economics 3456

1. Human Capital in the Solow-Swan Model: Suppose that the production function in the Solow-Swan model is given by Y(t) = AK(t)"H(t)L(t) 1-a-x where Y is output, A is technology, K is physical capital, H is human capital, and L is labour. Assume that 0 So 0. Output can be converted one-to-one into either investment in physical capital or human capital. Saving directed towards investment into physical capital is given by 0 0. Output can be converted onetoone into either investment in physical capital or human capital. Saving directed towards investment into physical capital is given by D 0 is a parameter. Jill uses the capital she got from her mother to produce consumption q, and investment is, according to the resource constraint: citi = VBk, + et, where B > 0 is a parameter, and a a random shock to the production function. The shock lakes different values in different periods. Jill knows e, once she is born, so for her it is just a constant. The capital that is left to Jill the daughter is determined by: kel = (1-6)k, + it, where the parameter s, the depreciation rate, is a number between zero and one. This just means that capital tomorrow is what is left over today after depreciation, plus investment. Compute Jill's decision of consumption and investment as a function of the parameters k and et- Exercise 9.5 (Moderate) If we want to examine the behavior of this model relative to the real world, the next step would be to set the parameters in a way that matches certain features of the real world. Since that is a complicated task, we will give some values to you. B is a scale parameter and does not affect the qualitative behavior of the model. Therefore we set it to B = .1. 6 is the depreciation rate, for which a realistic value is & = .05. A determines the relative size of c, and &, in equilibrium. A rough approximation is A = 4. Using these parameters, compare the reactions of c, and i, to changes in c- (Use calculus.) Exercise 9.6 (Moderate) In the last step, you will simulate business cycles in the model economy. All you need to know is the capital ki at the beginning of time and the random shocks c. As a starting capi- tal, use ki = 3.7. You can generate the random shocks with the random number generator in your spreadsheet. In Excel, just type "=RAND()", and you will get a uniformly distributed random variable between zero and one. Generate 50 such random numbers, and use your formulas for c and is and the equation for capital in the next period, kil = (1 -6) k, + it, to simulate the economy. Plot consumption and investment (on a single graph). How does the volatility of the two series compare? Plot a graph of GDP, that is, consumption plus in- vestment. How do the business cycles you see compare with the ones you found in the real world? You don't need to compute the length of each cycle, but try to make some concrete comparisons.Exercises Exercise 10.1 (Moderate) Answer: True, False, or Uncertain, and explain. 1. "Did you know that America's 22 million small businesses are the principal source of new jobs?" (Source: Web page of the Small Business Administration.) 2. "In the next century, 20% of the population will suffice to keep the world economy Exercises 93 going.... A fifth of all job-seekers will be enough to produce all the commodities and to furnish the high-value services that world society will be able to afford" the remaining 80% will be kept pacified by a diet of "Tittytainment". (Source: Martin, Hans-Peter and Harald Schumann. The Global Trap. New York: St Martin's Press. 1996.) Exercise 10.2 (Easy) The plant-level rate of employment growth is defined as: Zest where: A.Nest = Nest - Nes,i-1. That is, A Nest is the change in employment at plant e in sector s from t -1 to t. Show that fest = 2 for all plants that are born between t - 1 and t, and show that gest = -2 for all plants that die between t - 1 and t. Exercise 10.3 (Easy) Show the following: Ca = > (- ) gest, and: net = = > (2 ) Best. Here cat is the average rate of job creation of all plants in sector s. What does the term Zest / Zat mean? Exercise 10.4 (Moderate) For the purposes of this exercise, assume that you have data on annual national job creation G and job destruction D: for N years, so t = 1. .. N. Show that if annual national job reallocation R, and net job creation NET, have a negative covariance, then the variance of job destruction must be greater than the variance of job creation. Recall the definition of variance of a random variable X for which you have / observations, (2)M: var(X) = ~ _(x; -2), where z is the mean of X. Similarly, recall the definition of the covariance of two variables X and Y . If there are N observations each, {z;, y:}My, then: cov(X, Y) = = )( -2)(: -D). These definitions and the definitions of NET, and R, provide all the information necessary to answer this exercise