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ADEC3510 Mathematics for Economists

Final Exam Fall 2022

1. A consumer’s utility function is given by U (x, y) = (x − 2)(y − 2)(x + y − 2) where x and y are the quantities of two goods consumed. Find the bundle of these goods that maximize this consumer’s utility. Prove it is a maximum

2. Suppose a consumer’s utility function U(x, y) = xy. If 2x + y ≤ 2, then find the coordinates that maximize U(x, y) using Kuhn Tucker.

3. A firm’s Cobb Douglas production function is given by Q = Q(L, K) = L1/2 K1/2 where L is the quantity of labor employed and K is the quantity of capital employed in the production. Suppose the price of labor is 10 dollars and the price of capital is 5 dollars. If the producer’s costs are constrained to exactly 200 dollars, what is the maximum production level of this firm? Prove that it is a maximum.

4. A firm produces two products, the respective quantities of which are q1 and q2 . The firm has agreed to produce a total of 100 units of both products for a client. The total cost TC for the firm is TC(q1, q2) = 2q1 + q1q2 + 4q2 with demand functions p1 = 20 − q1 + q2 and p2 = 30 + 2q1 − q2 . Compute the firm’s maximum profit, where profit is (q1, q2 ) = p1q1 + p2q2 − TC(q1, q2). Use the lagrangian method. Prove that it is a maximum.

5. Let P1(Q) and P2(Q) be demand and supply curves, respectively, where P1(Q) = 100e-0.03Q and P2(Q) = 20e.05Q and where price P is in dollars for a quantity Q. Suppose a quantity regulation is set at 10 units.

A. What is the deadweight loss associated with this quantity regulation?

B. How much consumer surplus is lost from the quantity regulation?

C. If no quantity regulation was instituted, then what is the total gain from the trade as a function of the consumer and producer surpluses?

6. A landscaping company produces two types of mulch by the amount x and y pounds. Suppose its variable costs are given by the function VC(x, y) = 4x2 +3xy + 2y2 . The firm also incurs fixed costs in the amount of $75. The production capacity for both types of mulch together is equal to 30 pounds. Find the minimum total cost, where total cost is the sum of the variable costs and the fixed cost. Use the lagrangian method. Be sure to prove it is a minimum.

7. Suppose the supply function for a certain item is given by p= S(q)= q + 2. Suppose the rate of change in the demand function for this item is observed to be = − . Compute the consumer and producer surpluses when price p and quantity q are at equilibrium.

8. Suppose an economy is composed of a single consumer and one productive sector producing commodity Q. Let the consumer’s demand function be Q(P) = , where a > 0. Compute the consumer surplus up to the equilibrium quantity denoted by q*.