Provide solutions to the attached questions below.This question is complete.
Problem 2 (50 pts) In the late 90s it was observed that the relative price of equipment (capital) has declined at an average annual rate of more than 3 percent. There has also been a negative correlation (-0.46) between the relative price of new equipment and new equipment investment. This can be interpreted as evidence that there has been significant technological change in the production of new equipment. Technological advances have made equipment less expensive, triggering increases in the accumulation of equipment both in the short and long run. Concrete examples in support of this interpretation abound: new and more powerful computers, faster and more efficient means of telecommunication and transportation, robotization of assembly lines, and so on. In this problem we are going to extend the Solow Growth Model to allow for such investment-specific tech- nological progress. Start with the standard Solow model with population growth and assume for simplicity that the production function is Cobb-Douglas: Y = K" L, ", where the population growth rate is = n. Similarly, just as in the basic model, assume that investment and consumption are constant fractions of output It = sY, and Ct = (1 - s)Y. However, assume that the relationship between investment and capital accumulation is modified to: Kit1 - Ke = qthe - 6Kt where the variable q, represents the level of technology in the production of capital equipment and grows at an exogenously given rate y, i.e. 24 = y. Intuitively, when of is high, the same investment expenditure2. Economic Growth. Suppose that Real GDP is given by the following ag- gregate production function: Y, = A, KO3 NOT a) Suppose that, in a given year, output grows by 3%, capital grows by 2.67% and labour grows by 1%. What is productivity growth? Bonus points to students who can derive the growth accounting formula. Hint: You need to make use of the following approxi- mation for a growth rate in variable X: gx = In X - In X-1 b) Suppose that the savings rate, s, is 0.5, the depreciation rate, d, is 0.09 and population growth, n, is 0.01. Productivity, A, is equal to 2. i. Derive the per capita aggregate production function and plot it on a diagram with k, = ~ on the horizontal axis and y, = # on the vertical axis. ii. Plot savings per capita, M, on the same graph. iii. Plot the amount of investment required to keep capital per capita constant. c) Solve for a steady state equilibrium. In particular, find capital, output, consumption, savings and investment in per capita terms. Show this equilibrium on your graph from part a). d) The golden rule savings rate is the one where consumption per capita is maximized in the steady state. Find the savings rate which is associated with the golden rule consumption level. Is it higher or lower than the savings rate used in part c)? Show the golden rule steady state on your graph. Hint: Consumption per capita in the steady state is given by: c = (1 -$)A You will need to use calculus to find the savings rate that maximizes consumption. Take the derivative with respect to the savings rate and set the first order condition equal to zero. Make use of the product and chain rules for the derivative with respect to s. You do not need to check the second order condition.Question 5 (20 points) Consider the following three period economy, with time denoted by t = 0, 1, 2. The economy is populated by a continuum of measure 1 of individuals, each endowed with one unit of a storable consumption good. At t = 0 , individuals have two options with regards to how they can invest their endowment. They can either stuff it in their mattress, where it gets a gross return equal to 1 (i.e., 1 + r = 1), or they can invest it in a long-term project that yields a gross return R > 1 in period two. For example, an individual that invests an amount / will receive RI in period two, and has 1 - I stuffed under the mattress. In t = 1, individuals have the option of liquidating the long-term project at a penalty. If they liquidate, they only receive a return L - 1 (per unit invested) in period 1, rather than the return R in period 2. At time t = 1, a fraction * = 1/2 of the individuals receive a liquidity shock. These individuals are "impatient" and only value consumption in period one. The fraction 1 - " individuals that do not receive a liquidity shock are "patient" and only value consumption in period two. At time t = 0, all individuals have the same chance of being hit by the liquidity shock. Assume that individuals do not discount the future, so that their ex-ante expected utility is given by U = Tu(CI) + (1 - #)u(c2). where c and c, are the consumption in period 1 and 2, respectively, and u(c) = 1, with o > 0. a) Assume there are no financial markets available, so that individuals must simply invest on their own. Given that an individual has invested an amount / at time t = 0, what will be the optimal levels of consumption, C1, C2, if: (i) the individual receives a liquidity shock (i.e. is impatient); (ii) the individual does not receive a liquidity shock (i.c. is patient). Let c and & denote the consumption of an impatient individual in period 1 and of a patient individual in period 2, respectively. b) What is the optimal level of investment when individuals have to invest on their own? Denote this level by /. Hint: Show that there exists L, LE [0, 1] such that if L 2 L, the optimal level of investment is equal to 1, and if L - L, the optimal level of investment is zero. c) Suppose that when types are realized in period 1, this information is publicly observable. Suppose there exists a social planner that individuals entrust all of their endowment to at time 0. The social planner will pay impatient individuals cj in period 1 and patient individuals c; in period 2 (and zero otherwise). Solving the social planner's 9 problem, what is cf and c;? How much does the social planner invest? That is, what is I*? d) Assume that L = 1. Show that I* oz. In other words, show that the planner invests less than the individuals but it makes them face more consumption risk. e) Assume that L = 1 and o > 0. Now suppose an agent's type is private infor- mation, and the social planner can only offer a contract contingent on an individual's announcement of her type at time 1. Furthermore, at time 1, she meets each agent only once, with the meeting order randomly determined. If individuals report honestly, can the social planner achieve the same allocation as in question c)? Is it optimal for an individual to report honestly when everyone else does? f) Suppose that L = 1 and o w are claiming to the planner that they are impatient. Is it optimal for these agents to lie as well? Given your answer to this question, are runs possible when o
