PhD Macroeconomics question

THEORY OF COST

In production, research, retail, and accounting, a cost is the value of money that has been used up to produce something or deliver a service, and hence it is not available for use anymore. In business, the cost may be one of acquisition, in which case the amount of money expended to acquire it is counted as cost.

Outlay of fund for productive resources

What does cost mean to business/account?

There are many different costs, including fixed and variable, but they are all accounted for in the same way. Costs are recorded as expenses on the income statement during an accounting period and cleared- out in a closing entry at the end of the period.

Types of Costs:

A. Based on planning horizon:

There are two: Long-Run versus Short-Run Short run: a period of time during which one or more of a firm’s inputs cannot be changed, that means some inputs are fixed over the given production period).

Long run: a period of time during which all inputs can be changed (no inputs are fixed, and all are variable).

B. Based on Input nature

The concepts of long run and short run are closely related to the concepts of fixed inputs and variable inputs.

Fixed input: an input whose quantity remains constant during the time period in question.

Variable input: an input whose quantity can be altered during the time period in question.

Fixed Cost, Variable Cost, and Total Cost

In the short run, a firm will have both fixed inputs and variable inputs. These inputs correspond to two types of cost: fixed cost and variable cost.

Fixed cost (FC): the cost of all fixed inputs in a production process. Another way of saying this: production costs that do not change with the quantity of output produced.

Variable cost (VC): the cost of all variable inputs in a production process. Another way of saying this: production costs that change with the quantity of output produced.

TC = TFC + TVC

Definition

· Fixed costs are costs which do not vary with output, for example, rent. In the long run all costs can be considered variable. · Variable cost also known as, operating costs, prime costs, on costs and direct costs, are costs which vary directly with the level of output, for example, labor, fuel, power and cost of raw material. · Social costs of production are costs incurred by society, resulting from private production. · Average total cost is the total cost divided by the quantity of output. · Average fixed cost is the fixed cost divided by the quantity of output. · Average variable cost are variable costs divided by the quantity of output.

Cost Function:

The cost function is a  mathematical relationship between cost and output . It tells how costs change in response to changes in output. It is multivariate and derived function. It is derived from the production function.

Cost functions are derived functions. They are derived from a production function, which describes the available methods of production in a particular time horizon. Production economists are interested in distinguishing between short-run costs and long-run costs. Short-run costs are the costs over a period during which some factors of production (usually capital equipment and management) are fixed. The long-run costs are the costs over a period long enough to permit the change of all factors of production. In the long run, all factors become variable.

Total Cost (C) is multivariate function in both long-run and in short run. Symbolically,

C= f(Q)

C= f(Qi)

C= ƒ(Q,T,P,K) in short-run

C = ƒ(Q, T,P ) in long-run.

Where C= Total cost

Q = Output

T = Technology

P = Prices of factors

K= Fixed factor(s)

Graphically, costs are shown on two-dimensional diagram. And such curves imply that cost is a function of output i.e.

C = ƒ(Q); ceteris paribus

Traditional theory of cost

Traditional theory of cost is studied under the category of short run and long run. Short-run in economics implies for the time horizon in which some factor(s) of production is fixed. But in long- run all factors of production become variable. This section explains only about short-run cost of the traditional theory.

According to this theory, total cost is divided into two groups: total fixed cost and total variable cost.

TC = TFC + TVC

The total fixed cost (TFC) is graphically denoted by a straight line parallel to the output axis (fig.1)

Fig.1 Total fixed cost

The total variable cost (TVC) has broadly an inverse S shape (fig.2) which reflects the law of variable proportions. According to this law, the productivity of a firm increases as more of the variable factors employed at the beginning and the average variable cost falls.

Fig.2 Total variable cost

If the concept of TVC and TFC is added to obtain the TC, then the shape of the TC is basically influenced by the TVC and hence follows the shape of the TVC. TC will start from the point of TFC (Fig.3).

Fig.3 Combining TC, TFC and TVC

From the total cost curve, we obtain average cost (AC) curve.

AC =

The average fixed cost (AFC) is found by dividing TFC by the level of output:

Graphically, AFC is a rectangular hyperbola[footnoteRef:1] (Fig.4). The average variable cost (AVC) is similarly obtained by dividing the TVC with the corresponding level of output: [1: AFC curve is rectangular hyperbola because total fixed cost (TFC) remains constant at all points of AFC, i.e., the multiplication between AFC and production quantity at all points of AFC remains constant. It is the reason why AFC declines when production quantity increases. Elasticity remains unitary. ]

Fig.4.Average fixed cost

Graphically the AVC at each level of output is derived from the slope of a line drawn from the origin to the point on the TVC curve corresponding to the particular level of output. The short run AVC (SAVC) curve falls initially as the productivity of the variable factor(s) increases, reaches a minimum when the firm is operated optimally (with the optimal combination of fixed and variable factors), and starts rising beyond this point (Fig.5).

Fig.5. Short run average variable cost

The average total cost (ATC) is obtained by dividing TC by the corresponding level of output:

The shape of the ATC is like that of the AVC and derived in the same way as the SAVC. Both AVC and ATC are U-shaped. The U-shape of both AVC and ATC reflects the law of variable proportions or law of eventually decreasing returns to the variable factors of production.

The marginal cost is defined as the change in TC because of the unit change in output (Q).

Graphically the MC is the slope of the TC. The MC curve is U-shaped (Fig.6).

Fig.6.Marginal cost

AVC (SAVC) curve falls initially as the productivity of the variable factor(s) increases, reaches a minimum when the firm is operated optimally (with the optimal combination of fixed and variable factors), and start rises beyond this point.

Few Key Points

· Typical cost functions are either linear, quadratic, and cubic.

· A linear cost function is such that the exponent of quantity is 1. It is appropriate only for cost structures in which marginal cost is constant.

· A quadratic cost function, on the other hand, has 2 as exponent of output. It represents a cost structure where the average variable cost is U-shaped.

· A cubic cost function allows for a U-shaped marginal cost curve. The cost function in the example below is a cubic cost function.

Example of Cubic Cost Function and derivation of AC, AVC & MC

TC= 0.1Q3-2Q2+60Q+200

ATC= TC/Q= 0.1Q2-2Q+60+200/Q

The constant value in a total cost function represents the total fixed cost. Function for total variable cost can be arrived at by subtracting the constant value from the total cost function:

VC = TC-FC

VC= 0.1Q3-2Q2+60Q

AVC= VC/Q= 0.1Q2-2Q+60

Marginal cost equals the slope of the total cost curve which in turn equals the first derivative of the total cost function.

= 0.3Q2-4Q+60

Quadratic Cost Function:

Quadratic Cost Function: If there is a diminishing return to the variable factor the cost function becomes quadratic. There is a point beyond which TP is not proportionate. Therefore, the marginal physical product of the variable factor will diminish. And if TP falls MP will be negative.

TC= 45+10Q+4Q2

Example 3: A manufacturer has a monthly fixed cost of $150,000 and a production cost of $18 for each unit produced. The product sells for $24 per unit.

a. What is the cost function? Answer: C(Q)= 18Q+150000

b. What is the revenue function? Answer: R(Q)= 24Q

c. What is the profit function?

Answer: π(Q) = R(Q) - C(Q) = 24Q- (18Q + 150000)

= 6Q- 150000

d. Compute the profit(loss) corresponding to the production levels of 22,000 and 28,000 units

When output is 22,000 π(Q) = (6*220000)-150000 = -18000 i.e. LOSS

When output is 28,000 π(Q) = (6*280000)-150000 = 18000 i.e. Profit

e. How many units must the company produce and sell if they wish to make a profit of $40,000?

π(Q) = 40000

6Q- 150000 = 40000

Q = 31666.66 = 31667 unit must produce

Relationship Between AC and MC

· Both AC and MC are derived from total cost (TC). AC refers to TC per unit of output and MC refers to addition to TC when one more unit of output is produced.

· Both AC and MC curves are U-shaped due to the Law of Variable Proportions. The relationship between the two can be better illustrated through following schedule and diagram.

· The MC must cut AC and AVC at their lowest point.

• Whenever marginal cost is less than average total cost, average total cost is falling.

• Whenever marginal cost is greater than average total cost, average total cost is rising.

• The marginal-cost curve crosses the average-total-cost curve at an efficient scale.

•  Efficient scale is the quantity that minimizes average total cost

(Point S in above diagram)

Combining ATC, MC and MR curves

Fig. Interaction of MC, ATC and MR

You can combine cost curves to provide information about firms. In this diagram for example, we assume that the firm is in a perfectly competitive market. The marginal cost curve will cut the average cost curve to its lowest point. In a perfectly competitive market, a firm's profit maximizing price would be above the price at which the average cost curve cuts the marginal cost curve. If the marginal revenue is above the average total cost price the firm is deriving an economic profit.

Cost functions and relationship to average cost

In the simplest case, the total cost function and its derivative are expressed as follows, where f(Q) is a cost function relating cost to production volume and FC represents fixed costs:

TC = FC + f( Q)

See in the Tabular form:

Output	TC	MC= dTC/dQ	AC = TC/Q
0	20	-	-
1	30	10	30
2	38	8	19
3	45	7	15
4	50	5	12.5
5	57	7	11.4
6	68	11	11.3
7	80	12	11.45
8	96	16	12
9	114	18	12.7
10	135	21	13.5

Technological change and cost of production

Technological change in the production function is illustrated by a shift in the isoquant. The original point of production of output level Q1 is at B, with L2 amount of labor and K2 amount of capital. Technological change typically increases productivity and decreases the cost of production.

K

B

K2

Q1

A

K1

Q1’

L2

L1

L

Fig. Technological change and shift in isoquant

In figure above, we represent his type of technological change by a shift of the Q1 isoquant from Q1 to Q1’. The Q1’ is now tangent to an isocost line closer to the origin at point A, representing a lower total cost of production. Thus, the productivity has now increased, as the firm is able to produce output level Q1’ using only L1 amount of labor and input and K1 amount of capital input.

Derivation of supply curve in the short run from various cost curves

The supply curve of the firm may be derived by the point of intersection of its MC curve with successive demand curve. Assume that the market price increases gradually. This causes an upward shift of the demand curve of the firm. Given the positive slope of the MC curve, each higher demand curve cuts the (given) MC curve to a point which lies to the right of the previous intersection. This implies that the quantity supplied by the firm increases as price rises. The firm given its cost structure will not supply any quantity (will close) if the price falls below w, because at a lower price the firm does not cover its variable cost (Graph A). If we plot the successive point of intersection of MC and demand curve on a separate graph, we observe that the supply curve of the individual firm is identical to its MC curve to the right of the closing down point w. Below w, the quantity supplied by the firm is zero. As price rises above w the quantity supplied increases. The supply curve is shown in graph B.

Fig. A Supply curve derivation from cost relationship

MC curve is U-shaped having both negative and positive slopes while supply curve is positive sloping. So, we must not consider negative or downward sloping portions of the MC curve. The only rising portion (i.e., upward sloping) of MC is the supply curve. To be more specific, the rising portion of the MC that lies above the AVC curve is the supply curve of a competitive firm in the short run. The point w is called as “shut-down point” or the “closing-down point”. Note that the firm at the shut-down point is indifferent between operating and shutting down.

Fig. B. Supply curve of a firm in short run planning.

Rules for profit maximization

Profit = TR-TC

If TR is greater than TC, business is running in profit. That will show evidence that Benefit is exceeding Cost. Some say B/C ratio. B/C ratio greater than 1 is necessary condition for any business or program.

Thus, for profit maximization of output, we may state 2 important rules:

Rule I. One should not produce at all if the total revenue (TR) from selling the produce does not equal or exceed the TVC in the short run.

Rule II. Marginal revenue (MR) will be equal to marginal cost (MC) in the perfectly competitive market in the long run.

Profit maximization is the process by which a firm determines the price and output level that returns the greatest profit. There are several approaches to this problem. The total revenue -- total cost method relies on the fact that profit equals revenue minus cost, and the marginal revenue-marginal cost method is based on the fact that total profit in a perfectly competitive market reaches its maximum point where marginal revenue equals marginal cost in the long run.

Total cost-total revenue method

To obtain the profit maximizing output quantity, we start by recognizing that profit is equal to total revenue minus total cost. Given a table of costs and revenues at each quantity, we can either compute equations or plot the data directly on a graph. Finding the profit-maximizing output is as simple as finding the output at which profit reaches its maximum. That is represented by output Q in the diagram.

There are two graphical ways of determining that Q is optimal. Firstly, we see that the profit curve is at its maximum at point A. Secondly, we see that at point B that the tangent on the total cost curve (TC) is parallel to the total revenue curve (TR), the surplus of revenue net of costs (BC) is the greatest. Because total revenue minus total costs is equal to profit, the line segment CB is equal in length to the line segment AQ in the figure below.

Computing the price at which to sell the product requires knowledge of the firm's demand curve. The price at which quantity demanded equals profit-maximizing output is the optimum price to sell the product.

Fig. Profit maximization: TR-TC approach

Marginal cost (MC) -Marginal revenue (MR) method

If marginal revenue is greater than marginal cost, marginal profit is positive, and if marginal revenue is less than marginal cost, marginal profit is negative. When marginal revenue equals marginal cost, marginal profit is zero. Since total profit increases when marginal profit is positive and total profit decreases when marginal profit is negative, it must reach a maximum where marginal profit is zero - or where marginal cost equals marginal revenue. This is because the producer has collected positive profit up until the intersection of MR and MC (where zero profit is collected and any further production will result in negative marginal profit, because MC will be larger than MR). The intersection of marginal revenue (MR) with marginal cost (MC) is shown in the diagram below at point A. If the industry is competitive (as is assumed in the diagram), the firm faces a demand curve (D) that is identical to its Marginal revenue curve (MR), and this is a horizontal line at a price determined by industry supply and demand. The average total cost is represented by curve ATC. Total economic profits are represented by area PABC. The optimum quantity (Q) is the same as the optimum quantity (Q) in the first diagram. If the firm operates in a non-competitive market, minor changes will have to be made to the diagrams.

Fig. Profit maximization: MR-MC approach

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