# Consumer Theory (Finding demand functions)

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, 1 MU$_x=\frac{1}{2\sqrt{x}}$ and MU$_y=1$. Use $p_x$ to represent the price of food, $p_y$ 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 $I/p_y \geq p_y/(4p_x)$.

I'm having a really tough time working out how to solve this problem and any kind of help would be much appreciated.

Given the data:

• Utility function: $u(x, y) = \sqrt{x} + y$
• Income: $I > 0$
• Prices: $p_X > 0$ and $p_Y = 1$

Utility maximization problem is

\begin{eqnarray*} \max\limits_{x,y} & \ \ \sqrt{x} + y \\ \text{s.t.} & \ \ p_Xx+ y = I \\ \text{and} & \ \ x\geq 0, \ \ y\geq 0 \end{eqnarray*} We can substitute $y = I - p_Xx$ in the objective and convert it into a single variable optimization problem: \begin{eqnarray*} \max\limits_{x} & \ \ \sqrt{x} + I- p_Xx \\ \text{s.t.} & 0 \leq x \leq \frac{I}{p_X} \end{eqnarray*} Differentiating the objective with respect to $x$ yields: $\frac{1}{2\sqrt{x}} - p_X$ which is a decreasing function of $x$. The interpretation of the derivative is that it is the net marginal benefit curve of consuming X. As long as the value of the derivative is positive, it pays to spend more on X. Therefore, the utility maximizing choice will be to spend all the money on X if the net marginal benefit curve is still positive at $x = \frac{I}{p_X}$ i.e. $\frac{1}{2}\sqrt{\frac{p_X}{I}} - p_X>0$ holds, and the equilibrium choice will satisfy the property $\frac{1}{2\sqrt{x}} - p_X = 0$ otherwise. So, the demand function for X is: $\begin{eqnarray*} x(p_X, p_Y = 1, I) = \begin{cases} \frac{1}{4p_X^2} & \text{if } \frac{1}{2}\sqrt{\frac{p_X}{I}} - p_X\leq 0 \\ \frac{I}{p_X} & \text{if } \frac{1}{2}\sqrt{\frac{p_X}{I}} - p_X>0 \end{cases} \end{eqnarray*}$

First, your utility function is strictly increasing in both goods, so you know that your budget will bind so you can set I=pxx+pyx

Second, your utility function is known as "quasilinear", where the good y is linear in your utility and the other good is there in some increasing function not connected to y. That's why the marginal utility for y is just a constant.

Finally, solve for x, where x is a function of prices, then plug that into the budget constraint to solve for y. y should be a function of prices and income.

Since you showed some work:

Set the ratio of MUs to the ratio of prices: $MU_x/MU_y=p_x/p_y$, yielding a demand for $x=\frac{p_y^2}{4p_x^2}$

Now plug that into the budget, yielding the demand for $y= \frac{I}{p_y}-\frac{p_y}{4p_x}$

• so based off of this my demand function for good x would be equivalent to X=Py^2/4Px^2. Then plugging it into the budget constraint of I=PxX+PyY, Y would give Y=-(Py^2-4PxI)/4PyPx. Is this correct @frage_man? – T.H Aug 21 '16 at 3:00
• @T.H. Yes! I edited my answer to write it out. – VCG Aug 21 '16 at 3:16