#4971 - Consumer Surplus Mathematical Methods Model Analysis - Advanced Microeconomics

A consumer has the utility function u=x^0.5y^0.5.

(i) Find the consumers Marshallian demand curves for x and y

Thus

(ii) Suppose that initially the consumer has an income of 100, and faces Px = 1 and Py = 2. What will be the consumption of x and y?

(iii) Suppose that the price of y is now reduced to 1. Find an approximate value for the consumer surplus resulting from this price change, using the `rule of a half'

If the price of Y is reduced to 1 then Y=50.

Using the 'rule of a half'

Marshallian Consumer Surplus (MCS) = 0.5(P₁-P₂)(Y₁+Y₂)

0.5(2-1)(50+25) = 37.5

More accurately:

6

(iv) You are now asked to calculate the compensating and equivalent variations for this price change. Explain carefully how you would go about it (DO NOT ATTEMPT THE ACTUAL CALCULATION)

We need to minimise expenditure subject to the utility constraint.

From (i) and (ii)

From (iii)

(v) How would you expect the three measures of consumer surplus to compare in magnitude?

EV > MCS > CV in this case, where there is a price fall for a normal good.

Note that CV > MCS > EV where there is a price rise for a normal good.

COMPENSATING VARIATION (CV)

The amount of the money transfer necessary, following a price change, to maintain an individual’s welfare at its original level

EQUIVALENT VARIATION (EV)

The amount of the money transfer, which, in the absence of a price change, affords the individual an exactly equivalent change in welfare.

Neither is equal to Marshall's measure of consumer surplus due to the effective removal of the income effect by using compensating demand. Instead, the CV measure takes the initial income level as the base point; the EV measure takes the final income level.

(vi) In what circumstances is the Marshallian consumer surplus a good approximation to true willingness to pay / Discuss the pros and cons of each measure of consumer surplus?

The market place observes a Marshallian demand curve, so for a normal good, where demand increases with income, the following inequality holds for measuring consumer surplus: EV > MCS > CV. Different circumstances dictate as to which measure is most applicable. Firstly, I will observe MCS.

In the extreme case when consumer preferences are such that there exists no income effect between goods, a quasi-linear utility function yields EV = MCS = CV. In this scenario, Marshallian CS is as good approximation as any.

Willig suggests another possibility, by indicating error bounds for when the Marshallian measure is near enough the same as CV and EV. Using the Taylor approximation, the difference between CV (or EV) and Marshallian consumer surplus is a loose function of income elasticity, Marshallian consumer surplus and income. Willig surmises that EV MCS CV when income elasticity is zero:

Only when MCS/Y or e_y are large is the difference significant.

However, the Marshallian consumer surplus cannot account for multiple price changes. Let’s take a scenario where the price fall of good x will cause a shift in demand for good y, and this new level of demand must be used to evaluate consumer surplus after the price fall of good y. A money measure of utility can be ascertained easily enough, but what happens if the order of price falls is reversed? Even with the same nominal changes, if good y falls in price and then affects demand in good x’s market, then this measure of utility may differ. There is no reason to expect that the money measure of utility is similar. This is the path dependency problem.

No path dependency problem is faced when using the CV and EV measures. Intuitively, there is a substitution effect but not an income effect on the compensated demand of a price change. We remove the income effect by holding the consumer at a pre-specified utility level. Given that the change in utility from a price change is uniquely determined from the terminal values of the prices, we can now avoid the path dependency problem. The appropriate measure is determined by the utility level observed. If the consumer throughout remains at the initial welfare, the measure is compensating variation and is most applicable when analysing the implications for a compensation test. If the consumer is held at the after-price change level of welfare, the measure is equivalent variation where the changes in utility can be easily ascertained and thus be applied to a social welfare function....