Problem Set 4: Consumer Demand

Demand for Pumpkin Spice

Recall Kai Ryssdal from problem 3.2. His preferences over pumpkin spice lattes (good 1), pumpkin spice cookies (good 2), and pumpkin spice cheesecake (good 3) may be represented by the utility function \(u(x_1,x_2,x_3) = 3x_1 + 2x_2 + 6x_3\) Suppose Kai has $m = 36$ to spend on pumpkin spice things, the price of cookies is $p_2 = 8$, and the price of cheesecake is $p_3 = 18$. Derive and plot Kai’s demand function for pumpkin spice lattes. Remember: for the purposes of this class, goods are always perfectly divisible. It’s fine for Kai to buy half a latte, for example.

A New Utility Function

Consider the following utility function, which you haven’t seen before: \(u(x_1,x_2) = (8 + x_1)x_2\) Derive and plot the demand curve for good 1 if $p_2 = 4$ and $m = 120$.

Note: this was taken from an old exam…in an exam, what I often do is present a novel functional form and ask you to derive results from it. This helps to ensure you’re not just memorizing formulas, and gives you the tools you need to approach new kinds of functional forms…

Three’s a Crowd

(This is an optional, but easy, problem.) Let’s see how the Cobb-Douglas “rule” extends to three goods. Consider the following utility function: \(u(x_1,x_2,x_3) = a\ln x_1 + b\ln x_2 + c\ln x_3\) Find the demand function for good 1, $x_1^\star(p_1,p_2,p_3,m)$. Note: for this problem, you should set up the mathcal{L}angian, and use the budget constraint $p_1x_1 + p_2x_2 + p_3x_3 = m$.

CES Demand

Consider the utility function \(u(x_1,x_2) = \left(x_1^{-2} + x_2^{-2}\right)^{-{1 \over 2}}\)

What is the optimal bundle for this utility function if $p_1 = 8$, $p_2 = 1$, and $m = 80$? Illustrate both of these in a budget line/indifference curve diagram. Label the budget line $BL$, the optimal bundle $X$, and the indifference curve $U$.

What about if $p_1= 8$, $p_2 = 8$, and $m = 80$? Add this to your diagram from part (a), with labels $BL^\prime$, $X^\prime$, and $U^\prime$.

What happens to the optimal amount of goods 1 and 2 as a result of this price change? Does that indicate that these two goods are complements or substitutes?

Derive the demand function for good 1, $x_1^\star(p_1,p_2,m)$.

Derive and plot the demand curve for good 1 if $p_2 = 1$, and $m = 80$. Label your curve $d_1(p_1)$. Illustrate the quantity demanded at $p_1 = 8$.

Again suppose the price of good 2 rises from $p_2 = 1$ to $p_2 = 8$. What is the equation of the new demand curve? Add this new demand curve to your diagram as $d_1^\prime(p_1)$, and indicate the direction of the shift with an arrow. Be sure this new demand curve clearly shows the new quantity demanded at $p_1 = 8$!

How does the shift in the demand curve relate to your answer from part (c)?

Cobb-Douglas Demand and Indirect Utility

For the utility function \(u(x_1,x_2) = x_1^2x_2\)

Find the optimal bundle if $p_1 = 2$, $p_2 = 3$, and $m = 36$.

Derive the demand for good 1, $x_1^\star(p_1)$, if $p_2 = 3$ and $m = 36$. Confirm that the answer you found to part (a) is $x_1^\star(2)$.

Derive the general demand functions for good 1 and 2, $x_1^\star(p_1,p_2,m)$ and $x_2^\star(p_1,p_2,m)$, using the Lagrange method. (Do not transform the utility function!) What is the value of the Lagrange multiplier if $p_1 = 2$, $p_2 = 3$, and $m = 36$?

Find the indirect utility function, $V(p_1,p_2,m)$, by plugging your expressions for $x_1^\star(p_1,p_2,m)$ and $x_2^\star(p_1,p_2,m)$ back into the utility function. Take the partial derivatives of $V(p_1,p_2,m)$ with respect to $m$. What do you notice? (Hint: evaluate your expression at $p_1 = 2$, $p_2 = 3$, and $m = 36$…)

Practice with Duality (50Q only)

Consider an agent with utility function \(u(x_1, x_2) = \alpha \ln x_1 + (1 - \alpha) \ln x_2,\) and a budget constraint \(p_1 x_1 + p_2 x_2 = m,\) where $\alpha \in (0,1)$, $p_1, p_2 > 0$, and $m > 0$.

Marshallian Demand

Solve for the agent’s (Marshallian) demands for goods 1 and 2, $x_1^\star(p_1,p_2,m)$ and $x_2^\star(p_1,p_2,m)$, by maximizing utility subject to the budget constraint.
Also solve for the value for $\lambda$ which will be used in the last part of this problem.

Indirect Utility

Derive the agent’s indirect utility function $v(p_1,p_2,m)$ (the value of the maximization problem). In one sentence, explain what the indirect utility function represents.

Expenditure Function

Using duality, derive the agent’s expenditure function $e(p_1,p_2,\bar u)$, which gives the minimum expenditure needed to achieve utility level $\bar u$ at given prices.

Hicksian Demand

Apply an envelope theorem to find the Hicksian (compensated) demand functions $h_1(p_1,p_2,\bar u)$ and $h_2(p_1,p_2,\bar u)$.

Interpreting the Multiplier

Show that when the agent is optimizing, the derivative of the indirect utility function with respect to income equals the Lagrange multiplier from part 1.