Intertemporal Allocation of a Depletable Resource: Optimization Using the Kuhn-Tucker Conditions

Mathematical Economics Problems and Solutions - Section 3

See Mr. Stolyarov's complete list of Mathematical Economics Problems and Solutions.

Note: Here, I will present solve problems typical of those offered in a mathematical economics or advanced microeconomics course. The problems were authored by Dr. Charles N. Steele and are reprinted with his generous permission. The solutions to the problems are my own work and not necessarily the only way to solve the problems.

9. Intertemporal allocation of a depletable resource: suppose the demand for a resource in year t is P_t = a - bq_t, where t indexes the year, P is price, q is quantity demanded, and a and b are positive constants. Suppose also that the total cost of producing the resource in year t is TC(q_t) = cq_t, where c is a positive constant. Suppose also that the total amount of the resource is Q. The real interest rate is r. Assume that the objective is to maximize present value of net benefits to consumers consuming this resource (i.e., not a monopoly problem).

a. Set up the Lagrangian for this problem, assuming t = (0, 1, 2, ..., T)

b. Show what the Kuhn-Tucker FOC are for this problem.

c. Why are the Kuhn-Tucker conditions relevant, rather than equality constraints?

d. Solve this problem for a = 8, b = 0.4, c = 2, Q = 20, r = 0.05, and T = 2 (i.e., three period model.

e. The problem in (d) is a dynamic problem. What is meant by "dynamic?" How much would be consumed each period if Q = 100? What would be the value of relaxing the constraint Q = 20 in each period?

Solution 9a. For a time period t, net benefit to consumers can be expressed as

Total benefits - TC(q_t) = ₀^qt∫(a - bq)dq - cq_t = aq_t - (b/2)q_t²- cq_t

Lagrangian: L = _t=0^TΣ[(aq_t - (b/2)q_t²- cq_t)[1/(1+r)^t]] + λ[Q - _t=0^TΣq_t]

Solution 9b. FOC:
L_q0 = (a - bq₀ - c) - λ ≡ 0

L_q1 = (a - bq₁ - c)(1/(1+r)) - λ ≡ 0

L_q2 = (a - bq₂ - c)(1/(1+r)²) - λ ≡ 0

...

L_qT = (a - bq_T - c)(1/(1+r)^T) - λ ≡ 0

L_λ = Q - _t=0^TΣq_t ≥ 0, λ ≥ 0.