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Econ 101A – Problem Set 2
Due on Gradescope by 11:59pm, February 12th
Problem set must be uploaded and submitted on Gradescope by 11:59pm, February 12th.1 If, for any reason, you are unable to submit a problem set on Gradescope, please let one of the GSIs know before PS is actually due.
Show your work, write down the steps that you use to get a solution, write legibly (feel free to type your answers as well). If you cannot solve a problem fully, write down a partial solution. You are welcome to brainstorm with other students, but you must write a solution in your own words and hand in your own PS answers.
Problem 1. Univariate unconstrained maximization. Consider the following maximization problem: Let α > 1, and g(α) is a function of α
1. Write down the first order conditions for this problem with respect to x.
2. Solve explicitly for x
∗
that satisfies the first order conditions.
3. Compute the second order conditions. Is the stationary point that you found in point 2 a maximum?
Why (or why not)?
4. Is the function f concave in x?
5. As a comparative statics exercise, compute the change in x
∗ as α varies. In other words, compute dx∗/dα.
6. We are interested in how the value function f(x
∗
(α); α) varies as α varies. Use Envelope Theorem to compute df(x
∗
(α); α)/dα.
7. Now plug in your answer of x
∗
(α) into the objective function to obtain the value function, and derive it with respect to α. Does your answer coincide with the one from the previous item?
Problem 2. Multivariate unconstrained maximization. Consider the following maximization problem:
1. Write down the first order conditions for this problem with respect to x1 and x2.
2. Solve explicitly for x
∗
1 and x
∗
2
that satisfy the first order conditions.
3. Compute the second order conditions. Under what conditions for a1 and a2 is the stationary point that you found in point 2 a maximum?
4. Under what conditions on a1 and a2 is the function f concave in x1 and x2? When is it convex in x1 and x2?
5. Assume that the conditions for a1 and a2 that you found in item 3 are met. As a comparative statics exercise, compute the change in x
∗
2 as a2 varies. In other words, compute dx∗
2/da2. Compute it both directly using the solution that you obtained in point 2 and using implicit function theorem as seen in class. Compare results. Which method is faster?
6. We are interested in how the value function f(x
∗
1
(a1, a2), x∗
2
(a1, a2); a1, a2) varies as a2 varies. Use the envelope theorem to compute ∂f(x
∗
1
(a1, a2), x∗
2
(a1, a2); a1, a2)/∂a2.
Problem 3. Quasi-linear preferences. In economics, it is often convenient to write the utility function in a quasi-linear form. These utility functions usually look like the following:
u(x1, x2) = γ(x1) + x2
with γ
′
(x) > 0, and γ
′′(x) < 0. These preferences are called quasi-linear because the utility function is nonlinear in good 1 but linear in good 2. In this exercise we explore several convenient properties of quasi linear utility function. We will do so at first without assuming a particular functional form for γ(x).
Consider the following constrained optimization problem.
with p1 > 0, p2 > 0, I > 0.
1. Write down the Lagrangian function.
2. Write down the first order conditions for this problem with respect to x1, x2, and λ.
3. What do the first order conditions tell you about the value of λ? (Hint: Use the first order condition with respect to x2.) Does the value of λ depend on p1 or I? Why is this the case?
(Think of λ as the marginal utility of wealth)
4. Plug the value of λ into the first order condition for x1. You now have an equation that implicitly defines x
∗
1 as a function of the parameters p1, p2 and I. Does the optimal quantity x
∗
1 depend on income I? Is good 1 a normal good (∂x∗
1/∂I > 0), an inferior good (∂x∗
1/∂I < 0), or a neutral good (∂x∗
1/∂I = 0)? (If ∂x∗
i
/∂I = 0, then we say that good i is a neutral good - i.e. there is no income effect.)
5. Use the implicit function theorem to compute ∂x∗
1/∂p1.
6. Continue now under the assumption that u(x1, x2) = √
x1 +x2. Explicitely solve for x
∗
1 and then, using the budget constraint, solve for x
∗
2
.
7. Under what conditions for p1, p2, and I is x
∗
2 ≥ 0?
8. The indifference curves satisfy equation √
x1 + x2 = u or x2 = u −
√
x1. Draw a map of indifference curves in (x1, x2) space. What is special about these indifference curves? Compare them to the ones for Cobb-Douglas preferences.
9. Write down two budget lines: for (p1 = 1, p2 = 1, I = 1) and for (p1 = 1, p2 = 1, I = 2). Find, graphically, the optimal consumption bundles by tangency of the budget set and the indifference curve. You should find that x
∗
1
(1, 1, 1) = x
∗
1
(1, 1, 2); this means that there is no income effect in good 1 and the increase in income affects only the consumption of good 2. Interpret this result in light of your answer to point 4.
Problem 4. Addictive goods. In this exercise, we generalize Cobb-Douglas preferences to incorporate the concept of a reference point, a concept pioneered by behavioral economists. We can use such preferences to model the consumption of addictive goods. Consider the following utility function:
u(x1, x2; c) = (x1 − c)
αx2
β
with α + β = 1, 0 < α < 1, 0 < β < 1, and c > 0. Notice that the function is only defined for x1 ≥ c and x2 ≥ 0; assume that for x1 < c or x2 < 0 the utility is zero. The interpretation is that good 1 is an addictive good with addiction level c. (Examples of addictive goods are alcohol, drugs or... chocolate.) The more you
have consumed of good 1 in the past, the higher your addiction level c.
1. How does the utility function change as c changes? In other words, compute ∂u(x1, x2; c)/∂c. Why is this term negative? (Hint: If I have gotten used to drinking a lot of alcohol, then my utility from drinking three beers is...)
2. How does the marginal utility change as x1 changes? In other words, compute ∂u(x1, x2; c)/∂x1 for x1 >
c. How does this marginal utility change as c changes? In other words, compute ∂
2u(x1, x2; c)/∂c∂x1 for x1 > c. Why is this term positive? (Hint: If I have gotten used to drinking a lot of alcohol, then my utility from drinking one more beer is...)
3. Consider now the constrained optimization problem:
Write down the Lagrangian function.
4. Write down the first order conditions for this problem with respect to x1, x2, and λ.
5. Solve explicitly for x
∗
1 and x
∗
2 as a function of p1, p2, I, c, α, and β. You do not have to check the second-order conditions; take my word for it that they are satisfied as long as the condition in point 3 is satisfied.
6. What is the minimum level of income in order for the solution to make sense, i.e., so that x
∗
1 ≥ c and x
∗
2 ≥ 0?
7. Is good x1 a normal good for all values of I above the minimum level of income found in point 3?
8. How does x
∗
1 vary with c? Does your result make sense? Why do I consume more of good 1 if I am more addicted to it?
9. How does x
∗
2 vary with c? Does your result make sense? (Hint: Think of individuals who spend almost all their income on addictive goods.)
10. Use the envelope theorem to calculate ∂u(x
∗
1
(p1, p2, I; c), x∗
2
(p1, p2, I; c) ; c)/∂c and interpret the sign of this derivative.
