Homework #4
1 Aggregate Demand / Aggregate Supply
Long-run aggregate supply curve (vertical): YLR = 3000
Short-run aggregate supply curve (horizontal): PSR = 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 PLR,1 and YLR,1?
1.2 What is the velocity of money in initial long-run equilibrium (PLR,1, YLR,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 (PSR,2, YSR,2)?
1.4 If the aggregate demand and long-run aggregate supply curves are un-changed, what is the long-run equilibrium (PLR,2, YLR,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 = AKpα1Khα2Lα3
Here, Kp denotes physical capital; Kh human capital; L labor. The constant returns to scale condition α1 + α2 + α3 = 1 holds.
2.1 Write %ΔY in terms of %ΔA, %ΔKp, %ΔKh 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, %ΔKh = 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 = Kp/L;, kh = Kh/L. Write y as a function of lip and kh 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, %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 Kh is unobservable (can't gather data for it), we set %Δkh = 0 and try to estimate productivity growth using the equation: %,ΔA = %Δy - α1%Δkp). Assuming that our data on y, α1, and kp are accurate, will our estimated value %Δkp be too high or too low relative to the true %ΔA? (hint: you should have three cases based on %Δkh)
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:
Yt = Yt/Lt
kt = Kt/Lt
ct = Ct/Lt
it = It/Lt
Aggregate production function in per-worker terms:
yt = f (kt)
Labor force (population) growth equation:
Lt = (1 + n)Lt-1
3.1 Write ct and it as functions of kt.
3.2 Write the law of motion for kt, Δkt = kt+1 - kt.
3.3 What is the law of motion for ct, Δct ≡ ct+1 - ct, the law of motion for it, Δit ≡ it+1 - it? (hint: should be a function of kt and constants)
3.4 Assume a steady-state in kt, kt = k‾ = kss. What are Δkt, Δct, and Δit?
3.5 Continue to assume a steady-state in kt. Write Δkt/Kt, ΔLt/Lt, ΔYt/Yt, ΔCt/Ct and ΔIt/It 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 kt. What are the steady-state levels kss, yss, css, iss? (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(kt) = MI. (hint: maximize css as a function of kss from the previous part; your function for css shouldn't contain savings rate s; your answer should depend only on constants)
3.8 Solve for the savings rate sgr, that supports kgr 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 kgr and sgr.
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(kt) = ktα
δ = 0.02
n = 0.01
α = 0.3
s = 0.15
kt=0 = k0 = 2
General hints:
1. You can leave everything in per-worker terms. The -nkt term in the law of motion is all you need to include to account for Lt growing at rate n.
2. Make a table of t, kt, yt, ct, 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 ko and constants (again, refer to cells). Copy your row formulas for kt, yt, ct, and it down to run the model. Now you can see convergence to the steady-state!
4.1 Graph kt vs. time for 0 ≤ t ≤ 300. What is the steady-state level of the capital stock?
4.2 Let s = sgr, where sgr is the golden rule savings rate that you found in part (3.9). Graph kt 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 = sgr. At t = 300, a new production technology is introduced such that αt ≥ 300 = 0.4. Graph kt, ct, and it vs. time for 300 ≤ t ≤ 600. What is the new steady-state level of the capital stock?
4.4 Continue with s = sgr, and αt ≥ 300 = 0.4. A natural disaster at time t = 600 occurs such that δt ≥ 600 = 0.10 and nt ≥ 600 = -0.01. Graph kt, ct, and it 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.