PhD Macroeconomics question

Demand Function

Mathematical representation of demand function is more useful while understanding and analyzing the demand schedules of specific commodities in the given market. In its standard form a linear demand equation is

Q = a - bP.

Where, quantity demanded is a function of price. The inverse demand equation, or price equation, treats price as a function g of quantity demanded: P = f (Q). To compute the inverse demand equation, simply solve for P from the demand equation. The inverse demand function is the same as the average revenue function, since Price (P) = Average Revenue (AR). To compute the inverse demand function, simply solve for P from the demand function. For example, if the demand function has the form

Q = 180 - 2P

then the inverse demand function would be

P = 90 - 0.5Q

In economics, an inverse demand function is the inverse function of a demand function. The inverse demand function views price as a function of quantity (Varian, 1999).

Demand schedule:

Determinants of demand: Demand drives economic growth. Businesses want to increase demand so they can improve profits. Governments and central banks boost demand to end recessions. They slow it during the expansion phase of the business cycle to combat inflation. If you offer any paid services, then you are trying to raise demand for them.

So, what drives demand? In the real world, a potentially infinite number of factors impact each consumer's decision whether or not to buy something. In economics, however, the equation is simplified to highlight the five primary determinants of individual demand.

1. The price of the good or service.

2. The income of buyers.

3. The prices of related goods or services. These are either complementary (those purchased along with a particular good or service), or substitutes (those purchased instead of a certain good or service).

4. The tastes or preferences of consumers.

5. Consumer expectations. Most often, this refers to whether a consumer believes prices for the product will rise or fall in the future.

For aggregate demand, the number of buyers in the market is the sixth determinant. Likewise, weather is another determinant of demand for commodities.

D = f (price, income, prices of related goods, tastes, expectations)

Before breaking down the effect of each determinant, it's important to note that these factors don't change in a vacuum. All the factors are in flux, all the time. To understand how one determinant affects demand, you must first hypothetically assume that all the other determinants don't change.

When income rises, so will the quantity demanded. When income falls, so will demand. But if your income doubles, you won't always buy twice as much of a particular good or service. There's only so many pints of ice cream you'd want to eat, no matter how wealthy you are, and this is an example of "marginal utility." Marginal utility is the concept that each unit of a good or service is a little less useful to you than the first. At some point, you won’t want it anymore, and the marginal utility drops to zero.

What are the 4 basic laws of supply and demand ?

1) If the supply increases and demand stays the same, the price will go down.

2) If the supply decreases and demand stays the same, the price will go up.

3) If the supply stays the same and demand increases, the price will go up.

4) If the supply stays the same and demand decreases, the price will go down.

In 1890, Alfred Marshall's Principles of Economics developed a supply-and-demand curve that is still used to demonstrate the point at which the market is in equilibrium.

Factors Affecting Supply & Demand

· Price Fluctuations. Price fluctuations are a strong factor affecting supply and demand. ...

· Changes in cost structure and factors of production

· Income and Credit. Changes in income level and credit availability can affect supply and demand in a major way. ...

· Availability of Alternatives or Competition. ...

· Trends. ...

· Commercial Advertising. ...

· Seasons

· Global events: pandemic, wars, political changes etc.

Exceptions to the Law of Demand

Note that the law of demand holds true in most cases. The price keeps fluctuating until an equilibrium is created. However, there are some exceptions to the law of demand. These include the Giffen goods, Veblen goods, possible price changes, and essential goods. Let us discuss these exceptions in detail.

Giffen Goods

Giffen Goods is a concept that was introduced by Sir Robert Giffen. These goods are goods that are inferior in comparison to luxury goods. However, the unique characteristic of Giffen goods is that as its price increases, the demand also increases. And this feature is what makes it an exception to the law of demand.

The Irish Potato Famine is a classic example of the Giffen goods concept. Potato is a staple in the Irish diet. During the potato famine, when the price of potatoes increased, people spent less on luxury foods such as meat and bought more potatoes to stick to their diet. So as the price of potatoes increased, so did the demand, which is a complete reversal of the law of demand.

Veblen Goods

The second exception to the law of demand is the concept of Veblen goods. Veblen Goods is a concept that is named after the economist Thorstein Veblen, who introduced the theory of “conspicuous consumption”. According to Veblen, there are certain goods that become more valuable as their price increases. If a product is expensive, then its value and utility are perceived to be more, and hence the demand for that product increases.

And this happens mostly with precious metals and stones such as gold and diamonds and luxury cars such as Rolls-Royce. As the price of these goods increases, their demand also increases because these products then become a status symbol.

The expectation of Price Change/weather/cultural events

In addition to Giffen and Veblen goods, another exception to the law of demand is the expectation of price change. There are times when the price of a product increases and market conditions are such that the product may get more expensive. In such cases, consumers may buy more of these products before the price increases any further. Consequently, when the price drops or may be expected to drop further, consumers might postpone the purchase to avail the benefits of a lower price.

For instance, in recent times, the price of onions had increased to quite an extent. Consumers started buying and storing more onions, fearing further price rise, which resulted in increased demand.

There are also times when consumers may buy and store commodities due to fear of shortage. Therefore, even if the price of a product increases, its associated demand may also increase as the product may be taken off the shelf or it might cease to exist in the market.

Necessary Goods and Services

Another exception to the law of demand is necessary or basic goods. People will continue to buy necessities such as medicines or basic staples such as sugar or salt even if the price increases. The prices of these products do not affect their associated demand.

Change in Income

Sometimes the demand for a product may change according to the change in income. If a household’s income increases, they may purchase more products irrespective of the increase in their price, thereby increasing the demand for the product. Similarly, they might postpone buying a product even if its price reduces if their income has reduced. Hence, change in a consumer’s income pattern may also be an exception to the law of demand.

Price Elasticity of Demand

It is the ratio of the percentage change in demand to the percentage change in price of particular commodity. The formula for the coefficient of price elasticity of demand for a good is

Where P is the price of the demanded good and Q is the quantity of the demanded good. The above formula usually yields a negative value, due to the inverse nature of the relationship between price and quantity demanded, as described by the "law of demand". For example, if the price increases by 5% and quantity demanded decreases by 5%, then the elasticity at the initial price and quantity = −5%/5% = −1. The only classes of goods which have a PED of greater than 0 are Giffen goods[footnoteRef:2]. Although the PED is negative for the vast majority of goods and services, economists often refer to price elasticity of demand as a positive value (i.e., in absolute value terms). [2: Giffen goods are inferior goods that people consume more of as prices rise, and vice versa. Since a Giffen good does not have easily available substitutes, the income effect dominates the substitution effect.]

Point-price and Arc elasticity of demand

The point elasticity of demand method is used to determine change in demand within the same demand curve, basically a very small amount of change in demand is measured through point elasticity. One way to avoid the accuracy problem described above is to minimize the difference between the starting and ending prices and quantities. This is the approach taken in the definition of point-price elasticity, which uses differential calculus to calculate the elasticity for an infinitesimal change in price and quantity at any given point on the demand curve.

In other words, it is equal to the absolute value of the first derivative of quantity with respect to price multiplied by the point's price (P) divided by its quantity (Qd). However, the point-price elasticity can be computed only if the formula for the demand function, Q d = f (P), is known so its derivative with respect to price, d Qd / d P can be determined.

The arc elasticity is defined mathematically as;

=

This method for computing the price elasticity is also known as the "midpoints formula", because the average price and average quantity are the coordinates of the midpoint of the straight line between the two given points. This formula is an application of the midpoint method. However, because this formula implicitly assumes the section of the demand curve between those points is linear, the greater the curvature of the actual demand curve is over that range, the worse this approximation of its elasticity will be (Wall and Griffiths, 2008).

Relationship between Marginal Revenue and Elasticity of Demand

Marginal revenue is the change in total revenue due to change in output (sales) by one unit. In perfect competition, marginal revenue is always equal to average revenue or price, because the firm can sell as much as it likes at the going market Price. So, the firm is a price-taker.

For all markets other than purely competitive ones, the MR function is lower than the demand (or average revenue) functions for all units produced expect the first. In other words, MR is less than price. MR may be expressed as MR = dTR/dQ, where the dTR with respect to dQ is the first derivative of the total revenue function.

where MR = marginal revenue,

P = market price of the product, and

Ep = price elasticity of demand for the product

The above formula is very useful when the demand function has a known constant price elasticity. Business managers must estimate the value of MR in order to arrive at decisions about price and output.

Effect of Price Elasticity of Demand on total revenue

· When the price elasticity of demand for a good is perfectly inelastic (EP = 0), changes in the price do not affect the quantity demanded for the good; raising prices will always cause total revenue to increase. Goods necessary to survival can be classified here; a rational person will be willing to pay anything for a good if the alternative is death. For example, a person in the desert weak and dying of thirst would easily give all the money in his wallet, no matter how much, for a bottle of water if he would otherwise die. His demand is not contingent on the price.

· When the price elasticity of demand for a good is relatively inelastic (−1 < EP < 0), the percentage change in quantity demanded is smaller than that in price. Hence, when the price is raised, the total revenue increases, and vice versa.

· When the price elasticity of demand for a good is unit (or unitary) elastic (EP = −1), the percentage change in quantity demanded is equal to that in price, so a change in price will not affect total revenue, since EP = 1, then MR = 0.

· When the price elasticity of demand for a good is relatively elastic (−∞ < Ed < −1), the percentage change in quantity demanded is greater than that in price. Hence, when the price is raised, the total revenue falls, and vice versa.

· When the price elasticity of demand for a good is perfectly elastic (EP is − ∞), any increase in the price, no matter how small, will cause the quantity demanded for the good to drop to zero. Hence, when the price is raised, the total revenue falls to zero. This situation is typical for goods that have their value defined by law (such as fiat currency); if a five-dollar bill were sold for anything more than five dollars, nobody would buy it, so demand is zero.

Unit elastic zone

P

Elastic zone

Inelastic zone

Demand Curve

Q

Revenue

Decreasing ED

Fig 1. Figure 1. A set of graphs shows the relationship between demand and total revenue (TR) for the specific case of a linear demand curve. As price decreases in the elastic range, TR increases, but in the inelastic range, TR decreases. TR is maximized at the quantity where PED = 1

Income Elasticity of Demand

Income elasticity of demand (Em) is an economic measure of how responsive the quantity demanded for a good or service is to a change in income (M). Em is percentage change in quantity demanded for a good (X) to percentage change in income.

Em = *

The higher the income elasticity of demand for a particular good, the more demand for that good is tied to fluctuations in consumers' income. Businesses typically evaluate the income elasticity of demand for their products to help predict the impact of a business cycle on product sales.

If EM>0, X is normal good. 0<EM<1 then X is generally termed as income inelastic and necessity good.

Normal goods have a positive income elasticity of demand; as income rises, more goods are demanded at each price level. Normal goods whose income elasticity of demand is between zero and one are typically referred to as necessity goods, which are products and services that consumers will buy regardless of changes in their income levels. Examples of necessity goods and services include tobacco products, haircuts, water, and electricity.

Inferior goods have a negative income elasticity of demand; as consumers' income rises, they buy fewer inferior goods. A typical example of such a type of product is margarine, which is much cheaper than butter.

The income elasticity of demand for a particular product can be negative or positive, or even unresponsive.

Cross Price Elasticity of Demand

The cross elasticity of demand (Ec) is an economic concept that measures the responsiveness in the quantity demanded of one good when the price for another good changes. EC is percentage change in quantity demanded for a good (X) to percentage change in the price of another good (PY)

Ec = *

The cross-elasticity demand for substitute goods is always positive because the demand for one good increase when the price for substitute goods increases. For example, if the price of coffee increases, the quantity demanded for tea (a substitute beverage) increases as consumers switch to a less expensive yet substitutable alternative. 

Items with a coefficient of 0 (zero) are unrelated items and are goods independent of each other. Items may be weak substitutes, in which the two products have a positive but low cross elasticity of demand. This is often the case for different product substitutes, such as tea versus coffee. Items that are strong substitutes have a higher cross-elasticity of demand. Consider different brands of tea; a price increase in one company’s green tea has a higher impact on another company’s green tea demand. Toothpaste is an example of a substitute good; if the price of one brand of toothpaste increases the demand for a competitor's brand of toothpaste increases in turn.

Alternatively, the cross elasticity of demand for complementary goods is negative. As the price for one item increases, an item closely associated with that item and necessary for its consumption decreases because the demand for the main goods has also declined. For example, if the price of coffee increases, the quantity demanded for coffee stir sticks drops as consumers are drinking less coffee and need to purchase fewer sticks.

Summary: The cross-price elasticity for substitutes is always positive; the price of one good and the demand for the other always move in the same direction. Cross-price elasticity is negative for complements; price and quantity move in opposite directions for complementary goods and services. Finally, cross-price elasticity is zero, or nearly zero, for unrelated goods in which variations in the price of one good have no effect on demand for the second.

Usefulness of Ec: Business organizations utilize the Ec coefficient to set the prices to sell their products. Products with no substitutes can be sold at higher prices because there is no cross-elasticity of demand to consider. However, incremental price changes to goods with substitutes are analyzed to determine the appropriate level of demand desired and the associated price of the good.

Analyzing Consumer Demand and Behavior

These new approaches[footnoteRef:3] build on methods that marketing departments have traditionally used to analyze consumer behavior. [3: Farnham, Paul G. 2014. Economics for Managers. 3rd Ed. Pearson.]

1. Expert opinion

2. Consumer survey

3. Test marketing and price experiments

4. Analyses of census and historical data

5. Unconventional methods

6. Econometric approaches

Managerial rule of Thumb: when using expert opinion, consumer surveys, test marketing, and price experiments to analyze consumer behavior, managers must consider the following points:

· Whether the participating groups are truly representative of the larger population?

· Whether the answers given in these formats represent actual market behavior?

· How to isolate the effects of different variables that influence demand?

Consumer Demand and Behavior: Econometric Approach

1. Simple Regression Analysis

2. Multiple regression analysis

Suppose you’re a sales manager trying to predict next month’s numbers. You know that dozens, perhaps even hundreds of factors — from the weather to a competitor’s promotion to the rumor of a new and improved model — can impact the numbers. Perhaps people in your organization even have a theory about what will have the biggest effect on sales. Regression analysis is a way of mathematically sorting out which of those variables does indeed have an impact. It answers the questions: Which factors matter most? Which can we ignore? How do those factors interact with one another? And, perhaps most important, how certain are we about all these factors?

In regression analysis, those factors are called “variables.” You have your dependent variable — the main factor that you’re trying to understand or predict, monthly sales for example. And then you have your independent variables— the factors you suspect have an impact on your dependent variable, say advertisement spending.

Regression models are used to describe relationships between variables by fitting a line to the observed data. Regression allows you to estimate how a dependent variable changes as the independent variable(s) changes.

Simple linear/Multiple linear regression is used to estimate the relationship between one/two or more independent variables and one dependent variable.

Simple linear regression

Simple linear regression is modeled as:

· y is the predicted value of the dependent variable ( y) for any given value of the independent variable ( x).

· B0 is the  intercept, the predicted value of  y when the  x is 0.

· B1 is the regression coefficient – how much we expect  y to change as  x increases.

· x is the independent variable (the variable we expect is influencing  y).

· ε is the  error of the estimate, or how much variation there is in our estimate of the regression coefficient.

Linear regression finds the best fit line through your data by searching for the regression coefficient (B1) that minimizes the total error (ε)of the model.

Because performing linear regression by hand is a tedious process, researchers use statistical programs.

Assumptions of simple linear regression

Simple linear regression is a parametric test, meaning that it makes certain assumptions about the data. These assumptions are:

1. Homogeneity of variance (homoscedasticity): the size of the error in our prediction doesn’t change significantly across the values of the independent variable.

2. Independence of observations: the observations in the dataset were collected using statistically valid sampling methods, and there are no hidden relationships among observations.

3. Normality: The data follows a normal distribution.

Linear regression makes one additional assumption:

4. The relationship between the independent and dependent variable is linear: the line of best fit through the data points is a straight line (rather than a curve or some sort of grouping factor).

If your data does not meet the assumptions of homoscedasticity[footnoteRef:4] or normality, you may be able to use a nonparametric test instead, such as the Spearman rank test. [4: The assumption of homoscedasticity (meaning “same variance”) is central to linear regression models. Homoscedasticity describes a situation in which the error term (that is, the “noise” or random disturbance in the relationship between the independent variables and the dependent variable) is the same across all values of the independent variables. Heteroscedasticity (the violation of homoscedasticity) is present when the size of the error term differs across values of an independent variable. The impact of violating the assumption of homoscedasticity is a matter of degree, increasing as heteroscedasticity increases. ]

Example

Businesses often use linear regression to understand the relationship between advertising spending and revenue. For example, they might fit a simple linear regression model using advertising spending as the predictor variable and revenue as the response variable. The regression model would take the following form:

Revenue = β0 + β1 (Media Ad. Spending)

The coefficient β0 would represent total expected revenue when ad spending is zero.

The coefficient β1 would represent the average change in total revenue when ad spending is increased by one unit (say one dollar).

If  β1 is negative, it would mean that more ad spending is associated with less revenue.

If β1 is close to zero, it would mean that ad spending has little effect on revenue.

Decision rule:

If β1 is positive, it would mean more ad spending is associated with more revenue. Depending on the value of β1, a company may decide to either decrease or increase their ad spending.

One hypothetical regression output (Simple Linear Regression):

Residuals:

Min 1Q Median 3Q max

-2.0247 -0.4854 0.04078 0.4598 2.576

Coefficients:

Estimated Std. Error t-value Pr(>| t |)

(Intercept) o.20427 0.08884 2.999 0.219

Media Spend. 0.71383 0.01854 38.505 0.00001

Multi. R-squared: 0.7493, Adj. R-squared: 0.7488

Revenue = 0.204 + 0.713 Media Spending

Revenue = 0.204 + 0.713 Media Spending ± 0.018

Row 1 of the table is labeled (Intercept). This is the y-intercept of the regression equation, with a value of 0.20. You can plug this into your regression equation if you want to predict revenue values across the range of media spending that you have observed:

The next row in the ‘Coefficients’ table is Media Spending. This is the row that describes the estimated effect of spending on reported revenue.

The Estimate column is the estimated  effect, also called the  regression coefficient or r2 value. The number in the table (0.713) tells us that for every one unit increase in spending (eg one unit of spending = 10,00) there is a corresponding 0.71-unit increase in reported revenue (eg. 10,000).

The Std. Error column displays the standard error of the estimate. This number shows how much variation there is in our estimate of the relationship between revenue and spending. The t value column displays the test statistics. Unless you specify otherwise, the test statistic used in linear regression is the  t value from a two-sided t-test. The larger the test statistic, the less likely it is that our results occurred by chance.

The Pr(>| t |) column shows the P -value. This number tells us how likely we are to see the estimated effect of spending on revenue if the null hypothesis of no effect were true. Because the  P  value is so low ( p < 0.001), we can  reject the null hypothesis and conclude that spending has a statistically significant effect on revenue.

The last three lines of the model summary are statistics about the model as a whole. The most important thing to notice here is the  P  value of the model. Here it is significant ( p < 0.001), which means that this model is a good fit for the observed data.

Practice:

Fitted line: Sales (Y) = 125.8 + 171.5 spending

Multiple Regression:

Multiple linear regression is used to estimate the relationship between two or more independent variables and one dependent variable. You can use multiple linear regression when you want to know:

Example: 1. how rainfall, temperature, and amount of fertilizer affect crop yield?

2. How do promotion, geography, age, and education affect total

Sales/revenue of a firm?

Assumption: same as SLR

Multiple linear regression model can be presented as:

·  = the predicted value of the dependent variable

·  = the y-intercept (value of y when all other parameters are set to 0)

·  = the regression coefficient ( ) of the first independent variable ( ) (a.k.a. the effect that increasing the value of the independent variable has on the predicted y value)

·  = the regression coefficient of the last independent variable

·  = model error 

Interpreting the results: same as SLR

Practice Example:

Model used:

Heart disease (Y) = β0+ β1 biking + β2 smoking + ε i

Regression Output:

Residuals:

Min 1Q Median 3Q max

-3.0255 -0.4466 0.0362 0.623 1.097

Coefficients:

Estimated Std. Error t-value Pr(>| t |)

(Intercept) 14.987 0.08o34 184.55 0.0019

Biking -0.2002 0.01854 -146.53 0.00001

Smoking 0.1783 0.0035 50.39 0.00000

Multi. R-squared: 0.9796, Adj. R-squared: 0.9795

Fitted regression output:

heart disease = 15 + (-0.2*biking) + (0.178*smoking) or

heart disease = 15 + (-0.2*biking) + (0.178*smoking) ± e

Interpretation: There are significant relationships between the frequency of biking to work and the frequency of heart disease and the frequency of smoking and frequency of heart disease (p < 0.001 for each). Specifically, we found a 0.2% decrease (± 0.0014) in the frequency of heart disease for every 1% increase in biking, and a 0.178% increase (± 0.0035) in the frequency of heart disease for every 1% increase in smoking.

Reference:

Wall, Stuart and Griffiths, Alan (2008). Economics for Business and Management . Financial Times Prentice Hall. ISBN   978-0-273-71367-8 . Retrieved 6 March 2010.

Varian, H. R. 1999. Intermediate Microeconomics: A Modern Approach, 5th Ed. New York.

Farnham, Paul G. 2014. Economics for Managers. 3rd Ed. Pearson

2