Problem Assignment #4 Date Due: 22 October 2013

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1 Problem Assignment #4 Date Due: 22 October Chapter 4 question 2. (a) Using a Cobb Douglas production function with three inputs instead of two, show that such a model predicts that the rate of payment to human capital must be higher in developing countries. Explain, why this is a problem by comparing the wages of administrators, doctors, lawyers, and educators etc., across developed and developing countries. Answer (Ray): Suppose that output Y is constant returns to scale function of three inputs: human capital (H), physical capital (K), and labor (L). Then I can write Y = AH α K β L 1 α β, where A denotes the level of technological productivity in the economy. Now write down the marginal products of each of the inputs (which will be their factor returns under perfect competition). For H, the marginal product is Y H = αahα 1 K β L 1 α β, which is just equal to α Y H. thus the return to human capital, w H is w H = α Y H. Likewise, the return to capita, r and the return to labor w are: r = β Y K and w = (1 α β) Y L. Now if the technological coefficient A is the same across countries and developing countries have lower values of r and w, then it must be the case that Y K and Y L are lower for developing countries. But then the value of Y H must be higher, because (using constant returns to scale) we can not have all three of these fractions lower for one country to another and yet maintain the assumption of common technology. It follows that w H is higher. Therefore, this sort of theory poses a problem, because it suggests that while we can explain low rates of return to physical capital and unskilled labor (in developing countries), we only seem to be able to do this at the cost of an explanation that suggests that highly skilled labor in developing countries get even more than their counterparts in developed countries, which is counterfactual. (b) Adapt the Cobb Douglass specification in part (a) to allow for differences in technology across developed and developing countries. Now it is possible to generate situations in which the return to every input is lower in developing countries. Which input is likely 1

2 to have the lower return (in a relative sense)? Answer (Ray): If A is lower in developing countries, then it is easy to see from the equations in part (a) that it is possible to write down an example in which the returns to all three factors are lower in developing countries. Moreover, the lowest relative return will be for unskilled labor (because it is this input which is the most abundant in developing countries and therefore most subject to diminishing returns). The return gaps for skilled labor and for physical capital will be lower (relative to that of unskilled labor). (c) Provide some arguments for systematic technological differences across developed versus developing countries. If technology blueprints are known, why can they not be instantly imitated? Answer (Ray): Some explanations for systematic technological differences: (i) R&D will have large setup costs that can only be redeemed in large markets, and so it is likely that rich countries will invest more in research. (ii) The outcomes of such technical research will be most applicable to countries that have lots of physical capital and skilled labor, so that countries which have more unskilled labor will not be as wellplaced to use the technology even if it were to diffuse instantly. (iii) Even if technology blueprints are known, the technology might require support services and maintenance that are not available in developing countries. (iv) The technology might be protected by patents. 2. Chapter 4 question 9. The economy of Wonderland produces magic portions using capital (K) and labor (L). Total output is growing at a rate of 5% per year. The rental rate per unit of capital is 0.1 bottles of magic portion. The physical capital output ratio is 3:1. The stocks of capital and population are growing at the rate of 3 and 2% respectively. Assume that everyone works. (a) If all output is paid in wages and rent, calculate the shares of capital and labor in national income. Answer: (General) If all output is paid in wages and rent implies that the production function exhibits constant returns to scale. By Euler s adding up theorem (which I believe you have discussed in discussion section) another way to write a CRS production function is Y = F K K + F P P where F K and F P are the marginal products of capital and labor, respectively. Now divide through by Y to yield K 1 = F k Y + F P P Y = σ K + σ P, where σ K is the income share of capital and σ P that of labor. We are told that the marginal product of capital is 0.1 bottles of magic portion, and K/Y = 3. Thus, 1 = (.1)3 + σ P Hence, the share of capital is.3 while that of labor equals.7. Specific answer using the Cobb Douglas: on page 92 of Development Economics Ray 2

3 reports for the Cobb Douglas the share of capital income is Y K K Y = α The first term is the MP of capital which is.1 while the capital to output ratio is 3 which also yields an estimate of the capital share of income equal to.3. (b) Estimate the rate of technical progress (or TFP growth) in Wonderland. Answer: Letting z denote the percent change of z the growth accounting equation is: We now solve for TFPG: Or TFPG = or 2.7%. ẙ =σ K K + σ P P + TFPG 0.05 =.3(.03) +.7(.02) + TFPG (c) Suppose that doubling the inputs results in 2.5 times the original quantity of magic portion. How does this affect your answer to part (b)? Answer: We are told that doubling the inputs increases output by 2.5 times. Hence now the production function exhibits increasing returns to scale. You can see this by noting doubling the inputs means increases output by 2.5 times or Y = F(K, P) = K γ P β Y = F(λK, λp) 2.5Y = F(2K, 2P) = 2 γ+β K γ P β = 2 γ+β Y ln(2.5) = (α + β) ln(2) γ + β = ln(2.5)/ln(2) = 1.32 Thus the sum of the exponents on capital and labor now must sum to more than 1 (as 2.5 exceeds 2). Since labor is a factor of production (versus human capital) I assume that labor retains its 70% share. This implies that γ = =.52. The true MP K = γ (K/Y) or.52(y/k). Whereas in the CRS technology the MP K = α(y/k) =.3(Y/K). Thus, with increasing returns to scale the MP K is higher than measured by the income share. Using the factor share and the usual growth accounting scheme we will under estimate the productivity of the factors, underestimate factor accumulation and overestimate the rate of technical progress. (d) Suppose the owners of capital possess patent rights over their inventions and the rental rate reflects the monopoly price on capital rentals. How does this affect the growth accounting approach used in the foregoing answers? Answer: I have to say I found the question confusing, because I am not sure exactly what s being held constant. We know that owners of capital will charge the monopoly rental rate which (because of the monopoly power) is higher than the competitive rental 3

4 rate. Assuming the (observed) rental rate equals the competitive rental will under estimate the capital stock. Let r be the rental price of capital in a competitive market. As used above, in competitive markets factors are paid the value of their marginal product r = MP K. However the monopolist will charge r = r + δ, where δ > 0 is the monopoly premium so r > r. Thus, if we assume the market is competitive equating the marginal product of capital to r implies too low of an estimate for K. Thus, in the growth accounting, the factor accumulation will be underestimated and again too much of the residual will be ascribed to technical progress (and thus will overestimate the rate of technical progress). (e) Suppose that only half the labor population is engaged in production at any date. How does this affect your answer? If this proportion is increasing over time, what would happen to your estimate? textbfanswer: Once again I am not sure if we assume everyone works but in actuality only one half do. If so we overestimate the stock of labor and under estimate the rate of technological growth. If market participation is increasing and it is unrecognized, then we will underestimate the accumulation of labor and over estimate productivity growth. The message is that because the rate of technical progress is the residual any errors estimating the accumulation of factors or factor shares results in an erroneous estimate of rate of technological growth. 3. Chapter 5, question 2. Complementaries arise in all sorts of situations. Here is a tax evasion problem. Suppose that each of N citizens in a country needs to pay a tax of T every year to the government. Each citizen may decide to pay or to evade the tax. If an evader is nabbed, the law of the country stipulates payment of a fine of amount F, where F > T. However, the government s vigilance is not perfect, because it has limited resources to detect evaders. Assume that out of all the people who evade taxes, the government has the capacity to catch only one, and this person is chosen randomly. Thus, if n people have decided to evade taxes, each has probability 1/n of being caught. In what follows, we assume that people simply calculate the expected losses from each strategy and choose the strategy with the lower expected loss. (a) If the number of evaders is m, show that the average (expected) loss to an evader is F/m. This is to be compared with the sure loss faced by someone who complies, which is T. Answer (Ray): If I am an evader, then I will be caught with probability 1/m where m is the total number of evaders. E.g., if m = 3, then there are three evaders and the chance of my getting caught is one out of three or 1/3. If I am not caught, then I pay nothing. But if I am caught, then I pay a fine of F. Thus my expected payout is 1/m times F, or simply F/m. As a potential evader, I will compare this loss with the sure payment of T (if I do not evade), and take the course of action that creates smaller losses. (b) Why is this situation like a coordination game? Describe the complementarity created by one citizen s action. Answer: (Ray): This situation is like a coordination game because if one person becomes an evader, she makes it easier for other people to evade. This is because the probability of getting caught comes down, so that the expected losses from evasion come down as well. In terms of part (a), m goes up if an additional evader enters the scene, so that F/m comes down. Thus an evader causes complementarities for other 4

5 evaders. (c) Show that it is always an equilibrium for nobody in the society to evade taxes. Is there another equilibrium as well? Find it and describe when it will exist. Answer (Ray): To see that no evasion is an equilibrium, suppose that nobody in the economy is evading. You are a potential evader. If you pay your taxes you will pay T. If you evade, then m = 1 (which is just another way of saying that you will be caught for sure), so that your expected loss is simply F. But F > T by assumption. It follows that if nobody else is evading, you wonõt evade either. The same mental calculation holds for everybody, so that no evasion all around is an equilibrium. What about everybody evading? Suppose that this is indeed happening, and you are considering evasion. If you do evade, then m = N, so that your expected losses are F/N. It follows that if F/N < T, you will jump on the bandwagon and evade as well. Thus Òwidespread evasionó is also an equilibrium provided that the condition T > F/N holds. (9) You are introducing a new vacuum cleaner design in a tropical society which works well under humid conditions. The vacuum cleaner is produced under increasing returns, so that the unit cost of product declines with the price. Discuss how the following factors affect your likelihood of success: (a) the proportion of the population who already use vacuum cleaners; (b) the per capita income of the society; (c) the market for loans, and (d) the flow of information. If vacuum cleaners are produced under decreasing returns to scale, which of these factors continue to be important? Answer (Ray): Omitted discussed in the text. 4. Chapter 5, question #10 The country is considering the adoption of two different telephone networks. Network A is an older network. It involves initial capital expenditures of $2 million, but will pay off (i.e., yield benefits of) $120,000 per year for a lifetime of fifty years. The newer network B involves a capital expenditure of $5 million, but it will pay off at the rate of $200,000 per year over a lifetime of fifty years. Assume there is no discounting of the future so that to make comparisons, the government simply adds up all costs and revenues and looks at net worth, choosing the project with the higher net worth. (a) Which network will a government with no installed telephone network choose? Answer (Ray): If there is no discounting of the future, then the first project has a net return of $4 million dollars, while the second has a net return of $5 million dollars. Therefore project B will be chosen. (b) Suppose that a government had just installed network A (before the new technology B came along). Would it now shift to network B? Answer (Ray): If network A has just been installed, the choice is really between a net return of $5 million (from adopting B and scrapping A) and the gross returns from A (because the setup cost has already been sunk). If A has just recently been installed, the gross returns are $6 million. Therefore there will be no switch to B. (c) Explain why this example helps you understand why leapfrogging in infrastructure might occur. 5

6 Answer (Ray): This is an example of leapfrogging. Imagine that the technology A is an older technology and that a rich country has adopted it for its infrastructure. Then simply by virtue of the fact that the cost for A has been sunk, the country will be more reluctant to switch to a new infrastructure when the new technology comes along. On the other hand, a poor country which is thinking of installing an infrastructure from scratch will choose B over A. 11 Suppose that diamonds can be extracted for the cost of $100 per diamond. Consumers value diamonds according to the following formula, expressed in dollar equivalent units V = x where x is the number of diamonds extracted. (a) Plot the total cost and value of extracting diamonds on a graph, with x the quantity of diamonds, on one axis and total costs and values on the other. Answer (Ray): The graph should give you a familiar diminishing returns shape for the consumers valuation of diamonds (displaying diminishing marginal utility) and a straight line passing through the origin for the cost curve of diamond extraction (Cost = 100x). (b) What is the socially optimal extraction level of diamonds, assuming that consumer valuations and extraction costs are given as exactly the same weight? Calculate this amount and indicate the economic reasoning that leads you to this conclusion. Also show the socially optimal extraction level on the same graph that you used for part (a). Answer (Ray): The socially optimal rule says: diamonds should be extracted whenever the marginal benefit of doing so exceeds the marginal cost. This means that on your graph, you simply find the magic value x at which the marginal utility or marginal valuation (which is just the tangent to the valuation graph) equals the marginal cost, which is 100. If you know a little calculus, you can see that the marginal valuation is just dv dx, which is 5000x?1/2. This equals 100 (the marginal cost) when x = 2500 (just solve the relevant equation). (c) Show that a monopolist producer of diamonds would produce exactly the socially optimal level of diamonds provided that he had the power to charge different prices for each diamond sold. How is the total surplus divided among the consumers and the producer? Answer (Ray): If a monopolist had the extraordinary power of being able to charge different amounts for each diamond sold, he, too would produce the socially optimal amount, except that he would pocket all the social surplus from doing so. He would sell the first diamond at the maximum valuation that the market would bear, which is just the marginal benefit of the first diamond. Similarly, the second diamond would be sold at the marginal benefit of the second diamond, and so on all the way down. In this way, the monopolist will also produce diamonds until marginal benefit equals marginal cost (because each such diamond makes him a positive profit). (d) Now assume that diamonds are produced by many small producers who take the price of diamonds as given in making their extraction decisions. Assume that each producer has a very small production capacity (up to which he can extract at $100 per diamond), 6

7 but that an unlimited number of producers can enter. Show that the socially optimal level will be produced again. How is the total surplus divided among the consumers and the producers? Answer (Ray): In contrast, look at a market with many suppliers each of whom take the price of diamonds as given. As long as the price of diamonds exceeds $100, producers will rush in to produce, so that the equilibrium price of diamonds must be precisely $100 diamonds. At this price how many diamonds will be sold? Well, exactly the amount such that the marginal benefit of an extra diamond is $100. At this point we see again that marginal benefit has been equated to marginal cost (because we just showed that marginal benefit = 100 = extraction cost), so once again the socially optimal level of diamonds is produced. However, note that this time all the social surplus goes to the consumers (indeed, producers make zero profits because the equilibrium price equals the average cost of extraction). (e) Now return to part (c) and assume that the monopolists is not permitted to charge different prices for each diamond he sells, but must charge the same price for all diamonds. Now show that production will fall below the socially optimal level. This discrepancy can be understood by observing that the producer fails to fully internalized the marginal social gain from an additional diamond produced. Carefully verify that you indeed understand this. Answer (Ray): Since the monopolist can charge only one price, it is easy to see that production can not be at the socially optimal level. For suppose that it were. Then for all these diamonds to be sold, the price must be no greater than the marginal benefit of the last unit sold, which (at the socially optimal level) equals the constant extraction cost per unit. At this price the monopolist cannot make a profit! So he will hike the price of diamonds to a higher level. At this higher price, the demand for diamonds (and therefore the supply) will fall short of the socially optimal level. Another way to understand what s happening is to imagine a tiny price reduction by the monopolist. This has two effects: one, it affects the monopolistõs profit, but two, it increases consumer surplus. The monopolist reacts to the first effect but of course does not care about the second. A maximizer of social surplus, in contrast, would have cared about the second effect as well. This is why the monopolist, in failing to internalize the full social gain of a price reduction, fails to maximize social surplus. (f) Show these ideas can be used to extend and understand the discussion in the text regarding externalities and international trade. Answer (Ray): Now you can see how the externalities may appear in Hirshman s and in Rosenstein Rodan s world. An investment may be not made (or it may be undersupplied) because all the benefits from it may not be captured by the investor. For instance, the investment may cheapen the price of an input to someone else who will benefit as well, but the investor (here read the monopolist) does not internalize this benefit. 7