Homework #4

1 Aggregate Demand / Aggregate Supply

Long-run aggregate supply curve (vertical): Y_LR = 3000

Short-run aggregate supply curve (horizontal): P_SR = 1

Aggregate demand curve:

Y = 3M/P

Money supply: M = 1000

1.1 If the economy is initially in long-run equilibrium, what are the values of P_LR,1 and Y_LR,1?

1.2 What is the velocity of money in initial long-run equilibrium (P_LR,1, Y_LR,1)?

1.3 Now suppose a supply shock moves the short-run aggregate supply curve to P = 1.5 (still horizontal). What is the new short-run equilibrium (P_SR,2, Y_SR,2)?

1.4 If the aggregate demand and long-run aggregate supply curves are un-changed, what is the long-run equilibrium (P_LR,2, Y_LR,2) after the supply shock?

1.5 Suppose that after the supply shock, the Federal Reserve wants to hold output at its long-run level. What level of the money supply M' would be required to achieve this in the short-run?

2 Growth Accounting / Solow Residual

Assume the following functional form for the aggregate production function:

Y = AK_p^α1K_h^α2L^α3

Here, K_p denotes physical capital; K_h human capital; L labor. The constant returns to scale condition α₁ + α₂ + α₃ = 1 holds.

2.1 Write %ΔY in terms of %ΔA, %ΔK_p, %ΔK_h and %ΔL (hint: Δ log(x) ≈%Δx).

2.2 Based on your answer to the previous part, write %ΔA in terms of the other variables. Interpret this as the growth rate of the "Solow residual". In what sense is A a residual here?

2.3 For this part only, assume that %ΔY = 0.05, %ΔKp = 0.02, %ΔK_h = 0.04, %ΔL = 0.01, α₁ = 0.2, a₂ = 0.6, a₃ = 0.2. Compute %ΔA. What is the most important source of output growth here?

2.4 Now, let's write everything in per-worker terms. Define y = Y/L; kp = K_p/L;, k_h = K_h/L. Write y as a function of lip and kh only.

2.5 Write %Δy in terms of %ΔA, %Δk_p, and %Δk_h.

2.6 Based on your answer to (2.5), write %Δy in terms of %ΔA, %ΔKp, %ΔKh, and %ΔL.

2.7 Based on your answer to (2.5), write %ΔA in terms of %Δy, %Δkp, and %Δkh only. In what sense is A a residual here?

2.8 If human capital K_h is unobservable (can't gather data for it), we set %Δk_h = 0 and try to estimate productivity growth using the equation: %,ΔA = %Δy - α₁%Δk_p). Assuming that our data on y, α₁, and k_p are accurate, will our estimated value %Δk_p be too high or too low relative to the true %ΔA? (hint: you should have three cases based on %Δk_h)

2.9 Is it possible that %ΔA < 0? Interpret. Is this consistent with the idea that A is the level of technology available in an economy?

3 Solow Model with Population Growth

For this problem, try to keep time subscript t on all the relevant (i.e. changing over time) variables out of steady-state. This will help you for the next problem. Assume that the following variables are constant over time: depreciation rate (5; savings rate s; population growth rate n. Because population growth is non-zero, the labor force is now a function of time.
Definitions:
Y_{t =} Y_t/L_t

k_t = K_t/L_t

c_t = C_t/L_t

i_t = I_t/L_t

Aggregate production function in per-worker terms:

y_t = f (k_t)

Labor force (population) growth equation:

L_t = (1 + n)L_t-1

3.1 Write c_t and i_t as functions of k_t.

3.2 Write the law of motion for k_t, Δk_t = k_t+1 - k_t.

3.3 What is the law of motion for c_t, Δc_t ≡ c_t+1 - c_t, the law of motion for i_t, Δi_t ≡ i_t+1 - i_t? (hint: should be a function of k_t and constants)

3.4 Assume a steady-state in k_t, k_t = k‾ = k_ss. What are Δk_t, Δc_t, and Δi_t?

3.5 Continue to assume a steady-state in kt. Write Δk_t/K_t, ΔL_t/L_t, ΔY_t/Y_t, ΔC_t/C_t and ΔI_t/I_t in terms of constants. (hint: these are not per-worker quantities, so they will have non-zero growth rates)

3.6 Continue to assume a steady-state in k_t. What are the steady-state levels k_ss, y_ss, c_ss, i_ss? (hint: you can leave k„ in your answers)

3.7 Solve for the "golden rule" level of the (per-worker) capital stock, kgr, provided that f(k_t) = MI. (hint: maximize c_ss as a function of k_ss from the previous part; your function for c_ss shouldn't contain savings rate s; your answer should depend only on constants)

3.8 Solve for the savings rate s_gr, that supports k_gr from the previous part. (hint: use the law of motion for the capital stock; your answer should depend only on constants)

3.9 Let δ = 0.02, n = 0.01, and α = 0.3. Compute k_gr and s_gr.

4 Computational Solow Model

From parts (3.1) and (3.2) of Q3, you should be able to implement the Solow model with population growth in Excel.

Parameter values and initial conditions:

f(k_t) = k_t^α
δ = 0.02
n = 0.01
α = 0.3
s = 0.15
k_t=0 = k₀ = 2

General hints:

1. You can leave everything in per-worker terms. The -nk_t term in the law of motion is all you need to include to account for L_t growing at rate n.

2. Make a table of t, k_t, y_t, c_t, and it in the usual way. You already know how yo, co, and it) depend on loo; write analogous formulas in Excel. Your formulas should refer to cells (relative references), so don't substitute in the numbers directly. Fill in /co.

3. Use the law of motion to write k₁ in terms of k_o and constants (again, refer to cells). Copy your row formulas for k_t, y_t, c_t, and it down to run the model. Now you can see convergence to the steady-state!

4.1 Graph k_t vs. time for 0 ≤ t ≤ 300. What is the steady-state level of the capital stock?

4.2 Let s = s_gr, where s_gr is the golden rule savings rate that you found in part (3.9). Graph k_t vs. time for 0 ≤ t ≤ 300. Is the steady-state level of the capital stock now the golden rule level found in (3.9)?

4.3 Continue with s = s_gr. At t = 300, a new production technology is introduced such that α_t ≥ 300 = 0.4. Graph k_t, c_t, and i_t vs. time for 300 ≤ t ≤ 600. What is the new steady-state level of the capital stock?

4.4 Continue with s = s_gr, and α_t ≥ 300 = 0.4. A natural disaster at time t = 600 occurs such that δ_t ≥ 600 = 0.10 and n_t ≥ 600 = -0.01. Graph k_t, c_t, and i_t vs. time for 600 ≤ t ≤ 900. What is the new steady-state level of the capital stock?

4.5 In words, describe how you would modify your Excel file to account for s, n, and δ varying over time. You do not have to implement this in Excel.