Test-2 (Solutions)

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Page 1 of 18 MGEB02: Price Theory Summer 2023 Test-2 (Solutions) Instructor A. Mazaheri Instructions: This is a closed book examination. You are permitted to bring a non-programmable calculator. Show all your work otherwise you will not get full credit . Make sure you allocate time appropriately . You have 2 hours. Good Luck! Last Name: First Name: ID FOR MARKERS ONLY: The University of Toronto's Code of Behaviour on Academic Matters applies to all University of Toronto Scarborough students. The Code prohibits all forms of academic dishonesty including, but not limited to, cheating, plagiarism, and the use of unauthorized aids. Students violating the Code may be subject to penalties up to and including suspension or expulsion from the University. Q1 Q2 Q3 Q4 Q5 Total Marks Earned Maximum Marks Possible 45 20 15 10 10 100

Page 2 of 18 Answer all following 5 questions: Question-1 [45 Points] Answer the following Short Questions: a) [5 Points] Based on the following figure comment on returns to scale and marginal return of labor. Make sure to explain your answer: Solution: Returns to scale: Increasing Returns to Scale: 2,2 produces 75 while 3,4 produces 150 so 4.4 should produce more than 150. Marginal return of Labor: MPL is diminishing because if we keep capital constant (say at 1). Initial MPL is approximately ( 𝜟𝒒 𝜟? = ???−?? ?−? = ?? ) falling to ( 𝜟𝒒 𝜟? = ???−??? ?−? = ?? ) Alternatively if we keep K=2 and we add to the labour, the slope of the isoquant declines implying that as more labour is employed, more labour is needed to substitute for the same amount of capital => therefore the MPL is declining. Q = 75 Q2 = 120 Q3 = 150 2 3 4 4 2

Page 3 of 18 b) [5 Points] A firmʹs total cost function is given by the equation: TC = 4500 + 80Q + 20Q 2 . Find this firm output elasticity of cost and use it to identify its returns to scale. Solution: 𝐴? = 2400 ? + 80 + 20? ?? = 80 + 40? ? 𝑐 = ?? 𝐴? = 80 + 40? 4500 ? + 80 + 20? ? 𝑐 = 1 => 20? = 4500 ? => ? = 15(???) ? > 15, ? ? > 1(???) ? < 15, ? 𝑐 < 1(𝐼??)

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Page 4 of 18 c) (6 Points) Suppose we have the following production function q = min {6L, 30K}. Prices for K and L are 30 and 6 respectively. First find and graph the equation for the long run cost function. Then find and graph the equation for the short run cost function for a fixed capital of 10. Solution: Long Run (5 Points): => 6? = ? => ? = ? 6 => 30? = ? => ? = ? 30 ???? = ? + ? = 2? Short Run (3 Points): If K = 10, the only efficient production is Q = 30K = 300, which will need Q = 300 = 6L => L = 50 labor. This efficient production will cost (TC = 6*50 + 30*10 = 600). There will be no other efficient production so cost function will not be defined. LRTC q

Page 5 of 18 d) [5 Points] The short run production curve for a firm is shown in the following figure. On the lower diagram, graphically derive the AP L and the MP L . L q q

Page 6 of 18 e) [6 Points] In a perfectly competitive market, there are 100 firm split equally between the following short run cost functions: ? 1 (? 1 ) = 24q 1 2 + 20q 1 + 2500 ? 2 (? 2 ) = 8q 2 2 + 80q 2 + 3000 Find the total short run market supply curve and graph it. Solution: ? 1 (? 1 ) = 24q 1 2 + 20q 1 + 2500 SRMC = 48q 1 + 20, ??𝐴?? = 24? 1 + 20 ???? = ??𝐴??(? 1 = 0, ? = 20 − shoutdown point) ? = 48q 1 + 20(if P ≥ 20) ? 2 (? 2 ) = 8q 2 2 + 80q 2 + 3000 SRMC = 16? 2 + 80, ??𝐴?? = 8? 2 + 80 ???? = ??𝐴??(? 2 = 0, ? = 80 − shoutdown point) ? = 16? 2 + 80(if P ≥ 80) Market supply: ? 1 = 𝑃 48 − 20 48 (𝑖?? ≥ 20) ? 2 = ? 16 − 80 16 (𝑖??(≥ 80) ? = 0 (? < 20) ? = 50( ? 48 − 20 48 ) = 1.04P-3.125 (20 < ? < 80) ? = 50( ? 48 − 20 48 ) + 50( ? 16 − 80 16 ) = 4.17P-270.8 (? ≥ 80) 20 80

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Page 7 of 18 f) [6 Points] The long-run cost function of a firm in a perfectly competitive market is given by C(q) = 1000q-36q 2 +0.4q 3 , where q is firm output. Market demand is given by Q d = 10,000-2.9P Find the long-run equilibrium values of output per firm, price, & the number of firms in the market. Use a graph to demonstrate your solution. Solution: In long run equilibrium assuming identical firms we have: 1) P = MC 2) AC = MC 3) Qd = Qs = nq P = MC = 1000 – 72Q + 1.2q 2 AC = 1000 – 36Q + 0.4q 2 1), 2) => AC = MC => q = 45 => p = 190 3) Qd = 10,000-2.9*45 = 9449 = n*45 => n = 210 Graph (2 Points)

Page 8 of 18 g) [6 Points] A production is characterized by q = (L 1/2 + K 1/2 ) 2 . Suppose Let w = 8 and r = 10. Suppose in the short run capital is fixed at 49 units. What would be producer surplus if the market price is 4? Solution: L = [q 0.5 - 7] 2 => 𝑻𝑪 = ?[𝒒 ?.? − ?] ? + ??? 𝑷 = ?𝑪 = ? [? − ? 𝒒 ?.? ] 𝑽𝑪 = ?[𝒒 ?.? − ?] ? We know that in the short run fixed cost is sunk therefore: P = 4 > q = 196 VC = 392 PS = TR - VC = 4×196 – 392 = 392

Page 9 of 18 h) (6 Points) In the long-run equilibrium of a competitive market, the market supply and demand are: Supply: P = 200 + 2Q Demand: P = 2000 – 8Q, where P is dollars per unit and Q is rate of production and sales in hundreds of units per day. One firm in this market has a marginal cost of production expressed as: MC = 200 + 30q. Determine the economic rent that the typical firm enjoys and then briefly explain what Economic rent means? Solution: Supply = Demand 200 + 2Q = 2000 – 8Q Q = 180 (hundred per day) P = 200 + 2(180) = $560 / unit MC = 560 = 200 +30q q = 12 (hundred per day) PS = Economic Rent = (560-200)*12/2 = 2160 Economic rent: return to firm specific advantage (in labour, capital, location, etc)

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Page 10 of 18 Question-2 [20 Points] A consumer has the following utility function U=ln(8I) where I is the income. She is currently employed with an annual wage of $100,000. However, she might be fired and if so then she will have to find a second job that pays only $40,000 a year. The probability of him being fired is 40%. (p = 0.40) Note: Approximate to three digits. a) [8 Points] Depict the utility of this consumer on the following diagram. Calculate the expected utility for this consumer and identify it on the graph. Demonstrate that her utility of expected is higher than her expected utility . ?(𝐼) = 0.6 × 100,00 + 0.4 × 40,000 = 76,000 ?(?(𝐼)) = ln(8 × 76,000) = 13.318 ?(?) = 0.6 × ln(8 × 100,000) + 0.4 × ln (8 × 40,000) = 13.226 I U 100,000 40000 13.676 13.592 13.226 13.318 76000

Page 11 of 18 b) [4 Points] An insurance company is offering full insurance in the case of him being fired. What is the maximum that the consumer will pay for the insurance? Solution: ?(?) = 13.226 = ln(8 × ??) ?? = ? 13.226 8 = 69,314.5 Since this is a full insurance E(I) = 100,000, therefore: ?? = 100,000 − 69,314.5 = 30,685.5 Which is the maximum that the individual will pay for full insurance. c) [4 Points] Now suppose that the insurance company is offering a partial insurance of $15,000 (in the case she is fired he will receive a payment of $15,000 from the insurance company in addition to what she earns) for a premium of $4,000. Would the consumer insure herself? Solution: With the insurance the expected Utility is: ?(?) = 0.6 × ln(8 × (100,000 − 4000)) + 0.4 × ln(8 × (40,000 + 15,000 − 4000)) = 13.299 Which is higher than without => Will take the insurance.

Page 12 of 18 d) [4 Points] So far we have assumed that the probability of losing the job is 40%. Now suppose that is not necessarily the case. What would be the maximum probability such that she is indifferent between insuring and not insuring herself if she is offered the insurance as described in part (c). Solution: ?(?) 𝑤𝑖?ℎ = (1 − ?) × ln(8 × (100,000 − 4000)) + ? × ln(8 × (40,000 + 15,000 − 4000)) = 13.552 − 1.326? ?(?) 𝑤𝑖?ℎ = (1 − ?) × ln(8 × (100,000)) + ? × ln(8 × (40,000)) = 13.592 − 1.609? ?(?) 𝑤𝑖?ℎ = ?(?) 𝑤𝑖?ℎ??? 13.552 − 1.326? = 13.592 − 1.609? ? = 0.1439

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Page 13 of 18 Question-3 [15 Points] Suppose that the production function for chairs is given by: ? = 4?? 2 − 3? 3 where q is the number of chairs produced per year, K is the machine-hours of capital, and L is the man- hours of labor. a) [5 Points] Suppose that K = 1,500: Find average productivity of labor (AP L ) and the marginal productivity of labor (MP L ). At what level of labor does AP L reach its maximum? What is the level of output at this point? Solution: 6000 4 6 9 8 3 4 9 8 3 4 2 2 2 2 = = − = − = − = − = K L L L K L L K MP AP L L K MP L L K AP L L L L AP L is maximized when L = 1,000 L = 1,000, K = 1,500 => q = 3,000,000,000 b) [5 Points] Continuing to assume that K = 1,500: graph the AP L and MP L curves on the space below. Make sure to identify the points at which MPL = 0, the point where the output is maximized and the points where AP L and MP L are equal? ?? 𝐿 = 8? ̅ ? − 9? 2 = 0 ? = 0, ? = 1333.3 1000 1333.3

Page 14 of 18 c) [5 Points] Find MRTS. Does MRTS exhibit standard diminishing marginal returns to inputs? Graph the isoquant of q = 20. (Note: Identify at least two points on it) Solution: L L K L L KL MP MP MRTS L KL q K L 4 9 8 4 9 8 3 4 2 2 3 2 − = − = = − = MRTS exhibits standard characteristics (i.e. declining) only if 8K-9L > 0, K> (9/8)L. Assume q =20, If L=1 then K=23/4 = 5.75, if L=2 then K= 2.75. L K q = 20 1 2 5.75 2.75 Inefficient Production (When K< (9/8) L, MPL<0 & MRTS<0)

Page 15 of 18 Question-4 [10 Points] A firm faces the following production function: q =20 (K-10) 0.75 (L) 0.25 a) [5 Points] Find the expression for short run total cost, short run variable cost, and short run marginal cost. Assume that the rental cost of capital is 5 and the wage is 1 and that in the short run, capital is fixed at 16. Identify when the average cost is minimized and graph the marginal cost and the average cost. b) [5 Points] Find the long run cost function. Draw the firm’s expansion path. Solution: a) ? ̄ = 16 ? = 20(? ̄ − 10) 0.75 (?) 0.25 => ? = ( ? 56.57 ) 4 = ? 4 10240000 ???? = 𝑤? + ?? ̄ = ? 4 10240000 + 80 ???? = ? 4 10240000 ???? = 4 10240000 ? 3 AVC = MC => ? 3 10240000 + 80 ? = 4 10240000 ? 3 => 3 102400000 ? 3 = 80 ? => ? 𝑆? = 128.56 => ? 𝑆? = 0.83

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Page 16 of 18 b) ???? = (? − 10) 3? = 𝑤 ? = 1 5 ? = 3? 5 + 10 ? = (? − 10) 0.75 ? 0.25 => ? = ( 3? 5 ) 0.75 (?) 0.25 = 0.68? ? = 1.47?, ? = 0.88? + 10 ?? ???𝑔?𝑒𝑟? = 𝑤? + ?? = 1.47? + 5(0.88? + 10) = 5.87? + 50 L K 10 Expansion Path K=3L/5+10

Page 17 of 18 Question-5 [13 Points]: Suppose you are given the following information about a particular industry: ? ? = 45,000 − 800? ? 𝑆 = 3,700? ?(?) = 1000 + ? 2 100 Where the two equations represent industry demand and supply and the next stands for a typical firm cost structure. Assume that all firms are identical, and that the market is characterized by perfect competition. a) (4 Points) Would you expect to see entry into or exit from the industry in the long run? Explain. What effect will entry or exit have on market equilibrium? b) (6 Points) What is the lowest price at which each firm would sell its output in the long run? Is profit positive, negative, or zero at this price? Explain. What is the lowest price at which each firm would sell its output in the short run? Is profit positive, negative, or zero at this price? Explain. Solution: a) ? ? = ? 𝑆 => ? = 10 ?? = ? = 10 = ? 50 => ? = 500 𝜋 = 500 × 10 − (1000 + 500 2 100 ) = 1500 Since profit is > 0 firms will have incentive to enter. As a result, the market quantity will increase and the market price will fall. b) The minimum price can be found by equating the marginal cost with the average cost: ?? = ? 50 , 𝐴? = 1000 ? + ? 100 => ? = 223.6 => ? = ?? = ? 50 = 4.47 The firm will not sell for any price below 4.47. The long-run equilibrium price is therefore $4.47, and at a price of $4.47 , each firm’s economic profit equals zero because P = AC. ?? = ? 50 , 𝐴?? = ? 100 ?? = 𝐴?? => ? = 0 => ? = 0

Page 18 of 18 The firm will sell for any positive price, because at any positive price, marginal cost is above average variable cost. Profit is negative if price is below minimum average cost, or as long as price is below $4.37. Profit is zero if price is exactly $4.47, and profit is positive if price is greater than $4.47.

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Test-2 (Solutions)

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