Microeconomics (quasi-linear utility function)

Question

Suppose that Sally’s preferences over baskets containing food (good x), and clothing (good y), are described by the utility function $U(x,y)=\sqrt{x}+y$.

Sally’s corresponding marginal utilities are,

$U^\prime_x=\frac{1}{2\sqrt{x}}$ and $U^\prime_y=1$.

Use Px to represent the price of food, Py to represent the price of clothing, and I to represent Sally’s income.

Question 1: Find Sally's food demand function, and Sally's clothing demand function. For the purposes of this question you should assume that I/Py greater than or equal Py/4Px.

Clothing demand function : y= I/py - Py/4Px

Food demand function : x = P^2y/4P^2x

After deriving demand function for both Clothing and Food, the following question confused me.

Question 3: Now assume that I/Py less than Py/4Px. Find Sally's food demand function, and Sally's clothing demand function.

How do i get about doing this following question and getting the NEW Demand function for Food and Clothing?

I do know it might possible be a corner solution.. But how do i explain it in terms of the function as i am not given the price of food and price of clothing in numerical form?

Answer 1

This problem is a case of optimisation under inequality constraints, where the requirement $x,y \ge 0$ is not stated explicitly. Here is an answer in the language used for your question.

The demand functions are $x^*(p_x,p_y;I) = \dfrac{I}{p_x}$ and $y^*(p_x,p_y;I) = 0$.

Here is a proof. By the rule of weighted marginal utilities, at the margin it is better to purchase $x$ than $y$ whenever $$\frac{U^\prime_x}{p_x} \ge \frac{U^\prime_y}{p_y}$$ Substituting for $U^\prime_x$ and $U^\prime_y$ yields $$\frac{1}{2 \sqrt{x^*}}\frac{1}{p_x} \ge \frac{1}{p_y}$$ Replacing $x^* = I/p_x$, we get $$\frac{1}{2 \sqrt{Ip_x}} \ge \frac{1}{p_y}$$ Rearranging, this is precisely the condition $4IP_x \le p_y^2$. When this latter holds, it is always better to purchase $x$ rather than $y$.