Velocity of money in initial long-run equilibrium

Question

Problem 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 b e required to achieve this in the short-run?

Problem 2: Growth Accounting/Solow Residual

Assume the following functional form for the aggregate production function:

Y = AKp^α1 K_h^α2 L^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 %
Y in terms of %
ΔA , %
ΔKp , %
KΔ_h, and %
ΔL 

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 , %
KΔp = 0.02 , %
KΔh = 0.04 , %
ΔL = 0.01 , α₁ = 0.2, α₂ = 0.6, α₃ = 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 k_p and k_h only.

2.5 Write %
Δy in terms of %
ΔA , %
Δkp , and %Δkh .

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

2.7 Based on your answer to (2.5), write %ΔA in terms of %Δy, %Δk_p, and %Δk_h only. In what sense is A a residue 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 kp are accurate, will our estimated value %ΔA^ be too high or too low relative to the true %ΔA?

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.

Problem 3. Solow Model with Population Growth

For this problem, try to keep time subscriptt 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 δ; 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 (p opulation) 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 Δc_t ≡ c_{t +1} - c_t; the law of motion for i_t, Δi_t ≡ i_{t +1} - i_t;

3.4 Assume a steady state in k_t; k_t = k‾= k_ss. What are
Δk_t, Δc_t and Δit.

3.5 Continue to assume a steady-state in k_t. 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 constant.

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? 

3.7 Solve for the golden rule level of the (per-worker) capital stock, k_gr, provided that f (k_t) = k_t^α. 

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. Compute k_gr and S_gr
 
Problem 4 Computational Solow Model (do Q3 first)

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 = k_o = 2

General hints:

1. You can leave everything in per-worker terms. The

2. Make a table of t , k_t , y_t , c_t , and i_t in the usual way. You already know how y₀, c₀, and i₀ depend on k₀ ; write analogous formulas in Excel. Your formulas should refer to cells (relative references), so don't substitute in the numbers directly. Fill in k₀.

3. Use the law of motion to write k₁ in terms of k₀ and constants (again, refer to cells). Copy your row formulas for k_t , y_t , c_t , and i_t down to run the model. Now you can see convergence to the steady-stat

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 sto ck 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 t = 300, a new production 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 timet = 600 occurs such that α_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 mo dify your Excel le to account for s, n, and δ varying over time. You do not have to implement this in Excel.