Estimating Demand Elasticities and Compensating Variation

A student posed the following question:

Hi Respected Sir,
I am writing a report on Compensating Variation (CV) in case of more than two goods say 8 goods. My question is how can we estimate it in case of more than 2 goods.
Another Question is How can we write the equations for 8 commodities in Almost Ideal Demand System (AIDS) to calculate Own , Cross and Expenditure Elasticities of demand.
In which software both of the methods can be calculated?
I am waiting for your reply.
Thanks in anticipation

My response:

This question is more econometrics than it is intermediate micro.  So I am only going to provide a partial response and stick to the economic theory part, though I must say I'd only teach what I discuss below at the graduate level. 

The traditional approach to consumer choice is to start with preferences, specified by a utility function, u, and combine that with a budget constraint, that depends on a price system p = (p1,p2,...,pn) that specifies the prices of each good or service, and the consumers income y.  Together the price system and income determine the budget set.  The consumer's choice, or demand, or optimum, call it x*(u,p,y) solves the problem of maximizing utility subject to the budget constraint.   

There is an alternative approach called the duality approach, which is useful conceptually and in laying the foundation for the econometrics.  Two value functions are determined.  One, measured in dollar terms, is called the Expenditure Function.  It is analogous to the Cost Function developed in the theory of the firm.  The expenditure function maps indifference curves (or when there are more than two goods, level sets of the utility function) and price systems into an expenditure level - the least expenditure it takes to reach that indifference curve at the given prices.   The other value function, measured in utility terms, is called the Indirect Utility Function.  It maps budget sets into utility levels.  Alternatively, it gives the utility attained at the consumer's choice. 

One of the powerful results from duality theory is that you can recover the consumer demand's from these value functions.  The compensated demands (these measure the substitution effect only) are given by the first partial derivatives of the Expenditure Function with respect to the specific price.  The ordinary demands can be obtained in a similar way from the Indirect Utility Function, though the result, known as Roy's Identity, is a bit more complicated.   

Let me close with the little I know about the Almost Ideal Demand System.  Deaton and Mulbauer start with the Expenditure Function, express it in log form, and then linearize it locally, assuming it is some average of the expenditure at subsistence (the worst possible point) and bliss (the best possible point).  This makes it suitable for estimation. 

Good luck on your paper. 

Consumer Surplus

Serdar asked:

For the arguement  of "Compansated variation is always bigger than consumer surplus under all price changes", could you please discuss whether it is true or not by drawing the necessary graphs? And i would be pleased if you can give a numerical example to support the arguement (utility function is a Cobb-Douglas utility function). thank you in advance.

My response:

This is discussed in the video, CV EV and Change in CS.  The graph below is from the spreadsheet used to make that video.  Let's review the definitions of CV and CS and then consider the determinants of which is bigger.

CV - this is the area to the left of the compensated (Hicksian) demand curve for the original optimum between the original price and the new price.

Decrease in CS - this is area to the left of the ordinary demand curve between the original price and the new price.

Remember that the compensated demand measures the substitution effect only, but that the ordinary demand measures the substitution effect and the income effect in combination.  For a good where there is no income effect, CV = Decrease in CS.

More generally, what matters are:

(1) the direction of the price change, and

(2) whether the good is normal or inferior.

In the graph above the original price is given by the height of the dashed horizontal line.  Then the price rises and the new price is indicated by the height of the dotted horizontal line.  The blue curve is the ordinary demand curve.  The red curve is the compensated demand curve for the original optimum.  In this diagram, the blue curve is more elastic at the original price than the red curve.  That will be the case for a normal good.  The area to the left of the red curve between the two prices is greater than the area to the left of the blue curve between the two prices. Thus in this case CV > Decrease in CS.  

I leave it to you to consider the case of a price decrease and/or the case where the good is inferior.  By the way, if the utility function is Cobb-Douglas, then the good is normal.

The effect of a price of X change

Elly asked:

Hi Prof, can i ask you some questions regarding the budget line and indifference curve? When price of X falls, and X is normal, does it mean that normal good will always be on the right side of the budget line that has been separated by a point C? If that is the case, when price of X rises, and X is inferior, does the inferior good always falls on the right of budget line? Because from what i've known, when price of X rises, the budget line will rotate to the left from the original budget line, which means income decreases, so people will buy more inferior good, so inferior good will be on the right. Is that always true? I have also seen a few cases where the inferior good is on the left side even though Price of X rises and i cannot understand. Thank you for taking time to read my enquiries.

My response:

First, let's stick to the case where the price of X rises.  Afterward, the case where the price of X
falls can be worked through by doing the same analysis but in reverse.  Next, note that there are two effects to consider from a price change - a substitution effect and an income effect.  Let's consider those effects separately and then put them together.

Substitution effect

An increase in the price of X causes an increase in the relative price of X, because the price of Y has remained constant.  When a good's relative price has risen the substitution effect says less of the good (move to the left in the way Elly expresses it above).

Income effect

An increase in the price of X rotates the budget line inward around the Y intercept.  As long as some X was being consumed before the price change, that bundle is no longer affordable so this change means a reduction of real income.  The consequence of that income change on the amount of X consumed depends on whether X is normal or inferior. When X is normal the reduction of income leads to reduced consumption of X (again, that is a move to the left).  When X is inferior, the reduction of income leads to an increase in the consumption of X.

Overall

The substitution and income effects support each other when X is normal.  In this case the overall is to have less X consumed.  When X is inferior, however, the income effect offsets the substitution effect.  As an empirical matter we think that mainly the overall is determined by the substitution effect, so there still will be less X.  But it is logically possible for the income effect to win out, in which case the good is called a Giffen Good, named after the Scottish economist Sir Robert Giffen.