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1. (15 points) The representative firm has a production function Y - zN, where z is the labor productivity. The representative household has a utility function u(c, () = Vc+ 18vi. Let w denote the real wage rate. (a) (2 points) Suppose that z = 200. What is the representative firm's labor demand function? What must the wage rate be at the equilibrium? At this wage rate, how much profit can this firm send to the representative household? (b) (3 points) With this wage rate and non-wage income, what are the optimal time allocation and consumption for the representative household? (c) (1 point) At the equilibrium how big is labor input and how much output is produced? (d) (3 points) Then the productivity drops. That is z = 190. At the new equilibrium how big is labor input and how much output is produced? (e) (1 points) Calculate the percentage decrease of productivity, labor input, output, and consumption. (percentage change = After Before x 100%) Is the percentage decrease in output bigger or smaller than the percentage decrease in productivity? Can you explain your finding? (f) (3 points) Anticipating the productivity drop, the government implements a stimulus package by increasing the government spending from 0 to 8. The spend- ing is financed by government borrowing, equivalent to a lump-sum tax. After implementing this stimulus package, how big are labor input and consumption? How much output is produced? (g) (1 point) Calculate the percentage change of labor input, output, and consump- tion when productivity drops but with a stimulus package. (h) (1 point) Compare your results in part (e) and (g), What do you find?Section 1 [1 HW point] Real business cycle model and business cycle facts The economy has a 1-factor production function with labor as the only input: Y = Z . L where Z is the total factor productivity (or TFP, for short) whose fluctuations are driving the business cycle. There is just one household. Assume that the economy has an aggregate labor supply curve L(C, w) = 1 - w where v is weight on leisure, C is aggregate consumption and w is the current wage. a) Write down the profit maximization problem for the representative firm and solve for the labor demand curve. b) Solve for the labor market equilibrium: express equilibrium wage, w., and equilibrium labor quantity, L., through Z, C, v. c) Solve for the full market equilibrium. What is the market clearing condition for this economy? Combine the market clearing condition with the production function Y = ZL and the two expressions for L. and w. from part b) to solve for equilibrium quantities. That is, express Y., L., w., C. through two exogenous variables, v, Z. d) Now suppose that Z - the business cycle "driver" - varies over time, and Y., L., w., C. vary as well according to equilibrium relationships derived in part c). Evaluate whether this toy model fits business cycle facts by filling out the following table. Put a check mark in a cell if the model fits the fact, and a minus sign otherwise Fact Model fit Consumption is procyclical Consumption is less variable than GDP Employment is procyclical Average labor productivity is procyclicalCompetitive Equilibrium & TFP. Consider an economy with the representative consumer, representative firm, and government. Suppose that G = 12. The con- sumer's preferences over the consumption good and leisure are given by the utility function U(C,I) = Cl. Let h - 24. The firm's production function is Y = zN, where z = 2. Suppose that the government imposes a lump-sum tax on the representative consumer. (a) Write down the social planner's maximization problem and the social planner's optimality condition. Derive the Pareto optimal allocation for consumption, out- put, leisure, and labor. (b) In this economy the welfare theorems apply, and hence the Pareto optimal allo- cation can be supported as a competitive equilibrium. Find the real wage w that supports the Pareto Optimal allocation you found in part (b) as a competitive equilibrium. (c) suppose that the total factor productivity z increases from 2 to 3. How do equi- librium consumption, labor, output, and the wage rate change? What does your answer imply regarding whether the income or the substitution effect dominates for labor?4 One-Period Economy Consider a one-period economy where we have a representative consumer, a representative firm, and a government. The representative consumer derives utility from a consumption good (de- noted by C) and leisure (denoted by (), according to the separable utility function U(C. 1) = u( C ) + u(!) where both u(C) and v(() are increasing functions of their respective arguments. As usual, the consumer's endowment consists of h hours, which he can devote either to work or leisure. If he decides to work, he supplies labor to the representative firm and earns a wage rate w per hour worked. It also enjoys a dividend income * and has to pay a lump sum tax to the government of size T. The representative firm produces the consumption good according to a production function given by * F (K, N ) = > KON- where = stands for total factor productivity, h denotes capital, /V denotes labor input, and a is a positive real number. Finally, the government collects T in taxes and then uses these resources to finance government expenditures G a. Define a CE for this economy b. Let a = 1/2, a = 1, K = 1025 and h = 24 assume that consumer's preferences are given by U(C.D) = CM/2 where v(!) = 0 ( consumer does not value leisure). Suppose that the size of government expen- ditures is G - 40 Find the competitive equilibrium