Change in Consumer Surplus with Two Price Changes: Is it Path Dependent?

Question

The diagrams below showing demand for goods X1 and X2 are adapted from Johansson P-O (1991) An Introduction to Modern Welfare Economics (p 43). Demand for each good is assumed to be a function of its own price and the price of the other good. At the initial price P2i of X2, demand for X1 is D1i, and at the final price P2f of X2, demand for X1 is D1f. Similarly, demand for X2 increases from D2i to D2f when the price of X1 falls from P1i to P1f. (Thus the goods seem to be complements, but that’s incidental).

Question: What is the change in total (Marshallian) consumer surplus resulting from the combination of the two price changes.

Johansson argues that it depends on the sequence of the price changes. Suppose that the price of X1 changes first. Considering the change in price of X1 with the price of X2 remaining at P2i, the relevant demand for X1 is D1i and we can infer that consumer surplus increases by area A. Considering the change in price of X2, however, by assumption the price of X1 has already fallen to P1f, so the relevant demand is D2f and consumer surplus increases by area B + C. In total, therefore consumer surplus increases by A + B + C. But if the price of X2 changes first, then parallel reasoning leads to the conclusion that total consumer surplus increases by A + B + D.

It seems to me that this should instead be analysed as follows:

1. The change in total consumer surplus is the difference between a) the initial total consumer surplus and b) the final total consumer surplus.

2. The sequence of the price changes has no bearing on either (a) or (b).

3. (a) is E + F and (b) is (A + D + E + G) + (B + C + F + H).

4. Hence the change in total consumer surplus is an increase equal to (b) minus (a) or A + D + G + B + C + H.

Am I missing something?

Answer 1

Your reasoning is correct (i.e. the book is wrong). First, let's follow the book's logic a little more carefully step by step, beginning with the case where p1 changes first:

1. a fall in p1 leads to a gain of A in market 1 and of H in market 2 (because D2 shifts).
2. a subsequent fall in p2 leads to a gain of B + C in market 2 (we are now using the new demand curve in market 2); moreover, it causes D1 to shift so the consumer gains G + D.
3. thus, the total gain is A + H +G + D + B + C.

If p2 changes first:

1. a fall in p2 leads to a gain of B in market 2 and of G in market 1 (because D1 shifts).
2. a subsequent fall in p1 leads to a gain of A + D in market 1 (we are now using the new demand curve in market 1); moreover, it causes D2 to shift so the consumer gains H + C.
3. thus, the total gain is A + H +G + D + B + C.

We see that the order doesn't matter.

Another quick way to verify that the order doesn't matter is to note that these demand curves are coming from a consumer who chooses the optimal demand vector for given prices. If the prices are the same then the solution to the maximisation problem will also be the same.