ProblemSet_Production2_Solutions

ECO204Y1 Y LEC0301/0401 Tutorial (2023-24) Production II Exercise 1 Your convenience store has the following annual revenues and costs: Revenues 450,000 Supplies 25,000 Cost of goods sold 200,000 Electricity 6,000 Employee salaries 75,000 Your salary 80,000 You own the building, and can always close down the shop and rent the space 100,000 per year. You have a job offer to manage a local supermarket at a salary of 95,000 per year and a nearby restaurant at 65,000 per year. You can only work one job. What are your accounting costs? What are your economic costs? Should you shut down your shop? Solution: Revenues: 450,000 Accounting cost = Supplies + Cost of goods sold + Electricity + Employee salaries: 386,000 Your accounting profit = Revenues-Accounting Cost +Your salary: (64,000 + 80,000) Economic cost (includes (100,000 + 95000) opportunity cost): 581,000 Your economic profit = Revenues-Economic Cost: - 51,000 Yes, you should shut down your shop. Exercise 2 Your cost function is C ( Q ) = Q 3 − 20 Q 2 + 170 Q + 576 (a) What is MC ( Q )? What is ATC ( Q )?

ECO204Y1 Y LEC0301/0401 Tutorial (2023-24) Production II Solution: ∂C ( Q ) ∂Q = 3 Q 2 − 40 Q + 170 C ( Q ) Q = Q 2 − 20 Q + 170 + 576 Q (b) For what level of output is AV C ( Q ) minimal? Solution: AVC is minimized at the Q at which MC and AVC intersect. Q 2 − 20 Q + 170 = 3 Q 2 − 40 Q + 170 ⇐⇒ Q = 10 (c) You expect to sell your product at a price of 70. Assume that your fixed cost is sunk, should you produce the good? Would your answer be different if the fixed cost was not sunk? Solution: If you decide to produce, MC ( Q ∗ ) = 70 ⇐⇒ Q ∗ = 10. C (10) = 1276 70 · 10 − 1276 = − 576 If fixed cost is sunk, produce. Firm wants to minimize loss since it has to pay the fixed cost anyway, hence it chooses to produce. If it is not sunk, profit is zero and firm is indifferent between producing and not producing. Exercise 3 Assume the production function for home exercise ski machines is F ( K, L ) = K 2 3 + L 2 3 . (a) Determine whether the F ( K, L ) exhibits increasing, constant or decreasing returns to scale. 2

ECO204Y1 Y LEC0301/0401 Tutorial (2023-24) Production II K ( W, R, L ) = L W R 3 Production is given by the function: K 2 3 + L 2 3 = ¯ Q L W R 3 ! 2 3 + L 2 3 = ¯ Q Isolating for L , we get L ( ¯ Q, R, W ) = ¯ Q 3 2 R 2 W 2 + R 2 3 2 K ( ¯ Q, R, W ) = ¯ Q 3 2 R 2 W 2 + R 2 3 2 W R 3 C ( Q, R, W ) = WL + RK C ( Q, R, W ) = ¯ Q 3 2 R 2 W 2 + R 2 3 2 W 3 R 2 + W Plugging in the values, we get C (2600 , 5 , 25) =650000 C (2600 , 5 , 25) 2600 =250 (d) What is the long-run marginal cost of production at Q = 2 , 600? Solution: ∂C ( Q, R, W ) ∂ ¯ Q = − 3 2 ¯ Q − 5 2 R 2 W 2 + R 2 3 2 W 3 R 2 + W ∂C (2600 , 5 , 25) ∂ ¯ Q = − 375 4

ECO204Y1 Y LEC0301/0401 Tutorial (2023-24) Production II (e) Suppose that each ski machine now requires 100 units of steel, with P steel = 1$. How does the average and marginal cost of producing 2,600 change? Solution: Steel cannot be substituted with ℓ or k . C ′ (2600 , 5 , 25) 2600 = 350 ∂C ′ (2600 , 5 , 25) ∂Q = 475 Marginal cost and average cost are 100 higher with the additional requirement of steel. (f) Assume capital is fixed at ¯ K = 125 , 000, the quantity that minimizes costs when producing 2,600 ski machines when the wage is 25 and the price of capital is 5. What is the short-run cost of producing 3,146 units? Solution: Q ( ¯ K, L ) = ¯ K 2 3 + L 2 3 ⇐⇒ L ( Q, ¯ K ) = ¯ Q − ¯ K 2 3 3 2 C ( Q, ¯ K, L ) = R ¯ K + W Q − ¯ K 2 3 3 2 C (3146 , 5 , 25) =1035476 . 96 (g) What is the long-run cost of producing 3,146 units? Solution: Use the long-run cost equation from part c: C (3146 , 5 , 25) = 865150 5

ECO204Y1 Y LEC0301/0401 Tutorial (2023-24) Production II (c) Suppose that P = $6. For which value of F are the firm’s profits in (a) and (b) identical? Solution: π ( Q ∗ ) = π ( Q min ) 36 4 − F =6 √ F − 2 F F − 6 √ F + 9 =0 √ F = 6 ± √ 36 − 4 · 9 2 =3 F =9 Q min = Q ∗ =3 π (3) = 6 · 3 − (9 + 9) = 0 If F = 9, the firm makes zero profit. Why? MC ( Q ) = P and MC ( Q ) = ATC ( Q ) = ⇒ P = ATC ( Q ) Recall from the lecture that π ( Q ) = ( P − ATC ( Q )) Q, which is 0 if P is equal to ATC . Old Exam Question Given your level of capital, total product of labor ℓ given by Q ( L ) = 100 L +6 L 2 − L 3 . A unit of ℓ costs W . At what quantity of output does marginal cost switch from decreasing to increasing? A. The marginal cost is always increasing. B. Q = 105. C. Q = 216. D. Q = 327. 7

ECO204Y1 Y LEC0301/0401 Tutorial (2023-24) Production II E. Q = 432. Solution: When the second derivative is equal to zero, the marginal cost is at the in- flection point, i.e. switching from decreasing to increasing. ∂Q ( L ) ∂L =100 + 12 L − 3 L (marginal product of ℓ ) ∂ 2 Q ( L ) ∂L 2 =12 − 6 L ≥ 0 if L ≤ 2 Q (2) = 216 8