Midterm Exam. Monday, March hour, 30 minutes. Name:

Transcription

1 San Francisco State University Michael Bar ECON 702 Spring 2019 Midterm Exam Monday, March 18 1 hour, 30 minutes Name: Instructions 1. This is closed book, closed notes exam. 2. No calculators of any kind are allowed. 3. Show all the calculations, and explain your steps. 4. If you need more space, use the back of the page. 5. Fully label all graphs. Good Luck

2 1. (30 points). Consider the Solow model, briefly described as follows. This is a closed economy, with no government. Output is produced according to Y FK,L ), where K is capita stock and L is labor. Assume that F has all the properties of the Neoclassical production function. Capital evolves according to K 1 δk I, where δ is the depreciation rate and I is aggregate investment. People save an exogenous fraction s of their income. The population of evolves according to L 1 nl, where n is exogenous population growth rate. a. (10 points). Suppose that productivity is fixed. Derive the condition for steady state capital per worker, and provide economic intuition for it. The law of motion of aggregate capital: K 1 δk I Dividing by L : K L 1δK 1 nl sfk,l 1 nl k 1δk 1n where k K /L_t, and fk FK,L. Substituting k k k and rearranging, gives: k 1n 1 δk sfk k nδ sfk sfk 1n Economic intuition. The left hand side is the flow out of the stock of capital per worker, due to depreciation and the growing number of workers (dilution). The right hand side is the flow in to the stock of capital per worker, due to investment. At steady state, these flows are the same. 1

3 b. (5 points). Suppose that the production of output has Cobb-Douglas form: FK,L A K L, 0 θ 1, and productivity is constant at A. Solve for steady state capital per worker. Using the steady state condition derived earlier, we have k nδ sfk k nδ sa k k sa nδ c. (5 points). Suppose that the economy of Liberia has A 1, saving rate of 10%, population growth of 4%, depreciation rate of physical capital of 6%, and capital share 50%. Find the steady state level of capital per worker, output per worker, consumption per worker and investment per worker in Liberia. Plugging the given values in the formulas for k,y,c,i : k sa nδ y A k c 1 sy i s y

4 d. (5 points). Suppose that at time t Liberia received a gift in the form of physical capital, so its capital per worker is k kss. Describe the time path of capital per worker in Liberia, before and after the gift, in the following diagram. k k time e. (5 points). ). Suppose that in the data for Liberia, the GDP is the sum of unambiguous labor income (I ), unambiguous capital income (I ) and ambiguous income (I? ). Calibrate the capital share for an economy with I 80, I 120, under the assumption that the capital share in ambiguous income is the same as the capital share in GDP. Denote the capital share in your calculations by θ. θ GDPI θi? θ I GDP I? I I I % 3

5 2. (30 points). Consider the Neoclassical Growth Model discussed in class. There is a single representative household and a single representative firm, that live forever. The household s period utility function uc,l, and the lifetime utility is Uc,l β uc,l, where β is the discount factor, and 0ρ1 is the discount rate. The household has 1 unit of time, so the labor supply is h 1l. The household owns the capital stock k, with the law of motion k 1 δk x, where x is investment and k 0 is given. The household receives a real wage w per unit of labor supplied to the firm, a rental rate r per unit of capital stock rented to the firm, and π is the profit (dividend) from the representative firm. There is a single representative firm that produces the output in this economy, with production function y FK,L ), where K is capita stock and L is labor and the L t is labor. Assume that F has all the properties of the Neoclassical production function. The economy is closed and there is no government, thus the feasibility constraint is: c x y. a. (15 points). Derive the sufficient conditions for competitive equilibrium sequences c,h,k, i.e. t 0,1, 1 u c,1h u c,1h F k,h 2 u c,1h βu c,h F k,h 1δ 3 c k Fk,h 1δ Explain your steps clearly. Household s problem: max,,, β uc,1h s. t. c x w h r k π t k 1 δk x t Notice that the profit is zero since the production function has constant returns to scale. Also notice that we substituted the leisure from the time constraint: l 1h, and the choice variable now is h the labor supplied (work time). Combining the budget and law of motion of capital, gives c k w h r k 1 δk. 4

6 Lagrange function: Lβ uc,1h λ c k w h r k 1 δk First order conditions: c : β u c,1h λ 0 h : β u c,1h λ w 0 k : λ λ r 1δ 0 Combining the conditions for c and h, gives: HH : u c,1h u c,1h w Using the condition for consumption at time t and t 1 in k : HH : u c,1h βu c,h r 1δ First order conditions: Firm s problem: max π FK,L r K w L, F K,L r 0 F K,L w 0 Market clearing conditions: Substituting the market clearing conditions (k K, h L ), and combining the firm s optimality conditions with those of the household, HH and HH, gives equilibrium conditions (1) and (2). Combining the feasibility constraint with the law of motion of capital: c x Fk,h k 1 δk x This gives feasibility constraint (3). 5

7 b. (10 points). Provide economic intuition for the optimal investment condition (Euler equation). In your answer, briefly explain the economic meaning of expression on the left hand side and on the right hand side: u c,1h βu c,h r 1δ The left hand side is the pain (decline in utility) as a result of investing extra unit of income in physical capital (and therefore giving up 1 unit of consumption) in period t. Recall that the marginal utility of consumption is the change in utility resulting from 1 unit change in consumption. The right hand side is the utility gain from that investment. In period t 1, the return on this investment, in units of consumption, is equal to r the non-depreciated unit of capital originally created. To convert this return into utility we multiply by the marginal utility from consumption, and to convert to present value we multiply by the discount factor. Thus, the optimal investment condition requires balancing the marginal pain and the marginal gain from investment. Any model of investment must have a condition similar to this one. 6

8 c. (5 points). In the notes we proved that the saving rate on a Balanced Growth Path of the NGM is: δγ s δγργρ θ Where ρ is the discount rate in the discount factor β. Countries with lower discount rate are predicted to have higher/lower saving rate. Circle the correct answer, and provide economic intuition. Lower discount rate means that people put more weight on future utility, and therefore save (invest) more. 7

9 3. (30 points). Consider the neoclassical growth model with taxes discussed in class. There is a single representative household and a single representative firm, that live forever. The household s period utility function uc,l, and the lifetime utility is Uc,l β uc,l, where 0β1, is the discount factor. The household has 1 unit of time, so the labor supply is h 1l. The household owns the capital stock k, with the law of motion k 1 δk x, where x is investment and k 0 is given. The household receives a real wage w per unit of labor supplied to the firm, a rental rate r per unit of capital stock rented to the firm, and π the profit (dividends) from the firm he owns. There is a single representative firm that produces the output in this economy, with production function y FK,L, where K is capital and L is labor. Assume that FK,L satisfies all the assumptions of a neoclassical production function. There is a government that collects taxes τ,τ,τ,τ on consumption, investment, labor income and capital income. The government spends these taxes on government consumption g and lump-sum transfers τ, and we assume that the budget is balanced in every period t 0,1,2,. a. (5 points). Write the household s problem. No need to solve it. Household s problem: max,,, β uc,1h s. t. 1 τ c 1 τ x 1 τ w h 1 τ r k τ π k 1 δk x Substituting the law of motion of capital in the budget constraint, gives max,, β uc,1h s. t. 1 τ c 1 τ k 1 τ w h 1 τ r k 1 τ 1 δk τ Notice that the profit is zero, (π 0), since the production function has constant returns to scale. Also notice that we substituted the leisure from the time constraint: l 1h. 8

10 b. (5 points). We derived the following necessary conditions for competitive equilibrium: u c,1h 1: u c,1h 1 τ F 1τ k,h 2: u c,1h βu c,1h R 3: c k g Fk,h 1 δk where R 1τ 1τ F 1τ 1τ k,h 1τ 1 δ 1τ Provide economic interpretation for condition (1). In your answer, briefly explain the economic meaning of expression on the left hand side and on the right hand side of (1). LHS is the Marginal Rate of Substitution between leisure and consumption, MRS,. RHS is the Relative price of leisure to consumption. The price of leisure is after tax wage, 1 τ w. The price of consumption after tax is 1 τ 1. c. (5 points). Define distorting taxes in this model economy. Distorting taxes prevent the competitive equilibrium allocation from solving the social planner s problem: max,, β uc,1h s. t. c k g Fk,h 1 δk t 9

11 d. (5 points). List all the distorting and non-distorting taxes in this model economy. Distorting taxes in this economy: τ,τ,τ,τ Non-distorting taxes in this economy: τ, i.e. the lump-sum transfers. e. (5 points). The following figure uses the program TDCEsimulations.m to simulate the effect of anticipated permanent increase in consumption tax by 20% percentage points, starting 10 periods from the present. The results are presented in the next figure, with dashed lines indicating the current steady state levels of variables. path of c t c t h t 0.44 path of h t k t 4.5 path of k t 4 1 R t path of R t w t 1.45 path of w t 1.4 p kt path of p kt y t path of y t 0.95 r t path of r t 0.2 I t 0.15 path of I t i t 0.15 path of i t 1 g t t path of g t path of t -0.2 Explain intuitively why higher taxes on consumption cause a permanent reduction in worktime h in NGM with taxes? Higher taxes on consumption make the leisure relatively cheaper compared to consumption, and this substitution effect increases leisure and lower the time spent on working. 10

12 4. (10 points). The following Matlab commands are part of the script TDCEsimulations.m. A = [A0; ones(t,1)*a_ss]; g = [g0; ones(t,1)*g_ss]; tau_c = [tau_c0; ones(t,1)*tau_c_ss]; tau_x = [tau_x0; ones(t,1)*tau_x_ss]; tau_w = [tau_w0; ones(t,1)*tau_w_ss]; tau_k = [tau_k0; ones(t,1)*tau_k_ss]; tau_w(10:end) = tau_w_ss + 0.2; a. Describe the object ones(t,1). The object ones(t,1) is a column vector of 1s, of length T. b. What is the purpose of the semicolon ; at the end of each command? The semicolon suppress the input display on the monitor. Each of the commands creates a potentially long vector of values, and it is inefficient to display them all on the screen. c. What is the purpose of the command tau_w(10:end) = tau_w_ss + 0.2;? The above command increases the tax on labor income by 0.2, above the steady state level, starting from period 10 onward. d. What is the difference between the command in part c and the following command: tau_w(10) = tau_w_ss + 0.2;? The command tau_w(10) changes the tax on labor income only in period 10, and the tax remains at steady state level in all other periods. The command in part b changes the tax rate in all periods 10,11, e. What is the difference between the command in part c and the following command: tau_w(1:end) = tau_w_ss + 0.2;? The new command tau_w(1:end) changes the tax rate starting from the first period onward, while the command in part b changes the tax rate starting from period 10 onward. 11