5 (log linear lnQ= ln10+0.5lnK+0.5lnL)

Short run: if K= 2 (fixed), then Q = 10 ( 2 )0.5L0.5 = 14.14 L0.5 or Q= f(L) = 14.14 L0.5 .

Inside Business 5-1

Where does Technology Come from? What is the most important means for companies to acquire technology in the US? R & D? P. 163 (7th ed.).

Measures of Productivity

Here we define measures of productivity of inputs used in the production process. This is useful for evaluating the cost effectiveness of the production process and for making input decisions to maximize profit. The three most important measures of productivity include: total product, average product and marginal product

Total Product (short-run)

Suppose capital is fixed in the short-run, and then the production function is in the short run and is a function of labor only. Its graph is called the total product curve. In Table 5-1 (the Production Function), the maximum amount of output that would be produced with a given level of, for example, 5 units (hours or workers) of labor is 1,100 units of output, given K* = 2. This is a point on the total product curve, and so on.

Table 5-1: Production Function in the Short Run

(1) K* Fixed Input (Capital) [Given]	(2) L Variable Input (Labor) [Given]	(3) ΔL Change in Labor [Δ(2)]	(4) Q Output [Given]	(5) ΔQ /ΔL = MPL Marginal product of Labor [Δ(4)/ Δ(2)]	(6) Q/L = APL Average Product of Labor [(4)/(2)]
2	0	-	0	-	-
2	1	1	76	76	76
2	2	1	248	172	124
2	3	1	492	244	164
2	4	1	784	292	196
2	5	1	1,100	316	220
2	6 workers	1	1,416 units	316	236 units
2	7	1	1,708	292	244
2	8	1	1,952	244	244
2	9	1	2,124	172	236
2	10	1	2,200	76	220
2	11	1	2,156	-44	196

Average Product (APL)

In many instances, managers are interested in the average productivity of the input they used. For example, they may be interested in the (average) productivity of the average worker or average labor hour. This average productivity is measured by dividing the total product or output (Q) over the quantity of the input used such the number of workers or the number of labor hours.

APL =Q/L (this gives units of output per worker)

It is the output per worker or per hour. In Table 5-1, six workers together can produce 1,416 units of total output. This amounts to 1,416/6 = 236 units of output per worker (APL).

Marginal Product (MPL or MPK )

The marginal product of an input is the change in total output attributable to the last unit increase of an input. Thus MPL is thus the change in total output divided by the change in labor:

MPL = ∆Q/∆L

For example in Table 5-1, the marginal product of the 6th worker increases total output from 1,100 units of output to 1,416 units of output. Thus, its

MPL= (1,416-1,100)/(6-5) = 316 units of output.

The marginal product of capital (in the long run) is defined by

MPK = ∆Q/∆K

Fig. 5-1 below shows the relationship among total product, average product and marginal product for labor.

Fig. 5-1 Increasing, Decreasing and Negative Marginal Returns

It can be seen from Fig. 5-1 and Table 5-1 that MPL rises as labor increases from one to five workers or labor hours at point e. This increase in marginal labor productivity is a result of specialization. In Fig. 5-1 the total product curve between one and five units of labor or over the range A-E is convex or its slope increases as labor increases. This means that output increases at an increasing rate. This range is called the Increasing Marginal Returns to a single factor range for the short run. The single factor here is labor.

Over the second range from E to J or (labor units from 5 to 10), MPL is positive but decreasing, implying that output increases at a decreasing rate. In this range the total product curve is concave. This range is called the Diminishing Marginal Returns. The Law of Diminishing Returns to a Single Factor applies to this stage where Marginal Product of the variable input starts to decline. This is a short run concept.

Over the third range J to K, total output is decreasing because of labor crowding out and MPL is negative. This range is called the Negative Marginal Product range. No firm should employ resources in this range.

It can also be seen from Fig. 5-1 that as long as MPL exceeds APL, then APL is rising. Moreover, APL reaches its maximum when it intersects (equals) MPL.

Roles of Manager in the Production Process

The guiding role of production manger is two fold. (1) She should ensure that production is efficient or on the production function, which shows the maximum output given the available inputs (EFFICIENCY). To achieve this role, the manager should institute an incentive system that induces workers to perform well (e.g., profit sharing). (2) She should ensure that the firm uses the correct level of inputs or operates at the “right point” on the production function (OPTIMALITY). To do so, the manger should choose the input level according to the profit-maximizing input usage rule in the short run. This rule requires the manager to hire workers until:

Marginal benefit of the additional worker = Marginal cost of that worker.

To make this rule operational, marginal benefit is defined as the value marginal product of labor. Then

VMPL = Price of product*MPL = Marginal cost of that worker.

MC of labor is defined by the wage rate. For example, suppose the price of one unit of output sold is $3 and the cost of each unit of labor is $400. Using Table 5-2 below, how many units of labor should this manager hire? Or which point on the production function should she choose?

Table 5-2 The Value Marginal Product of Labor

(1) L Variable Input (Labor) [Given]	(2) P Price of Output [(2)]	(3) ΔQ /ΔL = MPL = Marginal product of Labor [Column 5 of Table 5-1]	(4) VMPL = P * MPL =Value Marginal Product of Labor [(2)*(3)]	(5) W Unit Cost of Labor [Given]
0	$3	-	-	$400
1	3	76	$228	400
2	3	172	516	400
3	3	244	732	400
4	3	292	876	400
5	3	316	948	400
6	3	316	948	400
7	3	292	876	400
8	3	244	732	400
Le = 9	3	172	516 >	400
10	3	76	228 <	400
11	3	-44	-132	400

The manager should hire 9 units of labor. This is the optimal labor (L* = 9) and optimal Q* =2,124.. See also Demonstration 5-2 for an algebraic solution in the short-run.

Max Profit = PxQ* -wXL* -rxK + ?? (K is constant and given in the short run).

Graphically, value marginal product curve is concave as in Fig. 5-2. Then using the profit-maximizing input usage rule, it gives the intersection between the unit labor cost and the VMPL as the point that the manager should choose to maximize profit. VMPL defines the demand for labor. It first slopes upward (because MPL slopes upward), then it slopes downward.

Fig 5-2: The Input Demand for Labor (optimal labor in the short-run)

Algebraic Forms of Production Functions

Linear Production Function: coefficients are slopes

Q = F(K, L) = ak + bL

where a and b are constants and equal:

a = ΔQ /ΔK = MPK and b = ΔQ /ΔL = MPL (these are slopes)

Thus the coefficients in linear production functions are the marginal products of L and K.

Example, Q = 5K + L (Linear)

This function says capital is five times more productive than labor or one machine does the work of five workers. If two machines and six workers are used then the total output produced is Q = 5(2) + 1(6) = 16 units of output.

Leontief Production Function

Q = F(K, L) = min(bK, cL) and fixed input proportion K/L = c/b in output.

This function is also called the fixed proportions production function because it implies that inputs are used in fixed proportions in the production process. Example: a word processor company where one machine (keyboard) is operated with one worker (keyboarder) is one to one. In this case b = c =1. Then the production function can be written as

Q = F(K, L) = min(K, L) and the fixed input proportion is K/L = 1/1.

where b = c =1 in this case.

Demonstration 5-1

Suppose the production function is given by the Leontief production function:

Q = F(K, L) = min {3k, 4L}

If K= 2 and L = 6 then the output Q = min {3(2), 4(6)} = min (6, 24) = 6 units of output. Fixed input proportion is K/L = 4/3.

Cobb-Douglas Production Function

Q = F(K,L) = AKaLb (exponential function).

This can be rewritten as: ln Q = ln A + a*ln K + b*ln L (log-linear),

where powers a and b are constants and can be proved to be percentages or elasticities:

a = Δ lnQ/ Δ lnK = % ΔQ /%ΔK = (ΔQ /ΔK)*K/Q = K-elasticity of output

b = Δ lnQ/ Δ lnL = % ΔQ /%ΔL= (ΔQ /ΔL)*L/Q = L-elasticity of output

An example of a Cobb Douglas is the production function for water desalination is

Q = F0 .6 H0 .4

where F denotes a group of inputs related to pumps and labor and H represents a group of inputs related to diem levels of heat. Output elasticities for F and H are 0.6 and 0.4, respectively. How much will output increase if input F increases by 10%? (6%?).How much will output change if input H increases by 10%? 4%. Which input is more important? Use the ratio of their elasticities: 0.6/0.4 > 1.Then input F is more….?

Regression ln Q = ln A + a*ln K+ b*ln L (log-linear) SUMMARY OUTPUT
Econ 322
Estimation of a Log Linear Production Function
Regression Statistics
Multiple R	0.9968049
R Square	0.9936
Adjusted R Square	0.9927692
Standard Error	0.0552893
Observations	18
ANOVA
	df	SS	MS	F	Significance F
Regression	2	7.1411349	3.570567447	1168.034	3.44E-17
Residual	15	0.0458536	0.003056905
Total	17	7.1869885
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
ln A	2.3434	0.0632274	37.06231409	3.63E-16	2.208589	2.478122	2.208589	2.478122
ln L	0.4527	0.0574264	7.883734499	1.03E-06	0.330333	0.575136	0.330333	0.575136
ln K	0.1882	0.0645684	2.914097063	0.010685	0.050534	0.325783	0.050534	0.325783
Ln Q = lnA + α*lnL + β*ln K
ln Q = 2.343 + 0.453 * ln L + 0.188 * ln K	R Square = 0.99
(37.06) (7.88) (2.91)
Labor elasticity α = % ΔQ / %ΔL = 0.45
Capital elasticity β = % ΔQ / %ΔK = 0.19
Returns to Scale =	α + β =	0.453 + 0.188	= 0.641
Decreasing returns to scale

Algebraic Measures of Productivity (i.e., MP and AP )

Cobb-Douglass production function: Q = AKaLb where A >0 is efficiency param.

* The marginal products of labor and capital can be derived as follows:

MPL = Δ Q/ Δ L = bAKaLb-1

MPK = Δ Q/ Δ K = aAKa-1Lb

Apply these formulae for the production function Q = 10K1/2L1/2 to calculate the marginal products. (e.g., MPL = 1/2*10K1/2L1/2 – 1 = 5K1/2L-1/2 = 5(K/L)1/2)

· Average product of labor: suppose 4 units of labor and 9 units of capital are used. Calculate the average product of labor for the above production function, Q = 10K1/2L1/2 .

Average product of labor = Q/L= {10(9)1/2 (4)1/2}/4 = 10*1.5 = 15 units of

output.

Linear production function: Q = F(K, L) = ak + bL

The marginal products as cited before are

MPK = ΔQ /ΔK = a

MPL = ΔQ /ΔK = b

The average products can be calculated by inserting the values of L and K in the definition of each average product Q/ L or Q/K.

Demonstration 5-2 (Calculating optimal labor L* in short run. See pages 68-69 for the optimality condition).

Assume the following Cobb-Douglass production function

Q = AK1/2L1/2 where A= 1 or Q = AK1/2L1/2

Suppose in the short-run K is fixed at one machine (K =1), the wage cost is $2 per unit of labor and the price of output is $10 per unit. How many units of optimal labor (L*) should the manager hire to maximize profit in the short run?

MPL = b*1*KaLb-1= b*1*( 1 )aLb-1 =0.5L-0.5

Recall the rule:

VMPL = W (profit-maximizing input usage rule in the short run)

P*MPL = ($10)*(0.5L-0.5) = $2

5L-0.5 = $2

Square both sides (5L-0.5 ) 2 = ($2)2

25 L-1 = 4

25 (1/L) = 4

25/4 = L*

Optimal labor input: L* = 6.24 units of labor (K = 1) in short run. Q*=(1)1/2(6.24)1/2

Max Profit = PxQ* -wxL* -rxK (and k is given in the short run).

Isoquants (Long Run)

Our next task is to derive both the optimal capital K* and labor L* in the long- run when these inputs are free to vary. The isoquant describes all combinations (L, K) that yield the same output. For example, an automobile manufacturer can produce 1,000 (= Q constant) cars per hour by using 10 workers and 1 robot. It can also produce the 1,000 (Q) cars using 2 workers (L) and 3 robots (K) and so on. Fig. 5-3 depicts a typical set of isoquants. Bundles or input mixes A and B produce the same level of output. The input mix A implies a more capital intensive process than the input mix B does. As we move in the Northeast direction in the figure, each new isoquant is associated with a higher level of output.

The slope of the isoquant is given by ∆K / ∆L. Since both MPL > 0 and MPK > 0 (both inputs are productive and increase output when they increase), then to keep output Q constant, it requires that an increase in labor (∆L > 0) must be matched by a decrease in capital (∆K< 0). Then all isoquants slope downward (∆K / ∆L < 0).

That is, the slope is negative. Moreover, the typical isoquants in Figure 5-3 are convex. This means that capital and labor are substitutes not perfectly substitutable as is the case in linear isoquants. This implies that as labor is substituted for capital it takes increasing amounts of labor to replace each unit of capital to keep output the same.

Substitution among Inputs (long-run):

Suppose the general form of the production function is given by

Q = f(L,K)

Then along a single isoquant, output Q is constant. That is,

_

Q = f(L,K).That is, when Q in the function is fixed, the function is called isoquant.

Then moving from point A to point B along this same single isoquant implies both increasing labor and decreasing capital without changing the output level Q.

The rate at which labor and capital can substitute for each other is called marginal rate of technical substitution. MRTSL,K (substituting L for K) is the absolute value of the slope of the isoquant ∆K / ∆L.

Slope of isoquant = MRTSLK

It can be shown that the absolute value of the slope of the isoquant is the ratio of marginal product of labor to marginal product of capital.

∆K / ∆L = MPL / MPK

Slope of isoquant = MRTSLK = ratio of marginal products.

MRTSL,K = the amount of capital that can be reduced when an extra unit of labor is used so that the output remains constant (or moving along the same isoquant).

Example: The table below contains data for an exponential production function

Using a production schedule (not a function) to calculate MRTS.

Combination	L	K	∆K / ∆L	MRTSL,K
A	1	5	-	--
B	2	2	-3/1	+3
C	3	1	-1/1	+1
D	4	1/2	-1/2/1	+1/2

MRTSL,K is diminishing which implies that the shape of this isoquant is convex.

Calculating MRTS using a linear production function. Example:

Q = f(L,K) = ak + bL. Example, Q = K + 2L. If Q is fixed then this production function becomes an isoquant which is a straight line. Recall MPK = a = 1 and MPL = b= 2. Slope of isoquant = ∆K/∆L = - MPL/MPK = - b/a = -2/1 = constant (for linear production function) as in Fig. 5-4. Here inputs L and K are perfect substitutes.

MRTSL,K = ∆K/∆L == -2 = + 2. Suppose Q = 20 units = K + 2L. Graph it.

Fig. 5-4: Linear Isoquant for Linear Production Function

L-shaped isoquants for Leontief production function

In this function inputs must be used in fixed proportion and they cannot substitute for each other. Therefore there is no MRTSLK. This implies that the isoquants of this function are L-shaped or right angled as in Fig. 5-5. Exemple: Q = 200*min (L, K)

Fig. 5-5: Leontief Isoquants for Leontief Production Function (one to one)

An example of a fixed proportions production function is the construction of a sidewalk, using one person and one jack hammer. Another example is film making where there is no substitution between cameras and actors. To produce more films, increase inputs (cameras, actors) proportionally. Inputs are perfect complements.

Convex isoquants (for example the Cobb-Douglas or exponential production function)

For most production functions, isoquants lie somewhere between straight line isoquants and the L-Shaped isoquants (the Fixed Proportion isoquants) or between perfect substitutes and perfect complements (no substitution). In this in-between case, the isoquants are convex and the inputs are just substitutable but are not perfectly substitutable as is the case in linear production functions. In Figure 5-6, moving from input mix B to input mix A, 1 unit of labor is substituted for 1 units of capital to produce 100 units of output. Now moving

Fig. 5-6: MRTS for a Convex Isoquant

Now moving from mix D to mix C for 1 unit of labor is 3 units of capital and vice versa . This marginal rate of substitution diminished as more labor substitutes for capital. This convex type of production functions satisfies the law of diminishing marginal rate of technical substitution.

Returns to Scale (long-run)

This is a long run concept. All inputs are variable and they change by the same proportion. If all inputs change by the same proportion, how does output change?

I. If output more than doubles when all inputs double, then there are increasing returns to scale (IRS). There are two types of firms that fit this category. Large firms that are capital intensive but need regulations (e.g., Public utilities). Emerging growth companies have IRS at early production stage. As the firm specializes, this increases productivity of all inputs.

II. If output doubles when all inputs double, then there are constant returns to scale (CRS). Size does not affect productivity of factors (e.g., banks in 1980s).

III. If output less than doubles, when all inputs double, there are decreasing returns to scale. Size leads to decreased productivity because of disorganization and distortion of signals going through layers of management levels.

Example 1: Q = 5L + 3K (linear production function with a straight-line, isoquant where labor and capital are perfect substitute).

If we double L and K, will Q double, i.e., Q′ = 2? Check.

Q′ = 5(2L) + 3(2K). will Q′ = 2Q? Factor out 2:

Q′ = 2 * [5L+3K] = 2Q (doubles) → CRS.

Example 2.: Q = 10 L0.8 K0.6 (Cobb–Douglas type production function)

where 10 is the efficiency co-efficient, 0.8 is labor elasticity of output and 0.6 is capital elasticity of output. 0.8/0.6 = relative importance of labor.

If %∆Q / %∆L = 0.8 then if %∆L = +10% it implies %∆Q = (0.8 *10% ) = +8%.

Which factor is more important in this production: L or K? L because …. See above.

What is the type of returns to scale? Double all inputs, how will Q change?

Q′ = 10(2L)0.8 (2K)0.6 (double all inputs)

Q′ = 20.8 *20.6 * [10L0.8 K0.6]. Substitute Q for [10L0.8 K0.6]

Q′ = 20.8 +0.6 *Q = 21.4*Q > 2Q (output more than doubles)

Q′ > 2Q. Therefore, IRS.

For the Cobb-Douglas type only Q = ALα Kβ., we can follow the following rules:

If alpha + Beta > 1 (where alpha is L-elasticity and Beta is K-Elasticity), then IRS

If alpha + Beta < 1 , then DRS

If alpha + Beta = 1, then CRS

Isocosts (long-run)

Similar to an isoquant, an isocost line includes all input combinations that will cost the firm the same amount ($C). The formula for an isocost line is for constant C given by

C = w*L + r*K

where w is the wage rate and r is the rental price of capital. Both w and r are constant.

Graph of Isocost Line:

Since the equation for the isocost line is linear, then we only need to locate the two-end points and then connect them with a straight line to graph the isocost line.

__

Let K = 0 then C = w*L and the maximum amount of L = C/w. This determines a point on the horizontal axis.

__

Next, let L = 0 then C = r*K and the maximum amount of K = C/r. This determines the endpoint on the vertical axis. If we connect these endpoints with a straight line, we get the isocost line associated with cost level $C.

Different $ costs (C ) give different isocost lines. There are two levels of $C: C0 andC1. Each endpoint on these isocost lines is defined as the ratio of the $C over the respective price of input, w or r.

Slope of the Isocost Line

We can express the above formula for the isocost line in terms of the intercept and the slope.

C = wL + rK.

Move K to the left hand side as it is on the vertical axis in the graph, and then solve for K:

rK = C – wL

K = C/r – (w/r) L

This is the equation for the isocost line expressed in terms of the intercept and the slope.

Thus, along an isocost line, capital K is a linear function of input L with a vertical intercept of C/r and a slope of –w/r.

The last graph in Fig. 5-7 below represents a change in the slope of the isocost line. In this graph the wage rate w is increased from w0 to w1while the rental price of capital r is kept the same. This represents an inward rotation in the isocost line around the vertical endpoint. This means the isocost line has become steeper.

Fig.5-7: Isocosts

Cost Minimization (long-run): Calculating both Optimal L* and K*

Some organizations such as the non-profit ones do not maximize profit but they minimize costs. Therefore, since labor and capital are free to vary in the long run, we need to find a rule that will allow us to choose the optimal input mix of both labor and capital.

Given the isoquant representing the given output Q1 and the three isocost lines C0, C1 and C2, in order to minimize cost and find optimal L* and K*, we must look at the lowest isocost line that is tangent to this isoquant ( i.e., the minimum cost that can produce output Q1). Input mix B costs more than input mix A although both points lie on the isoquant and can produce the given output Q1. The isocost line that can finance the production of the given output is isocost line C1. The tangency point between this lowest isocost line and the given isoquant determines the optimal input mix (L*, K*). This means that the optimal input choice is characterized by the condition that

Slope of isoquant = Slope of isocost line (tangency point)

Or

∆K / ∆L = - (w/r)

∆K / ∆L = w/r

Or MRTSL,K = w/r which is the condition for cost-minimization input for optimal inputs.

Recall: it can be shown that MRTSL,K = MPL / MPK

Then the equilibrium condition for producer’s input choice can be rewritten as:

MPL / MPK = w / r

Or MPL / w = MPK / r

Demonstration Problem 5-3 (Optimal inputs in the long run)

Example: Calculating optimal input choice (producer equilibrium) L* and K*.

Suppose the Cobb-Douglas production function is given by: Q = 50KL2 , the given output Q1 = 4,800 units, w = $20 and r = $60.

First, use the equilibrium condition to determine the optimal labor/capital ratio.

MPL / MPK = w/r

MPL = ∆Q / ∆L = (2) 50 KL2-1 = 100KL

MPK = ∆Q / ∆K = (1) 50 K1-1L2 = 50L2

(now set MPL/MPK = w/r). That is, 100KL / 50 L2 = $20 / $60

K*/L* = 1 / 6 or K* = (1/6) L* where 1/6 is the optimal capital labor ratio.

This means each worker is managing 6 machines (6 K*= 1L).

Second, determine. K* and L*. Substitute K* into the production functions and let Q 1 = 4,800 units which becomes an isoquant equation (that is, when you fix Q).

4,800 = 50*(1/6 L*) L*2

(4,800 * 6) / 50 = L*3 (take the cubic root of both sides and solve for L*).

(576)1/3 = (L*3)1/3 or (576)1/3 = L*

L* = 8.320 labor hours

Substitute L* into the optimal K/L ratio or K* = 1 / 6 * (8.32) = 1.387 machine hours.

Third: Determine minimum cost:

Substitute the values for L* and K* into the isocost equation.

Min cost C = w*L* + r*K*

= ($20)*(8.32) + ($60)*(1.387) = $249.62

Minimum cost = $249.62. for producing the given output 4,800 units.

Optimal Input Substitution (long-run) after an increase in price of an input

A change in the price of an input will lead to a change in the cost-minimizing (optimal) input mix. Suppose that the initial isocost line in Fig. 5-9 is FG and the firm is cost-minimizing at input mix A, producing Q0 units of output. Now suppose the wage rate w increases so that if the firm spent the same $ amount on inputs, its isocost line would rotate inward to FH. With this new isocost line the firm cannot produce the same output Q0. To produce this output and taking into account the new higher wage rate, the isocost line should be parallel to FH and also tangent to the isoquant defined by output level Q0. This isocost line is IJ which is tangent to the isoquant at input mix B. In this case due to the increase in price of labor the firm substituted capital for labor and moved from input mix A to input mix B.

Fig. 5-9: Substituting Capital for Labor, Due to Increase in Wage Rate

THE COST FUNCTION

Cost functions summarize information in the production function and they can along with total revenue be used to find the output level (Q) that maximizes profit (= TR-TC). They are functions of output that defines an isoquant, and the cost (C) associated with this isoquant is the minimum cost.

C = F (Q).

Short Run Costs

Short-run: is the time period during which at least one of the inputs is fixed. This means that there is a fixed cost which is the cost of the fixed input, usually capital..

Fixed Cost (FC) : Expenditures for plant maintenance, insurance, minimal number of employees, principal and interest payments, property taxes. FC does not change with output.

Variable Cost (VC) : Expenditures for wages, salaries and raw materials. VC increases with the size of output. It starts from the origin.

Total Cost (TC): Sum of VC and FC:

In the short run, TC starts where FC starts. When output is zero, TC = FC. In the graph, the difference between TC and VC is FC and, thus constant at all output levels.

Fig 5-11 illustrates the cost of producing with the same technology used in Table 5-1 as can be seen in the first three columns. Price of capital = $1000 per hour and w = $400.

Table 5-3: The Cost Functions

(1) K Fixed Input (Capital) [Given]	(2) L Variable Input (Labor) [Given]	(3) Q Output [Given]	(4) FC Fixed Cost [$1,000*(1)]	(5) VC Variable Cost [$400*(2)]	(6) TC Total Cost [(4)+(5)]
2	0	0	$2,000	$0	$2,000 =FC
2	1	76	$2,000	400	2,400
2	2	248	$2,000	800	2,800
2	3	492	$2,000	1,200	3,200
2	4	784	$2,000	1,600	3,600
2	5	1,100	$2,000	2,000	4,000
2	6	1,416	$2,000	2,400	4,400
2	7	1,708	$2,000	2,800	4,800
2	8	1,952	$2,000	3,200	5,200
2	9	2,124	$2,000	3,600	5,600
2	10	2,200	$2,000	4,000	6,000
2	11	2,156	$2,000	4,400	6,400

Fig. 5-11 illustrates the relations among total cost (TC), variable cost (VC) and fixed cost (FC). FC is a horizontal line because it does not change with output even if output is zero. On the other hand, variable cost is zero if output is zero and it increases with the increase in the level of output. Total cost equals fixed cost when output is zero and then it increases with output, as does variable cost.

Fig. 5-11: The Relationship among Costs

Average and Marginal Costs

Average Costs

Average fixed cost (AFC).

AFC = FC/Q

When the fixed cost (FC) is spread out over a larger quantity of the output (Q), the fixed cost per unit or the average fixed cost AFC declines as shown in column 5 of Table 5-4 in the textbook.

Average variable cost (AVC).

AVC = VC/Q

The typical variable cost per unit of output declines first then it reaches a minimum and then it begins to increase as shown in column 6 of Table 5-4 (It is U-shaped). In this table the AVC reaches a minimum of 1.64 between 1,708 and 1952 units of output.

Average total cost (ATC).

ATC = TC/Q

Or

ATC = AFC + AVC.

In this case

AFC = ATC – AVC.

ATC is analogous to AVC. It initially declines and then reaches a minimum before it begins to increase. This U-shaped pattern for ATC reflects the battle between AFC and AVC. Initially, AFC wins the battle and ATC declines and reaches a minimum. Then after that the rising AVC dominates the declining AFC and ATC begins to rise. Column 7 in Table 5-4 in the book gives the ATC.

Marginal Cost (MC): is the increase in total cost resulting from producing an additional unit of output. It is the most important cost concept.

MC = ∆TC / ∆Q = ∆(VC + FC) / ∆Q = (∆VC +0)/ ∆Q = ∆VC / ∆Q (because FC does not change when Q changes. That is, MC in the short run can be calculated from either total cost or variable cost because FC is a constant and FC cancels out.

MC declines initially then it starts to increase as shown in column 7 of Table 5-5.

Table 5-5: Derivation of Marginal Cost

(1) Q [Given]	(2) ΔQ [Δ(1)]	(3) VC [Given]	(4) ΔVC [Δ(3)]	(5) TC [Given]	(6) ΔTC [Δ(5)]	(7) MC= ΔTC/ΔQ [(6)/(2) or (4)/(2)]
0	-	0	-	2,000=FC	-	-
76	76	400	400	2,400	400	400/76 = 5.26
248	172	800	400	2,800	400	400/172 = 2.33
492	244	1,200	400	3,200	400	400/244 = 1.64
784	292	1,600	400	3,600	400	400/292 = 1.37
1,100	316	2,000	400	4,000	400	400/316 = 1.27
1,416	316	2,400	400	4,400	400	400/316 = 1.27min
1,708	292	2,800	400	4,800	400	400/292 = 1.37
1,952	244	3,200	400	5,200	400	400/244 = 1.64
2,124	172	3,600	400	5,600	400	400/172 = 2.33
2,200	76	4,000	400	6,000	400	400/76 = 5.26

It is the reciprocal of Marginal Product of Labor (MPL = ∆Q/∆L) when there is only one input (labor) is variable in the short run (VC = w*L). Change both sides with respect to Q

∆VC / ∆Q = w*∆L/∆Q = w*(1/MPL) or MC = w/ MPL

where w is the wage rate or the price of labor and is constant. That is, there is an inverse relationship between MPL and MC, given w.

Example: (the stage of increasing marginal returns to single factor which is labor because MPL is increasing)

MPL	w	MC= w/MPL
2 Units	$6 / hr	$6 /2 = $3 per 1 unit of output
3	$6	$2 per 1 unit
6	$6	$1

MC is the $ labor cost per unit of output. It is decreasing in this example.

Relations among Costs

Fig. 5-12 depicts the relations between AFC, AVC, ATC and MC. Note that MC crosses the AVC and the ATC curves at their minimums.

Fig. 5-12: The Relationship among Average and Marginal Costs

Fixed and Sunk Costs

Fixed cost is a cost that does not vary with output. Sunk cost is a cost that is paid and lost forever. For example,

Demonstration 5-4

ACME paid $5,000 to lease a railcar from the Reading Railroad. The lease contract says that only 1,000 of this fixed payment is refundable if the railcar is returned within two days.

1. Upon signing the contract how much is ACME’s fixed costs? $5,000.

2. Suppose one day after receiving the railcar, ACME has realized that it does not need it. Farmer Smith offered to lease it for 4,500 that day and AMCE accepted. How much is ACME’s sunk cost? $500.

3. Suppose ACME’s refused to lease it and the two days passed. How much is ACME’s sunk cost? $5,000.

4. Suppose ACME returned the railcar to Reading Railroad within two days. How much is the sunk cost? $4,000.

Unlike the variable and total costs, sunk cost doesn’t affect the optimal decisions of the firm. It, however, affects total profitability.

Algebraic Forms of Cost Functions

Cubic cost function:

C(Q) = f + aQ + bQ2 + cQ3 .

where FC = f and VC = aQ + bQ2 + cQ3..

Example: TC = 20 + 5Q – 4Q2 + 6Q3 , where a = 5, b = -4, c = 6.

In this cost function

FC = $20

VC = 5Q – 4Q2 + 6Q3

AFC = FC/Q = 20/Q

AVC= VC/Q = (5Q – 4Q2 + 6Q3)/Q = 5 - 4Q + 6Q2

ATC = TC/Q = AFC + AVC = 20/Q + 5 - 4Q + 6Q2 (where 20/Q is AFC)

MC = ∆TC / ∆Q = ∆VC / ∆Q = 0 + 1*aQ1-1 + 2bQ2-1 + 3cQ3-1

= a + 2bQ + 3cQ2

where a = 5, b = -4, c = 6. Apply it to the example above,

MC = 5 – 8Q + 18Q2 OR

MC = ∆TC / ∆Q = 0 + 1 *5Q 1-1 – 2 *4Q 2-1 + 3 * 6Q3 -2 =5 – 8Q + 18Q2

Long Run Costs

Suppose the firm is unsure about future demand and is considering three alternative sizes Q0 < Q1 < Q2 (small, medium and large).

· Three plant sizes with S/R average costs: ATC0 , ATC1 and ATC2.

· Suppose Min ATC0 > Min ATC2 > Min ATC1 , the medium size has the lowest min. Short run MC (SMC) goes through the minimum respective ATC.

Fig. 5-13 Optimal Plan Size and Long-Run Average Cost

· If the firm expects output to be Q0 ( small size ), it will consider the smallest plant because this is the size that gives the lowest cost per unit possible. If Q1 (middle size), then the firm will choose second or medium size and so on.

· That is, the long run average cost LRAC or (LAC) with the three firms is the cross-hatched portions of the three S/R average cost-curves because these portions show lowest cost of production for any of the three output levels.

· If there are infinitely many plant sizes that can be built, then RLAC (or LAC) will be the envelope that touches infinitely many short-run ATCs and this will generate a smooth U-shaped long run average cost. Each point on LAC is a min point on a short-run ATC.

· Efficient plant sizes correspond to where the short run ATC curves touch the envelope LRAC curve, usually at the min S/R ATCs.

Scale Economies (long-run)

This concept relates changes in output to changes in cost without any restrictions on input proportions, as is the case under returns to scale. It concentrates on changes in long run average cost, LRAC (or LAC). Returns to scale are a special case of scale economics.

· If doubling output implies doubling cost, then LRAC (= 2*LTC / 2*Q) will not change and there are constant returns to scale (CRS). LAC is a straight line.

· If doubling output implies less than doubling cost (that is, LRAC will decline), then there are economies of scale. LAC is a declining curve

· If doubling output implies more than doubling cost (that is, LRAC will rise), then there are diseconomies of scale. LAC is an increasing curve

Examples of cost functions:

C = Q0.8 (an exponential function and the exponent is an elasticity).

where 0.8 = %∆TC / %∆Q = 8%/10% or the cost elasticity of output. This means that the 8% change in cost is less than the 10 % change in output. There are economies of scale.

Question: Suppose C= Q1.2. Determine the type of scale economies (%∆TC = ? if %∆Q = 10%/)

Economic Cost versus Accounting Cost

Accountants:

· Take a retrospective view of a firm’s finances

· Their purpose is to evaluate past performance

· Equate costs with actual expenses and depreciation expenses

· Depreciation expenses are calculated according to tax rules

Economists:

· Take a forward-looking view of the firm’s finances.

· Purpose to evaluate future profitability

· Equate costs with actual expenses and opportunity costs (including actual costs) because the firm rearranges resources to minimize cost and increase expected profitability. The cost = actual expenses + opportunity costs of own time, money, materials and buildings .

· Depreciation expenses = actual wear or tear.

Example :

Owner/manager of a pizza restaurant in his/her own building

Accounting costs Economic costs

Owners/managers salary = 0 Owners / managers salary = opportunity cost > 0

Own building rent = 0 Own building rent = opportunity cost > 0

Workers wages > 0 Workers wages > 0

Cheese > 0 Cheese > 0

Flour > 0 Flour > 0

Other expenses > 0 other expenses > 0

Total accounting cost < Total economic cost

Total economic Cost = Explicit $cost + Implicit cost

Explicit $cost = accounting cost (out of pocket expenses).

Implicit cost = forgone own salary + forgone interest on own money + forgone own rent

In this case, the implicit cost is the sum of opportunity costs.

Based on that:

Accounting profit = TR – accounting costs

Economic profit = TR – economic costs

Multiple Output Cost function

Here, we focus on firms that produce multiple outputs. GM, for example, produces different types of cars and different types of trucks. In this case, the cost function of the multi product firm depends on all levels of all output types . Suppose the firm produces two types of products: product 1, Q1, and product 2, Q2. Then the multiple output cost function is represented by: C(Q1, Q2).

Economies of Scope

Economies of scope exist if

C(Q1, Q2) < C(Q1) + C(Q2)

or C(Q1) + C(Q2) - C(Q1, Q2) > 0. It can also be written in percentage as

S = [C(Q1) + C(Q2) - C(Q1, Q2)]/ [C(Q1) + C(Q2)] > 0

That is, producing the two products (say steak and chicken) from two separate plants cost more than producing them from one restaurant. If the two products are produced from two separate restaurants will be a duplication of the cost of building, equipment and maybe labor. ( This concept E. of Scope uses Total Cost and not MC ).

Cost Complementarity

Cost complementarity exists in a multi-product cost function when the marginal cost of producing one product (Q1) is reduced when the product of another output (Q2) is increased. ( Cost complementarity uses MC and not TC ).

Let C(Q1, Q2) be the cost function for a multi (two)-product firm.

Let MC1(Q1, Q2) = ∆C/ ∆Q1 be the marginal cost of producing the first product.

The cost function exhibits cost complementarity if

∆MC1(Q1, Q2) / ∆Q2 < 0.

That is, if an increase in the output of product 2 decreases the marginal cost of product 1.

Similarly, the cost function exhibits cost complementarity if

∆MC2(Q1, Q2) / ∆Q1 < 0.

That is, if an increase in the output of product 1 decreases the marginal cost of product 2.

Example of cost complementarity is the production of doughnuts and doughnuts holes. The firm can produce these products jointly or separately. But the (MC) cost of making doughnut holes additional to making doughnuts is lower when workers roll out the dough, punch the holes and fry both the doughnuts and the doughnut holes instead of making the holes separately.

Multi-product Quadratic Cost Function: Example

C(Q1, Q2) = f + aQ1Q2 + (Q1)2 + (Q2)2, where f is the fixed cost and aQ1Q2 is the interaction term for producing the two products under one roof. We hope that this term is negative to reduce cost.

The single product cost functions for Q1 and Q2 separately are:

C(Q1) = f + (Q1) 2

C(Q2)= f + (Q2)2

Marginal costs for products Q1 and Q2 in the multiproduct cost function are:

MC1 = ∆C/ ∆Q1 = a Q2 + 2Q1 (MC is for product 1 and a can be positive or negative), and

MC2 = ∆C/ ∆Q2 = a Q1 + 2Q2 (MC for product 2 and a can be positive or negative)

1. Examine whether economies of scope exist for this quadratic multi-product cost function. Check if this condition for total costs holds:

C(Q1) + C(Q2) - C(Q1, Q2) > 0.

[ f + (Q1)2 ] + [ f + (Q2)2 ] – [ f + aQ1Q2 + (Q1)2 + (Q2)2 ]

[ f + (Q1)2 ] + [ f + (Q2)2 ] – f - aQ1Q2 - (Q1)2 - (Q2)2

Things cancel out and we have

f - aQ1Q2 > 0

(where a can be positive or negative and f is FC). You can have many scenarios for f and the interaction costs. Obviously, the higher the fixed cost, the greater that economies of scope exist.

Then if f > aQ1Q2 (FC is greater than the interaction term), then there are economies of scope. THIS is THE CONDITION YOU CHECK FOR ECONOMIES OF SCOPE. You do not need to go over the whole math if the functions are quadratic. Just use the result above to check for econ of scope.

Special case : If a < 0, then Economies of Scope exist because f , Q1 and Q2 in

f - aQ1Q2 are always positive, and in this case - aQ1Q2 is also a positive number. If a is positive, then -aQ1Q2 is negative. Then one has to calculate the difference

f - aQ1Q2 and see if it is positive.

2. Check if the quadratic cost function exhibits multi-product cost complementarity (use here marginal costs to check and not TC). That is, check whether ∆MC1/ ∆Q2 < 0.

MC1 = ∆C/ ∆Q1 = aQ2 + 2Q1

Then ∆MC1/ ∆Q2 = a. If a < 0, there are cost complementarities.

If you have cost complementarities, you will have economies of scope because … . The opposite is not always true.

Demonstration 5-7

Suppose the quadratic cost function of firm A which produces two goods is given by

C = 100 - 0.5 Q1Q2 + (Q1)2 + (Q2)2 (note here a = -0.5)

The firm wishes to produce 5 units of good 1 and 4 units of good 2.

1. Do complementarities exist? Do economies of scope exist?

MC1 = ∆C/ ∆Q1 = - 0.5Q2 + 2Q1

For complementarity check whether: ∆MC1/ ∆Q2 = a = -0.5 < 0.

Here a = -1/2 is negative. Then cost complementarities exist.

Check for Economies of Scope. Check whether f - aQ1Q2 > 0. Note that a = -.5, and substitute 5 units of good 1 and 4 units of good 2.

f - aQ1Q2 =

100 – (-.5)(5*4) = 110 > 0. Yes

Example 2:

C = 50 + .8 Q1Q2 + (Q1)2 + (Q2)2

And Q1 = 15 units and Q2 = 10 units.

Do we have economies of scale? Cost complementarity?

f - aQ1Q2

MC1 = ∆C/ ∆Q1 = +0.8Q2 + 2Q1

∆MC1/ ∆Q2 = a = 0.8 >0 ???

Final Remark:

If Economies of Scope exist then there is a benefit of merging two distinct firms into a single firm because there will be a reduction in costs relative to the costs of the separate firms. The additional cost that occurs as a result of joint production under Economies of Scope may not be significant. Look at it the other way around. Selling off unprofitable subsidiaries when Economies of Scope exist could only result in minor reductions in costs. Example: C(Q1)=$100, C(Q2) = $80 but C(Q1, Q2) = $110.

Chapter 7: The Nature of Industry

Much of the material in this chapter is factual and is intended to acquaint the students with aspects of the “real world” related to Managerial Economics. These statistics on industries are important for managers and they affect how those mangers make decisions. Although those numbers change over time they are still informative and they can explain how information affects managerial decisions.

In this chapter we will discuss the factors that affect market structure across industries. We will also examine the conduct or behavior as well as performance across industries.

MARKET STRUCTURES

Market structure refers to several factors including: number of firms in the market; size of firms; size distribution or degree of market concentration; technological and cost conditions; and ease of entry and exit in the market or industry. Different industries may have different structures and these structures affect managerial decisions.

Market Power and Market Structure

One Producer Two Producers Few producers

Homogeneous Product Many producers with Many Producers.

Or Differentiated Product Differentiated products homogeneous

Free entry/exit

Equilibrium Conditions:

MR = MC MR = MC MR = MC MR = MC P = MC P=equation P=eq P=eq P=equation P=constant

The following subsection provides a summary of the major structural factors

Firm size

Some firms are larger than others. Table 7-1 lists the sales of the largest firm in each of 26 industries. General Motors is the largest firm in the motor vehicles and parts industry, with sales of over $184 billion in 2001. In contrast, the largest firm in the furniture industry is Leggett and Platt, with sales of only 4.3 billion. One important lesson that can be derived from the table is that some industries naturally give rise to larger firms than other industries.

Industry concentration

This factor deals with the size distribution or concentration within an industry or a market. Some industries are dominated by few large firms. There are two measures of share concentration.

The four–firm concentration ratio: This ratio measures the fraction of total industry sales produced by the four largest firms in the industry or market. Let S1, S2, S3 and S4 denote the $ sales of the four largest firms in an industry. Additionally, let ST represent the $ total sales of all firms in the industry or market. This ratio is given by

C4 = (S1 + S2 + S3 + S4)/ST

This ratio can also be expressed in terms of market shares (%):

C4 = (S1/ST) + (S2/ST) + (S3/ST) + (S4/ST) or

0 < C4 = w1 + w2 + w3 + w4 ≤ 1

where wi = (Si/ST) (i = 1,2,3,4) are the four firms’ market shares. If C4 is close to zero it indicates there are many small sellers, giving rise to much competition (see wood containers and pallets C4 = 6% in Table 7-2). If it is close to one, it implies little competition (see breweries C4 = 90%). When there are four or less companies in the industry, then C4 =1.

Demonstration 7-1

Suppose the industry has six firms. The four largest firms have sales of $10 each and the remaining two firms have sales of $5 each.

Total industry ST = (4*10) + (2 * 5) = $50

The four–firm concentration ratio is = (4*10) /$50= $40/$50 = 0.80

This means the four largest firms account for 80% of total industry sales.

The Herfindahl–Hirschman index (HHI)

Let firm i’s share of total industry output denoted by

wi = Si / ST

HHI is defined as the sum of the squared market shares of all firms in an industry.

HHI = [ {(w1)2 + (w2)2 + ….. + (wn)2 }*10,000]

The multiplication by 10,000 is to eliminate the need for decimals, squaring the shares means giving higher weights to higher shares in the index.

0 < HHI ≤ 10,000

If HHI = 10,000 it means there is a single firm in the industry and w1 = 1 (monopoly). A value close to zero means there are many very small firms in the industry (competition). The government cutoff point for high concentration is 1,800. In this case the industry is considered “highly concentrated” and the Justice Department may block a horizontal merger if increases the HHI by more than 100 points. It will challenge it depending on the values of those two statistics. More information on this index is given below under “horizontal integration”. Here, we have two statistics.

The difference between HHI before and after the merger is:

(HHI = [2wiwj]*10,000 where firms i and j want to merge. Suppose firm 3 with a market share of 20% and firm 4 with a market share of 23% proposed to merge. How much is potential (H? If (HHI>100, then this is another statistical evidence for the government to question the merger. [2*0.2*0.23]* 10,000 = ? Thus, the government uses two statistics: HHI and (HHI, in addition to other factors.

Demonstration 7-2

Suppose an industry has three firms. The largest firm’s sales are $30 and the remaining two have sales of $10 each. Calculate both the HHI and the four-firm concentration ratio.

HHI = 10,000*[(30/50)2 + (10/50)2 + (10/50)2 ] = 4,400

The four-firm concentration ratio is:

C4 = (30+10+10)/50 = 1,

because the three firms account for all industry sales.

On balance, the HHI and C4 usually signal the same pattern of concentration (see Table 7-2). However there are exceptional cases where they are not in synch, as can be seen in the two industries: tires and the snack food in Table 7-2. Why is it possible that these two industries can give un-similar pattern? HHI covers all the firms in the industry while C4 includes the four largest firms. Another reason is that HHI is biased toward the larger firms because of the squared shares.

Limitations of Concentration Measures

1. Global markets: The indices take into account the national firms and ignore foreign firms operating in the domestic industry. With fewer firms included, this leads to overestimation of concentration (see example, the brewery industry)

2. National, Regional and Local Markets: Consider, for example, the market for gas stations. Suppose we are interested in the local gas station market in Kansas City. The national or regional gas stations are not relevant for the local market in Kansas City. If the local concentration ratio for gasoline in Kansas City is measured at the national level, then this measure underestimates concentration because it will have too many irrelevant firms included.

3. Industry Definitions and Product Classes: In constructing indices of market structure, there is considerable aggregation across product classes. Consider for example, the soft drink industry. C4 for this industry is 47%. This number may seem surprisingly low when one considers how Coca-Cola and Pepsi dominate the product class for cola. However, the soft drink industry as defined by the Bureau of Census includes many more types of bottled and canned drinks including birch beer, root beer, fruit drinks, ginger ale, iced tea, lemonade, etc. Cross price elasticity is used to determine close substitutes that belong to a product class.

Technology

Some industries are very labor intensive, while others are very capital intensive and require large investments. The differences in technologies give rise to differences in production techniques across industries. In the petroleum-refining industry, for example, firms utilize about one employee for each $ 1 million in sales. In contrast, the beverage industry utilizes about 17 workers for each $1 million in sales.

Technology is also important within a given industry where one firm has superior technology and it dominates the industry (e.g., Intel).

Demand and market conditions

Industries can also differ with respect to demand and market conditions. Industries (e.g., refrigerator or elevator) with low demand may be able to sustain few firms, while those with strong demand (e.g., shoes) may require many firms to produce the output.

Information available to consumers may vary across markets or industries. In some industries such as the airlines it is easy to find the lowest prices. In contrast, it is much more difficult to get information on a used car. Market structures and decisions of managers will vary depending on the amount of information available in the market.

Finally, industry elasticity of demand will vary from one industry to another. Moreover, within the same industry, the individual firm’s demand elasticity may be much more elastic than that of the industry as a whole because of the availability of substitutes from similar firms within the same industry (see Inside Business 7-2). For example, for the whole food industry price elasticity is -1.0 and for the representative firm it is -3.8.

One measure of elasticity of industry demand for a product relative to that of an individual firm is the Rothschild index. This index is defined as the sensitivity of quantity demanded of the whole industry to the price of the product group (industry’s demand elasticity) relative to the sensitivity of the quantity demanded of the individual firm to its own price (firm’s demand elasticity). That is,

Rothschild index (R) = ET/EF and 0 =< R =<1,

(where closer to zero means more competition and closer to one means more monopoly)

where ET the industry’s demand elasticity and EF is the firm’s own demand elasticity. This index takes on a value between 0 and 1. When the firm’s elasticity is much greater than the industry’s elasticity when there are many substitutes, the R-index is close to zero. But if the firm’s elasticity is the same as that of the industry, the index is one (Tobacco) and there is monopoly power. In case of perfect competition, the index is zero.

Table 7-3 provides estimates of the firm and industry’s elasticities and the Rothschild indices for 10 US industries. Notice these indices for the tobacco and chemical are unity. What do these indices mean in terms of substitution? What does the index of (0.26) mean for the individual food firm?

Demonstration 7-3

The industry elasticity for airline travel is -3 and the elasticity for an individual carrier is -4.Calcutale the Rothschild index for this industry. R = -3 /-4 = 0.75.

Table 7-3: Market and Representative Demand elasticities and Rothschild Index

for Selected US industries

Potential for Entry

In some industries, it is relatively easy for new firms to enter the markets with high competition, while in other less competitive markets it is more difficult because of barriers to entry. There are many factors that create barriers to entry including high explicit costs (such as capital investment), patents and economies of scale. In some industries (e.g., public utilities) only one or two firms can exist in the industry because of economies of scale. Other firms cannot enter because they cannot generate the scale or volume that will give the low average cost (LAC) associated with economies of scale.

CONDUCT OR BEHAVIOR

Industries differ not only in terms of market structure but also in terms of conduct or behavior regarding pricing, production, advertising, R& D, merging …etc. Some industries charge higher markups than other industries. Some industries are more susceptible to mergers or takeovers than others.

Pricing Behavior (Behavior #1)

Firms in some industries charge higher prices than firms in other industries. The index that economists use to measure pricing behavior and market power is the Lerner Index which is given by

L = (P – MC)/P (= cents per $1 of sales and 0 =< L =<1 measures market power), where P is the price of a product and MC is the marginal cost of producing an incremental unit of the product. This index defines the markup level as a percentage of the price (it gives cents of markup per dollar of sales). If the typical firm sets price equal to marginal cost as is the case in perfect competition, where firms are very small and price-takers, then the index equals zero . In contrast, in highly monopolized industries where firms do not compete for customers the index takes on a value of one. Firms in other industries come in between.

We can express this index as a markup factor by rearranging the variables:

P = [1/(1-L)]MC = (Markup factor)* MC

where 1/(1-L) is the markup factor. When the markup index L is zero, the markup factor is 1 and the price is exactly equal to MC. If the markup index is 1/3, the markup factor is 1.5. If index is 0.5 then the markup factor is 2 times MC. Try it if the index is 2/3. The higher the Lerner index, the higher the markup factor, and the price as a multiple of marginal cost.

Table 7-5 provides estimates of the Lerner index and the markup factor for 10 US industries. There are considerable differences in these measures across industries. The tobacco industry has the highest Lerner index (76 cents markup per each $1 of sales) and markup factor of (4.17). The textiles industry has the lowest Lerner index (with a markup of 21 cents per $1 of sales) and a markup factor of 1.27. The goal in this section is to help the manager determine the optimal markup for a product.

Demonstration 7-4

Suppose; P = $300, MC = $200. What are the Lerner index and the markup factor?

Lerner index or mark up: L = (P – MC)/P = (300-200)/300 = 1/3.

Markup factor = 1/(1-L) = 1/(1-1/3) =1.5

Integration and Merger Activity (Behavior # 2)

Integration refers to uniting of productive resources and it can occur through a merger or unification of two or more existing firms into a single, larger firm. Integration can also occur during the formation of a firm. Of course, integration results in larger firms. There are three types of mergers: vertical, horizontal and conglomerate.

Vertical integration:

Various stages in the production of a single product are integrated out in a single firm. Example, a firm that produces leather merges with a firm that produces clothes. Another example of vertically integrated firm is the automobile manufacturer that produces its own steel, uses the steel to make car bodies and engines and finally sells the single product automobile. How about the merging of a semiconductor company with a PC company? Thus, a vertical merger is the integration of two or more firms that produce components for a single final product. Firms vertically integrate to reduce the transaction costs associated with acquiring inputs which are outputs of other firms.

Horizontal Integration

This integration refers to merging the production of similar products into a single firm. For example, horizontal integration occurs if two computer companies merge into a single company. Another example is the Merging of Exxon and Mobil. How about two banks?

The primary reason for firms to engage in horizontal integration is:

1. To enjoy the cost saving of economies of scale and scope.

If horizontal integration allows for cost savings then these types of horizontal mergers are socially beneficial (Social benefits).

2. To enhance their market power.

Since this merger reduces the number of firms that compete in the market. This tends to increase both C4 and HHI (Social costs).

The social benefits due to cost savings should be weighed against the social costs associated with a more concentrated industry. Under its current Merger Guidelines, the Justice Department views industries with HHI in excess of 1,800 to be “highly concentrated” and may block the horizontal merger if it will increase HHI by more than 100 points. However, the Justice department permits the merger in industries that have high HHI if there is evidence of significant foreign competition, an emerging new technology, increased efficiency or when one of the firms has financial problems.

Industries with HHI below 1,000 are generally considered “unconcentrated” by the Justice Department and mergers are usually allowed. If HHI is between 1,000 and 1,800 (moderately concentrated) the Justice Department relies on other factors such as economies of scale.

Conglomerate Mergers

This means merging firms that produce different products into a single company. An example is merging a cookie manufacturer with a cigarette maker and a soft drink maker into one single company. The advantage of conglomerate mergers is that they can improve firms’ cash flows because revenues derived from one product at a time when it has a high demand can be used to generate working capital when demand for another product is low. This reduces the variability of a firm’s earnings and gives it better access to capital markets. Example, GE? How many divisions does it have?

The Link between Market Power and Market Concentration

In the case of a single firm, the incremental market power for one firm is defined by the Lerner’s markup rule index as

L= (P- MC)/P = - 1/EPD > 0 (*)

where P is the price, MC is marginal cost and EPD = %(Q/%(P = ((Q/(P)*P/Q is the (direct) market price elasticity of demand, which is negative. More elastic demand implies less market power because of the availability of substitutes.

In the case of multiple firms i = 1, 2 , .., N, the ith firm’s monopoly power is defined by

(P- MCi)/P = - wi /EPD > 0 (**)

where wi is the market share of firm i (that is, wi = Si/ST where Si is firm i’s sales in dollars and ST is the total industry sales in dollars).

To express Equation (**) in terms of

HHI = [ (w1)2 + (w2)2 + … + (wn)2] (without 10,000)

which is a measure of market concentration without the multiplication by 10,000, we will follow the following mathematical manipulation.

Multiply both sides of Equation (**) by the ith market share wi, we have

wi*(P- MCi)/P = - (wi)2/EPD (***)

Sum both sides in Equation (***) over i = 1,2 , …, N, we have

ΣNi= 1 wi*(P- MCi)/P = - ΣNi= 1 (wi)2/EPD

Notice that in this case HHI = ΣNi= 1 (wi)2

Upon substitution of HHI, we have

ΣNi= 1 wi*(P- MCi)/P = - HHI/EPD (****)

Notice that ΣNi= 1 wi*P = P* ΣNi= 1 wi = P*1 = P because the sum of the shares is equal to 1.

Denote ΣNi= 1 wi*MCi = MC as the weighted average MC for the industry.

Then Equation (****) can be rewritten for the industry as

(P- MC)/P = - HHI/EPD

1> = (P- MC)/P = - HHI/EPD >= 0 (*****)

Market power for the average firm = - market concentration/demand elasticity

Recall that EPD or price elasticity of demand is negative . So the RHS is positive. (1) the higher HHI or market concentration, the greater the market power. (2) More elastic demand is, the less the market power because of many available substitutes. For example, the market power is lower with demand elasticity = -2.1 than with elasticity =

-1.7.

Example , Assume there are six firms in the industry whose individual sales are as given in the table below. Assume that market EPD = - 4.1. How much is the market power for the average firm? (Hint: Since we do not have information on P and MC, we can use the right-hand side of Equation (****)).

Firm	Si ($ m)	wi= Si/ST	MCi ($)	wi*MCi	(wi)2
1	$10 m	1/5			(1/5)2
2	$10	1/5			(1/5)2
3	$10	1/5			(1/5)2
4	$10	1/5			(1/5)2
5	$5	1/10			(1/10)2
6	$5	1/10			(1/10)2
Total	$50	ΣNi= 1 wi = 1			HHI = ? 0.18000???

Then use the formula:

0 <  (P- MC)/P = - HHI/EPD = -0.18000???/-4.1 = ?? <= 1

Advertising (behavior # 3)

Firms in certain industries spend considerably more money on advertising than firms in other industries. For example, firms in the food industry such as Kellogg spent about 9% of their sales revenues on advertising in 2000, while firms in the rubber and plastic products such as Goodyear spent less than 2% of their sales revenues (see Table 7-6).

PERFORMANCE

Performance refers to both the profit and the social welfare (sum of consumer and producer surplus) that result in a given industry. It is important for future managers to recognize that those two measures of performance vary considerably across industries.

Profit

Profit varies from one industry to another. Moreover, big firms do not always earn big profits as percentage of sales. In Table 7-6, Ford generated more sales than any other firm on the list. Yet, its profit as a percentage of sales is one of the lowest listed (2.5%).

Social Welfare

This is defined as the sum of consumer and producer surplus. Dansby and Willig proposed a useful index for measuring performance in terms of social welfare. The Dansby-Willig (DW) index measures how much social welfare would improve if firms in an industry increased output in a socially efficient manner. If the DW index is zero , it means that consumer and producer surplus is maximized and there is no social benefit increase from altering output . On the other hand, industries with a large index value show low performance and they can generate improvement in social welfare if they expand output.

The DW index can be used to rank industries in terms of their abilities to improve social welfare if they alter their outputs. If the DW is large and the industry is ranked low, then it means that the industry shows low performance and thus can alter output. Industries operating under high competition, they usually exhibit high efficiency and have low DW index.

Table 7-7 below shows that the textiles industry has the lowest DW index among the nine industries listed. It thus has the best social welfare performance on the list. Chemicals, petroleum and paper have the worst performances on this list.

OVERVIEW OF THE REMAINDER of the BOOK

In the remaining chapters of the book, we examine the optimal managerial conduct (e.g., pricing, output, advertising, etc) under a variety of market structures. There are four basic market structures.

Perfect Competition

Under this market structure, there are many buyers and sellers in any given market. The firms produce homogeneous (identical products) and each has no perceptible impact on the price which is determined by the market as a whole. The concentration ratios (such as C4 and HHI), the Rothschild index, Lerner index and the Dansby-Willig index for industries characterized by this market structures are close to zero.

Monopoly

In this market structure, there is only one firm that produces a product that does not have close substitutes. An example of industry that has this market structure is public utilities operating in a certain region or city which enjoy considerable economies of scale. These public utilities constitute a local natural monopoly. In this market structure, the monopolist restricts output and charges higher prices.

The C4 concentration ratio and Rothschild index are equal to unity (1) for monopolies. Moreover, the Lerner index is close to unity and the social welfare performance is low and the DW index is high.

Monopolistic Competition

In this market structure, there are many small firms and consumers just as in perfect competition but the products are differentiated. The products are substitutable but ate not perfect substitutes. Thus, the concentration ratio C4 or HHI is close to zero. However, unlike under perfect competition, each firm under monopolistic competition produces a product that is slightly differentiated and is not homogeneous. An example of monopolistic competition is the restaurant industry in a city or a metropolitan area. Therefore, because of imperfect substitutes the Rothschild indexes are greater than zero (the firm’s elasticity is not infinity) and higher than in perfect competition whose products are perfect substitutes and firms’ are infinity.

Because the products are differentiated the monopolistically competitive firm has some market power or control over prices. Lerner index is greater than zero . When the firm increases its price some of its customers have brand loyalty and won’t switch to other brands. But some will switch to other brands. For this reason, firms in this market structure often spend considerable sums on advertising in an attempt to convince consumers that their brands are better than other brands.

Oligopoly

In an oligopoly market structure, there are few firms that dominate the market, giving rise to high concentration of market share. Examples of this market include the airline, automobile, and aerospace industries. One firm’s actions affect the other firms’ profitability and leads to reactions from those firms. Thus the distinguishing feature of an oligopoly market is mutual interdependence among firms in the industry.

The interdependence of profits in this market structure gives rise to strategic interaction among firms. So a manager of an oligopolistic firm should consider how managers of the other rival firms in the industry would react to her decisions and make her strategic plan accordingly. Therefore, it is very difficult to manage firms operating in oligopolistic markets.

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M/P2f M / P1f M / P0f F

Pf

P2f

P1f

P0f

Df

F*A F*B F*D F

P

P

5

4

3

2

1

P

5

4

3

2

1

DM

5

4

3

2

1

D1

D2

1 2 3 4 5 Q

Individual 2

1 2 3 4 5 Q

Individual 1

1 2 3 4 5 6 7 Q

Market

Rule: Price of product*MPL = W

To derive optimal input L* in short run

K

Ka

Kb

A

B

_

Q

L

La Lb

K

20

Slope = -2

isoquant

10

Assume Q = 20 = K + 2L (isoquant)

units

K and L are perfect substitutes

L

No substitution between L and K

Q1 Q2

K

3 2 1

MPK = 0

Q2 = 400

MPL = 0

Q1 = 200

1 2 3 L

K and L are imperfect substitutes

K

__

K = $C0/r

Isocost Lines

L

__

L = $C0/W $C1/W

K

K*

Q1

L* C0/w C1/w C2/w L

B

Isocost lines

C0 < C1 < C2

A

SMC2

SMC1

SMC0

ATC2

ATC1

ATC0

LMC

ATC

Q

LRAC

LRAC

LRAC

CRS

Diseconomies of Scale

Economies of Scale

Q

Duopoly

Perfect

Competition

Monopolistic Competition

Oligopoly

Monopoly