# Revealed preference when there is no substitution effect

I've started reading about revealed preference, and in particular, how it works in the context of price changes.

I understand that revealed preference can show us the general direction of the substitution effect. However, can revealed preference help us if there is no substitution effect present? As in, is there any information we can discern from it about the final choice?

For example, consider a scenario where Px and Py increase by the same amount. (I've drawn this diagram below). If they initially chose Bundle X1, then they would choose it again on the "hypothetical" bundle after income is compensated for. However, what does this mean for the final choice, if anything? Would they consume at X = 4 also? Or can we not know anything else, as the 0 substitution effect means we can't know the direction of the final choice?

My apologies if I've missed something obvious, but it's a problem that's been bugging me for a while!

• With "by the same amount" you mean the same absolute or the same relative increase? In your graph, this happens to be indistinguishable since your initial prices $P_x$ and $P_y$ are identical. Apr 14, 2021 at 8:01
• @VARulle My apologies - it's both. The prices both start at the same amount, and increase by the same amount.
– Mary
Apr 14, 2021 at 21:06

The short answer is no. Reveal preference only determines which combination of goods is preferred, it does not affect in any way the allocation of goods, which has more to do with the functional form of the utility function, the types of goods and the types of changes made. For example, the 2 budget lines represent different budget sets when there is a price change in either good. The weak axiom of revealed preference (WARP) posits that $$x^i$$ is preferred to $$x^j$$ insofar as good 1 is revealed preferred to good 2. It does not give any information on the allocation of the consumption bundle, however. If there is a simultaneous change in both the price of good 1 and 2 so that there exists another budget line that is parallel with $$p^i$$, the allocation of good in this case has more to do with the optimization of utility.

I have attempted to derive the change here but I am not entirely sure if this is the correct way to do it or whether I have done all the steps correctly so take it with a pinch of salt.

Assuming the consumer is maximizing his/her utility function, which is standard in consumer theory, we want to find the highest point tangent to the budget line. The slope of the indifference curve, after the price adjustment, depends on the types of adjustment made, the type of goods, and the functional form of the utility function. As I understand it, you are asking about the case of relative price change (since the slope of the budget line remains the same). In this case, the total change in each type of goods can be derived from the Slutsky equation

\begin{align} {d x_A\over dp} &= {\partial x_A\over \partial p_A} + {\partial x_A \over \partial p_B}\\ &= \underbrace{\partial h_A\over \partial p_A}_{\text{substitution effect good A (?)}}-\underbrace{{\partial x_A\over \partial m}x_A}_{\text{income effect good A (?)}} + \underbrace{\partial h_A\over \partial p_B}_{\text{substitution effect good B (?)}}-\underbrace{{\partial x_A\over \partial m}x_B}_{\text{income effect good B (?)}}\\ \end{align} where $$x_i,h_i,p_i$$ and $$m$$ respectively denote the Marshallian demand, Hicksian demand, the price of good $$i$$, and the total budget. The ?'s in the parentheses are the sign of the term. As in the case here where the types of goods are ambiguous, I can say little about the direction of change.

In the case of normal good (both are normal goods) and standard Cobb-Douglas utility, the substitution effect of own good (A to A) is negative, the income effect of own good (A to A) is negative, the substitution effect of the other good (A to B) is positive while the income effect of the other good (A to B) is negative. The substitution effect of own good (A to A) and of the other good (A to B) cancel out each other, leaving only the negative terms. This generally translates into a lower Marshallian demand for good A, and similarly for good B. The total change is then, $${dx_A\over dp}=-{\partial x_A\over \partial m}(x_A+x_B)$$. In levels, $$\Delta x_A=-{\partial x_A\over \partial m}(x_A\Delta p_A+x_B\Delta p_B)\\ \Delta x_B=-{\partial x_B\over \partial m}(x_A\Delta p_A+x_B\Delta p_B)$$ Taking the ratio, $${x_A-\Delta x_A\over x_B-\Delta x_B}={x_A+{\partial x_A\over \partial m}(x_A\Delta p_A+x_B\Delta p_B)\over x_B+{\partial x_B\over \partial m}(x_A\Delta p_A+x_B\Delta p_B)}$$ This will remain unchanged if and only if $$\Delta x_A/\Delta x_B=x_A/x_B$$, which simplifies to $${\partial x_A\over \partial m}/{\partial x_B\over \partial m}={x_A\over x_B}$$. This holds true when with the CES form of Cobb-Douglas utility wherein the coefficient of good A and B are equal to 0.5.

• This is wonderful, thank you so much! Your derivation was particularly interesting to read through.
– Mary
Apr 14, 2021 at 21:09