# 1. True or False. If the marginal product of labor is decreasing, then the average product of labor must also be decreasing. Explain.

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1 ECO 220 Intermediate Microeconomics Professor Mike Rizzo Second COLLECTED Problem Set SOLUTIONS This is an assignment that WILL be collected and graded. Please feel free to talk about the assignment with your friends or with your group and I strongly encourage you to submit your assignment as a group. Assigned: Due: Monday, April 4 th Monday, April 11 th 1. True or False. If the marginal product of labor is decreasing, then the average product of labor must also be decreasing. Explain. This is false. The marginal product of labor tells us how much output CHANGES when I add one more worker. We know that if the change in output is greater than the average level of output, then the new average must be increasing and vice versa. It is certainly possible that the marginal product can be decreasing, but still have a value that is greater than the average value. MP, AP AP L 0 MP L 0 L 1 L 2 Workers (L) 2. Nicholson, Chapter 6, Problem 6.7. a) In the short run, firms can only adjust variable inputs. Recall that to minimize costs across two different plants at the same firm that the extra output produced from the last dollar spent on labor should be the same at all plants. Recall that this condition is expressed as MPL / w. Therefore, to minimize costs, the entrepreneur should choose Q such that: MPL1/w1 = MPL2 / w2 Since the wage rates are the same for both firms, this results in setting the marginal product of labor equal at both sites. MP L1 = 5/(2L 1^0.5) and MP L2 = 5/L 2^0.5. Setting these equal and solving for L will yield L 2 = 4L 1. Each firm has a similar production function, but firm 1 uses 25 units of K while firm 2 uses 100 units of K. Simply plug in the amount of labor into each to find out how much each firm should produce. 1

2 q 1 = (25 x L 1 )^0.5 and q 2 = (100 x L 2 )^0.5 q 1 = 5 L 1 and q 2 = 10 4L 1 = 20 L 1 Therefore, q 2 = 4q 1. For every unit of output produced at plant 1, 4 units should be produced at plant 2. b) STC(plant1) = fixed costs + variable costs = rk 1 + wl 1 = (\$1 x 25) + (\$1 x L 1 ). Note that the cost functions we learned about in class tell us how much it costs to produce a given amount of output and note that from part a above that L 1 = q 1^2/25. So STC(plant1) = 25+ q 1^2/25. Similarly for plant 2, STC = q 2^2/100. The STC for the entire firm is simply the sum of these two: STC = STC 1 + STC 2. Also note that total output produced by the firm is q = q 1 + q 2. In this case, q = q1 + 4q1 = 5q1. Therefore q1 = 0.2q. Similarly, q2 = 0.8q. Substitution yields: STC = (q/5)^2 / 25 + (4q/5)^2 / 100 STC = q 2 /125 ATC = STC / q = 125/q + q/125 MC = d(stc)/d(q) = 2q/125 MC(100) = \$1.60; MC(125) = \$2.00; MC(200) = \$3.20. c) In the long run because of constant returns to scale, can change K, so it doesn t really matter where production occurs. One could easily split output across each plant, or could move entire production process to one plant. LTC = rk + wl = K + L = 2q (take a look at production function to see this) LAC = LTC / q = 2 LMC = d(ltc)/d(q) = 2 d) If there were diminishing returns to scale, then both firms should have equal shares of production. Since average costs and marginal costs are not constant (would be increasing functions of output), you would not want either plant to grow to be too large. 3. Nicholson, Chapter 6, Problem

3 a) Since w/r = \$10/\$10 = 1, the expansion path would be unchanged. All costs would now be twice what they were before: TC = 2q; AC=MC = 2. b) If wages rise to \$20 and rents stay at \$5, w/r = 4 and the firm will operate on a new expansion path. Since cost minimization requires that the slope of the isoquant equals the slope of the isocost line (i.e. RTS = w/r), then: The production function is q = 10 KL and the RTS = (dq/dl) / (dq/dk) = K/L in this case. So cost minimization requires that K/L = w/r = 4. K = 4L. Therefore q = 10 KL = 20L = 5K TC = 5K + 20L = q + q = 2q. Hence, AC = MC = 2. Multiplication of the wage by 4 only doubles costs because the firm will substitute capital for labor in the long run. 4. Nicholson, Chapter 7, Internet Exercise #2. Without an incentive to economize on usage, congestion can become quite serious. Indeed, the problem is more serious for data networks than for many other congestible resources because of the tremendously wide range of usage rates. On a highway, for example, at a given moment a single user is more or less limited to either putting zero or one cars on the road. In a data network, however, single user at a modern workstation can send a few bytes of or put a load of hundreds of Mbps on the network. Within a year any undergraduate with a new Macintosh will be able to plug in a video camera and transmit live videos home to mom, demanding as much as 1 Mbps. Since the maximum throughput on current backbones is only 45 Mbps, it is clear that even a few users with relatively inexpensive equipment could bring the network to its knees. They suggest a pricing scheme based on users revealing their preferences for type of usage. Cocchi, Estrin, Shenker, and Zhang (1992) point out that it is useful to look at network pricing as mechanism design problem. The user can indicate the #type# of his transmission, and the workstation in turn reports this type to the network. In order to ensure truthful revelation of preferences, the reporting and billing mechanism must be incentive compatible. The field of mechanism design has been criticized for ignoring bounded rationality of human subjects. However, in this context, the workstation is doing most of the computation, so that quite complex mechanisms may be feasible. 5. A firm purchases capital and labor in competitive markets at prices of r = 6 and w = 4, respectively. With the firm s current input mix, the marginal product of capital is 3

4 12 and the marginal product of labor is 18. Is this firm minimizing its costs? If so, explain how you know. If not, explain what the firm ought to do. We know that a necessary condition for cost minimization is that the additional output gained from the last dollar spent on labor must be equal to the additional output gained from the last dollar spent on capital. In other words, MP L /w = MP K /r. In this example, MP L /w = 4.5 and MP K /r = 2.0, the firm CANNOT be minimizing its costs. This is the case because the firm gained 4.5 units of output from the last dollar spent on a worker and only 2 units of output from the last dollar spent on a machine. Therefore, the firm can buy less capital and more labor in fact it can produce the same quantity at a lower cost by doing this. 6. True or false. If marginal cost lies below average fixed cost, the firm should shut down in the short run. Explain. FALSE. The firm should shut down if and only if its price is below average variable costs. Marginal costs can lie below average fixed costs at the same time price lies above average variable cost. 7. Suppose all firms in a competitive industry are operating at output levels for which price is equal to long-run marginal cost. True or false and explain. This industry is necessarily in long-run equilibrium. FALSE. Firms will produce where the price equals marginal costs. If all firms have u- shaped cost curves, it is possible that firms are operating at points where the marginal cost curve (and price) exceeds the average cost curve and firms therefore are making economic profits. Entry will occur in this industry until the price falls down the long-run average total cost, and we know that at this point firms make zero economic profit and price will equal long run marginal cost at the minimum point on the AC curve. 8. The marginal and average cost curves of taxis in Metropolis are constant at \$0.20 per mile. The demand curve for taxi-trips in Metropolis is given by P = Q, where P is the fare in dollars per mile, and Q is measured in miles per year. If the industry is perfectly competitive and each cab can provide exactly 10,000 miles per year of service, how many cabs will there be in equilibrium and what will be the equilibrium fare? The taxi industry supply curve must be a horizontal line at the point where MC=AC = \$0.20 per mile and we know that long-run competitive equilibrium occurs when prices equal the break-even point for firms minimum average total costs. Therefore the price must be \$0.20. You can draw the picture or solve the demand curve for Q when the price = \$0.20 to find out the total number of miles driven in equilibrium in this market. Rearranging the demand curve gives an expression for Q = 1-P; Q = 100,000(1-P); Q = 100,000(0.8) = 80,000 miles per year. If in equilibrium 80,000 miles per year are driven and each taxi can supply a maximum of 10,000 per year, then we should expect to see 8 taxis serving the city. 4

6 include foreign producers into our calculation. It seems arbitrary to me that we do not consider the well being of individuals across a somewhat arbitrary boundary line. 12. True, False and Explain. If an excise tax is imposed on a factor of production in perfectly inelastic supply (e.g. land in Wyoming), there will be no effect on social welfare. What will such a tax do? Because land is in fixed supply (for argument s sake), placing a tax on it will not result in any decreased output. Therefore, an excise tax on land will fall entirely on the owners of the fixed factor of production and will result in a transfer of surplus from the owners of the land to the recipients of the taxes. One fly in the ointment that taxing land entails no loss in social welfare is that land is not equally valuable in all uses. When landowners garner rents, they let their land to the people that will pay the most for it. If land is not allocated precisely to those people that value it the most, there is a diminution of social welfare. Imposing a tax on landowners is akin to telling them to allow designated individuals to use their land at a lower price (note that the supply and demand curves are unchanged, so it is possible that land will be rented to folks that value it less than other folks did before the tax). 6

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