Problem Set 5: Inequality-Constrained Maximization
1. Suppose an agent has the following objective function:
u(x1 , x2 ) =
√
x1 +
√
x2
and faces a linear constraint
1 = px1 + x2
and non-negativity constraints x1 ≥ 0, x2 ≥ 0.
i. Verify the objective function is concave in (x1 , x2 ).
ii. Graph the budget set for various prices p, and some of the upper contour sets of the objective
function.
iii. Solve the agent’s optimization problem for all p > 0. Check the second-order conditions are
satisfied at the optimum. Explain how you know, as a consequence of i, that the solution will be
characterized by the FONCs.
iv. Sketch the set of maximizers as a function of p, x1 (p).√
√
v. Solve the agent’s problem when u(x1 , x2 ) = x1 + x2 + 1 for all p > 0. Is it possible to
get corner solutions in this case? Explain the mathematical difference between the old objective
function and the new one.
2. Suppose a firm hires capital K and labor L to produce output using technology q = F (K, L),
with ∇F (K, L) ≥ 0. The price of capital is r and the price of labor is w. Solve the cost minimization
problem
min rK + wL
K,L
subject to F (K, L) ≥ q̄, K ≥ 0 and L ≥ 0.
i. Under what additional conditions is an interior critical point a local maximizer?
ii. How does (K ∗ , L∗ ) vary in r and q̄ at each solution?
iii. How does the firm’s cost function vary in q̄?
iv. Use the complementary slackness conditions to explain clearly when the firm selects a corner
solution, hiring only capital or only labor.
3. Consider a household that maximizes its utility u(x), x ∈ RN
+ , subject to the constraints
that p · x ≤ w and x ≥ 0. Assume u(x) is twice-differentiable.
i. Does a solution to the maximization problem exist?
ii. Suppose ∇u(x) 0. Explain clearly why this implies p · x = w. Assume ∇u(x) 0 for the rest
of the question.
iii. Suppose u(x) is weakly quasi-concave. Prove that this implies the set of maximizers is convex.
If u(x) is strictly quasi-concave, how many solutions are there?
iv. When is the optimal x∗i > 0? x∗i = 0? Use the complementary slackness conditions to clearly
explain.
v. Derive conditions that ensure a unique interior solution, so that x∗ 0.
4. Suppose an agent gets utility from consumption, c ≥ 0, and leisure, ` ≥ 0. He has one unit
of time, which can also be spent working on hours working h, which satisfies 1 ≥ h ≥ 0. From
working, he gets a wage of w per hour, so his total income is I = wh. His utility function is
u(c, `),
which satisfies ∇u(c, `) 0. Normalize the price of consumption to 1.
i. Characterize the optimal choice of consumption and leisure by providing first-order necessary
and complementary slackness conditions. What conditions ensure that he consumes a strictly
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positive quantity of consumption, but not necessarily a strictly positive quantity of leisure?
ii. Suppose now that he faces taxes, which take the form
0
, I < I0
t(I) =
τ (I − I0 ) , I ≥ I0
so that that up to I0 income, he is untaxed, but after I0 he is linearly taxed at rate 0 < τ < 1.
Sketch the agent’s budget set. Is it a convex set? Show that the budget set is equivalent to the
two inequality constraints
c ≤ w(1 − `)
c ≤ I0 + (wh − I0 )(1 − τ )
Characterize the optimal choice of consumption and leisure by providing first-order necessary and
complementary slackness conditions. How would you carry out the second-order test?
iii. How does the optimal choice of c vary with τ ?
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