Final Exam Solutions

Transcription

1 Final Exam Solutions Fall 2014 Kevin Proulx Problem 1 For the United States in 2009, mean household income was $67,976 and median household income was $49,777. Thus, the median was approximately 73.3% of the mean. Problem 2 Following a sudden increase in the depreciation rate in the closed economy Solow model Consumption will decrease rapidly and then gradually settle into its new lower steady state. The capital output ratio, K Y, will decrease rapidly and then gradually settle into its new lower steady state. C) Total depreciation, δk, will jump up as δ increases, but then will gradually decrease to its new lower steady state level as K declines. Problem 3 The number of 50 year olds is greater than the number of 40 year olds. This follows from the fact that the net rate of reproduction (NRR) is below the population replacement level, 1. Recall that the NRR is the number of daughters that each girl who is born can be expected to give birth to. The NRR is 3/4, because 1/2 probability of having any children x 3 children x 1/2 children being girls. Problem 4 Gasoline consumption will be higher under scenario B. This is because consumers are comfortable changing capital goods and making fixed investments when the price decrease is expected to be permanent. For example, with a permanent price drop consumers will substitute into less fuelefficient cars. Problem 5 The Gini coefficient for this country is.50. Although a graph is not provided here, the general way to calculate the Gini coefficient would be to divide the area between the line of equality and the Lorenz curve by the total area under the line of equality. The correct calculation for this example.25 is = 0.5. An alternative solution is to note that because there are only two income levels the Gini coefficient is equal to the fraction of all income from the high group, 1, minus the fraction of households in the high income group,.5. 1

2 Problem 6 Plan A is not optimal and not feasible. After the bureaucratic mess up, the stock will increase above its optimal level and its growth rate will be below the maximum sustainable yield, 500. In which case, harvesting 600 initially will deplete the stock below the optimum and will not be restored by simply reducing the harvest to equal the maximum sustainable yield. Rather, a period of harvesting below the maximum sustainable yield would be required to restore the stock to the optimal level. This plan is feasible, but not optimal. The 100 extra tons that were not harvested due to the mess up will never be recovered under this plan because a harvest of 500 will not bring the stock directly back to its optimal level. It is best to calculate the first harvest to immediately return the stock to its optimal level, and then return to harvesting the maximum sustainable yield of 500 thereafter. Problem 7 Substituting the given data into the production function and solving for A gives A Z A P = 2. = 1 and The relative efficiency of Zerg to Protoss is 1. Proof. Since A = T x E, the relative levels of efficiency can be written as E Z E P = A Z T P = 1 T P T Z A P 2 T Z Given that efficiency has been constant, we know that technology has been growing at the same rate as productivity, 1%, for a very long time. Therefore, since Protoss is 70 years ahead of Zerg, we can write the relative levels of technology as T P = T Z (1.01) 70 = T P T Z = (1.01) 70 The rule of 70 implies that it takes approximately 70 years for the relative technology to double when the growth rate is 1%. Therefore, the level of technology in Protoss is twice the relative technology in Zerg, and E Z = 1 2 E P 2 1 = 1. Problem 8 In the year 2000, country 1 is the technological leader because A1 A 2 of technology and output in country 1 will be = 2. Therefore, the growth rate  1 = γ A,1 µ i =.1 10 =.01 Similarly, since country 2 is the follower, its growth rate for technology and output will be.05. Since the growth rates are linear and the labor forces devoted to R&D are constant, we know that at some point country 2 will overtake country 1 as the leader. In fact, once country 2 becomes 2

3 technological leader, its output will grow faster than country 1 s indefinitely. Below I will prove that country 2 overtakes country 1 at approximately t = 17.5, at which point the growth rate will fall to.025 for country 2 and jump to.02 for country 1. Proof. We are looking for a time period, t, such that A 1,2000+t = A 2,2000+t. This implies A 1,2000 (1 +.01) t = A 2,2000 (1 +.05) t ( ) t 1.05 = 2 = = ln(2).70.04t 1.01 = t 17.5 Problem 9 If y < 2, the steady state output per capita is 1, the steady state population level is 9, and the equilibrium is stable. If y 2, the steady state output per capita is 3, the steady state population level is 1, and the equilibrium is unstable. A proof for the second case is provided below. Proof. Recall that the growth rate of population is zero in the steady state. Therefore, ˆL = y = 0 = ȳ = 3 Solving for the steady state population level in terms of steady state output per capita Y = 9 1/2 L 1/2 = y = 3L 1/2 ( y ) ( ) = L = = 3 y = L = 1. For stability, first suppose that there is a shock that reduces the population level to 1/2. This causes output per capita to rise instantaneously to 3 (1/2) and for the growth rate of population to become negative. Since L t+1 < L t, output per capita will continue to rise. Now consider what happens if L suddenly doubles. This causes output per capita to fall instantaneously to 3 2 and for the growth rate of population to become positive. Since L t+1 > L t, output per capita will continue to fall. This proves that the equilibrium is unstable because there is no correction towards the initial steady state. Problem 10 The pattern of specialization you should expect to see is skilled workers specializing in book production, unskilled workers specializing in sandwich production, and either both producing haircuts or only skilled workers producing haircuts. This is because skilled workers have a strong comparative advantage in book production relative to sandwich production. I will prove in part G) that only skilled workers will produce haircuts. Note that your solutions to parts through F) were not impacted if you chose the pattern of specialization where both types of labor produce haircuts. From the solution to, the relative wage of skilled to unskilled workers is pinned down by the relative productivity of skilled to unskilled workers in producing haircuts, which is 2. 3

4 C) The price of sandwiches, p(s), must equal the total unskilled labor cost, a(s)w u, where a(s) is the unit labor cost of producing sandwiches for an unskilled worker. Therefore, p(s) = w u. Hence, using sandwiches as the unit of account, w u = 1 sandwich per hour. D) The price of haircuts, p(h), must equal the total skilled labor cost, a(h)w s, where a(h) is the unit labor cost of haircuts for a skilled worker. Therefore, p(h) = 2 sandwiches per haircut. E) Since the relative wage of skilled to unskilled workers is 2, while w u = 1, the skilled wage must be w s = 2 sandwiches per hour. F) The price of books, p(b), must equal the total skilled labor cost, a(h)w s, where a(h) is the unit labor cost of haircuts for a skilled worker. Therefore, p(h) = 2 sandwiches per haircut. G) Assuming 6 hours work days and the case where only skilled workers produce haircuts: Good Daily Consumption of Unskilled Daily Consumption of Skilled Sandwiches 2 4 Haircuts 1 2 Books 1 2 Good Daily Production of Unskilled Daily Production of Skilled Sandwiches 6 0 Haircuts 0 3 Books 0 3 This is an equilibrium because total supply equals total demand for each good. To see why the other allocation of workers is not an equilibrium notice that all unskilled labor must be allocated to making sandwiches to meet demand. This leaves no unskilled workers to produce haircuts. 4

5 Appendix 5