Helllp me here
A perfectly competitive firm has a Cobb-Douglas production function f (X1, X2) = X1X2. Suppose that input prices are w1 = 1 and W2 = 1. The firm wants to find the cheapest way of producing y = 32. a. Suppose that in the short run the quantity of input factor 2 is fixed at X2 = 8. Solve the firm's short-run cost minimization problem to derive the optimal input quantity x1. b. Derive the corresponding costs Cs of producing y = 32 in the short run. c. Now consider the long run, in which both input factors are variable. Set up the Lagrangian function for this firm's long-run cost minimization problem. d. Derive the first-order conditions of the above long-run cost minimization problem. e. Solve the above first-order conditions to derive the optimal input quantities x1 and X2. f. Derive the corresponding costs c* of producing y = 32 in the long run.9.1 Explain the Solow Growth model. Include the following: a. Write the per-capita production function and the per-capita capital accumulation equations. Define the variables that you use. b. Graph the production, savings, and capital maintenance functions. Show what happens when savings increases. Show what happens when population increases. Explain your graph. c. Graph the output, capital, consumption, rate of change in capital, and the growth rate. Explain your graph.d) Suppose the monetary policymaker wants to implement the optimal policy from part (c) using an interest rate rule for 4. The policymaker is considering a rule which sets is equal to 150(1 - P)(1 - p)gt, the natural real interest rate in this model. Briefly explain why this will not work. Furthermore, suggest a modification to the proposed rule that would successfully generate the outcomes in part (c). (Hint: you do not need to derive anything. You should be able to answer this from your knowledge of the model) e) Now consider two modifications to the model: (i) government spending provides utility where the utility function is: 1 + xlog G - 14 (ii) the decrease in govern- ment spending is initially used to lend money to households rather than to cut taxes (although, over time, the government reverses this policy and cuts lump sum taxes later). Discuss how these changes might affect your answers in parts (a) and (b). You do not need to derive anything, just explain the economic intuition. 8 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 I 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(C1) + (1 - #)u(c2), where c, and c, are the consumption in period 1 and 2, respectively, and u(c) = _", 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 & and 2 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 I. 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 S 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 OMarket equilibrium: Problems 4-4 and 4-6 in Chapter 4 of Borjas. Labor supply: Problems 2-3, 2-6, 2-8, and 2-14 in Chapter 2 of Borjas. B. Data analysis High rates of overall economic growth are generally believed to be associated with strong real wage growth: a rising tide lifts all boats. Some observers have suggested that this relationship changed around 1980, however, perhaps because labor unions became weaker or because of increased international trade. 1. Collect relevant data from the Economic Report of the President or another similar source and assess the claim of a changed relationship between wage growth and overall economic growth since around 1960. To do this, you will need to (i) Plot real the time series of real wage growth and real GDP growth (ii) Plot annual real wage growth against annual real GDP growth separately for the periods before and after 1980. (iii) Perform regressions of real wage growth on real GDP growth separately for the two time periods. "iv) Pool the data from both time periods and regress real wage growth on real GDP growth, a post- 1979 dummy, and the interaction of real GDP growth and the post-1979 dummy. (v) Discuss your results from (i) - (iv) and use these to assess the claim that the relationship between wage growth and overall economic growth changed since 1980. 2. Does the relationship between economic growth and wage growth have a clear theoretical interpretation in a supply-and-demand framework? How should your answer here affect the interpretation of the results from part 1? C. Analytical problems 1. Show that if labor supply is elastic, we might be able to raise tax revenue by lowering taxes. 2. In recent years, many immigrants have come to Silicon Valley to work in the software industry. Assume there are two types of programmers: natives (Type 1) and immigrants (Type 2). Initially, assume that there are n, native programmers in the labor force and no immigrants. The aggregate labor supply function of native programmers is nS,(w,), where S,(w,) is the labor supply function of an individual programmer. (i) Suppose that immigrant programmers have labor supply functions identical to native labor supply functions. Further, assume that firms treat immigrant and native programmers as perfect substitutes. Using a graph, show the effect on native programmers of an influx of n,=n immigrants. Compare this with the effect of immigration if immigrant labor supply is perfectly inelastic. (ii) Extra Credit: Using the mathematical methods in the Johnson (1980)' paper, find the effect of dn, on native employment in the two immigrant-elasticity scenarios described in (i), above
