Professor Scholz Posted: 10/20/2009 Economics 101, Problem Set #7 Due: 10/27/2009 Costs, Adding Demand Curves and Perfect Competition Please complete work on sheet and SHOW your work! Problem 1 The following chart represents the production function and cost curves for a firm. a) Please fill in the open squares given the information provided, and answer the related questions below. Assume that labor is paid a constant wage, i.e. our firm is a price-taker in the labor market. Hint: use your definitions! L K Q MPL VC FC TC AVC AFC ATC MC 0 10 0 --- 0 36 36 --- --- --- --- 1 10 1 1 4 36 40 4 36 40 4 2 10 4 3 8 36 44 29 11 1.33 3 10 9 5 12 36 48 1.33 4 5.33 0.8 4 10 12 3 16 36 52 1.33 3 4.33 1.33 5 10 14 2 20 36 561.43 2.57 4 2 6 10 15 1 24 36 60 1.6 2.4 4 4 The following formulas can be used to calculate the necessary values for the chart. • MPL (Marginal Product of Labor) = (change in Q)/(change in L) In this case, labor is the only input that varies and we assume a constant wage. • VC (Variable Cost) = L*wage • FC (Fixed Cost) = TC – VC • AVC (Average Variable Cost) = VC/Q • AFC (Average Fixed Cost) = FC/Q • ATC (Average Total Cost) = TC/Q or AFC + AVC • MC (Marginal Cost) = (change in TC)/(change in Q)b) What is the relationship between ATC and MC, if ATC is decreasing as output increases? Specifically, can we tell if one is higher than the other? Why? When ATC is decreasing, MC is less than ATC. This is because the cost of producing each additional unit, represented by marginal cost, is less than the average cost of production for all the units already produced, represented by average total cost. So when the cost of producing the next unit is less than the average unit cost, then the average must fall when this next unit is produced. So ATC must decrease when MC is less than ATC. Problem 2 Suppose the production function is given by Q=3KL, where K denotes capital and L labor. The rental rate of capital is $2/hr, the wage is $24/hr, and capital is fixed at 4. a) Write down the total cost equation, in terms of capital and labor. TC = rK + wL (where r = rental price of capital and w = wage) TC = 2K + 24L or TC = 8 + 24L (since K = 4) b) Write down the total cost equation in terms of quantity. From the production function we know that Q = 3KL but that K is fixed at 4, so the production function simplifies to Q = 12L. This means that L = (1/12)Q Substituting this equation into the cost equation above gives us: TC = 8 + 2Q c) Graph the total cost (TC), Variable cost (VC) and Fixed cost (FC) for this production process. Put quantity on the X axis and dollars on the Y axis.Problem 3 Consider the following demand for Pizza by Jeff and Michael. Michael’s demand for pizza is P = 10 – 2Q, and Jeff’s demand for Pizza is P = 8 – Q. a) Draw each individual’s demand curve separately. b) Draw the market demand for Pizza. Problem 4 The market for study desks is characterized by perfect competition. Firms and consumers are price takers and in the long run there is free entry and exit of firms in this industry. All firms are identical in terms of their technological capabilities. Thus the cost function as given below for a representative firm can be assumed to be the cost function faced by each firm in the industry. The total cost and marginal cost functions for the representative firm are given by the following equations: TC = 2qs2 + 5qs + 50 MC = 4qs + 5 Suppose that the market demand is given by: PD = 1025 - 2QD Note: Q represents market values and q represents firm values. The two are different.a) Determine the equation for average total cost for the firm. ATC for the firm is TC/q, so dividing the total cost equation above by q gives us: ATC = 2qs + 5 + 50/qs b) What is the long-run equilibrium price in this market? (Hint: since the market supply is unknown at this point, it’s better not to think of trying to solve this problem using demand and supply equations. Instead you should think about this problem from the perspective for a firm. Specifically, a long run equilibrium occurs where ATC = MC = Price) In a long-run equilibrium, ATC equals Marginal Cost and profits equal zero. Setting the two equations equal: ATC = 2qs + 5 + 50/qs = 4qs + 5 = MC 50/qs = 2qs 50 = 2qs2 25 = qs2 Take the square root of both sides and find: 5 = qs However, the question wants us to find long run prices. We know that the firm produces were Price = MR = MC, so if we can determine the firm’s MC, then we can determine the equilibrium price in the market. We know that: MC = 4qs + 5 And solved for: 5 = qs Substituting: MC = 4(5) + 5 = 25 The equilibrium price in the market is 25. c) What is the long-run output of each representative firm in this industry? We solve for this in the previous part. 5 = qsd) When this industry is in long-run equilibrium, how many firms are in the industry? (Hint: firms are identically sized). Now we should determine the market quantity Q from the market demand curve, given that we know the market price is 25. Market demand is given as: PD = 1025 - 2QD And we know that market price = 25, so: 25 = 1025 - 2QD 1000 = 2QD 500 = QD Since each firm is making 5 units (as we found in parts b and c), there must be 100 firms, since they are all identically sized. Now suppose that the number of students increases such that the market demand curve for study desks shifts out and is given by, PD = 1525 - 2QD e) In the short-run will a representative firm in this industry earn negative economic profits, positive economic profits, or zero economic profits? (Hint: You can solve this without calculation.) The demand curve has shifted to the right. Given what we learned earlier in the semester, we should know that the market price will increase. If market prices are increasing, then firms are earning higher marginal revenues than they earn in a long-run equilibrium. This means that firms are earning positive economic profits. f) In the long-run will a representative firm in this industry earn negative economic profits, positive economic profits, or zero economic profits? (Hint: again, no calculation required) In the long-run economic profits are always zero since there is free entry/exit in a perfectly competitive market. Firms will either enter the industry until …
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