# Complicated utility function

I am trying to answer this past paper question on microeconomics and a rather complicated utility function question. The question is below as well as my answer.

In my answer, I use the fact that the marginal rate of technical substitution (MRS) is equal to their price ratio. I think that I derive both demand functions for pizza and apple. I am not sure if they are right and I don't know how to answer:

How James responds to an increase in the price of Pizza?

Just looking at the price of pizza (p1) in the demand functions would mean that an increase in p1 would result in a decrease in demand for apples.

I feel like I have done something wrong and that I would have to consider substitution and/or income effects? I'm terribly confused.  The demand functions you have derived are correct.

The Marshall demands are given as

$$x^\star(p_x,p_y,M) = \sqrt{\frac{M}{p_y}} \\[8pt] y^\star(p_x,p_y,M) = \frac{M - p_x \sqrt{\frac{M}{p_y}}}{p_y},$$

where $$M$$ is income and $$p_x$$ is pizza price and $$p_y$$ is the apple price.

To answer the question of how the agent responds to change in pizza price it would be standard to find derivatives and/or elasticities of demand with respect to the price $$p_x$$ for pizza.

The derivative of pizza demand with respect to own-price is given as $$\frac{\partial x^\star(p_x,p_y,M)} {\partial p_x} = 0,$$ which implies that the elasticity is

$$El_{p_x}x^\star(p_x,p_y,M) = \frac{p_x}{x^\star(p_x,p_y,M)}\frac{\partial x^\star(p_x,p_y,M)} {\partial p_x} = 0,$$

so you can conclude that demand is perfectly inelastic. When the price increase the consumer simply keeps demand constant and substitutes away from apples. This is manifest from the fact that

$$\frac{\partial y^\star(p_x,p_y,M)} {\partial p_x} = - \sqrt{\frac{M}{p_y^3}},$$

which is clearly negative and by implication the cross-price elasticity $$El_{p_x}y^\star(p_x,p_y,M) <0$$.

This is "unusual" if you expected apples and pizza's to be substitutes with the demand of one increasing when the cross-price increase.

• Thank you Jesper, I like the way that you applied the reasoning to show that it will only substitute away from apples and showed the quantity changes with changes in price of pizza px Dec 12 '20 at 12:15