MACROECONOMIC THEORY ASSIGNMENT
Q1. Consider the Ramsey-Cass-Koopmans model, and assume that output is produced according to the following production function:
yt = kt1/4 (1)
where yt is output per effective worker and kt is capital per effective worker. Households' lifetime utility is:
U = Bt=0∫∞e-βt ct1-θ/1 - θ dt
where ct is consumption per effective worker, B is a parameter, and β = ρ - n - (1 - θ)g. The discount factor is ρ = 0.9 and the coefficient of relative risk aversion is θ = 0.75.
Assume that the technology factor At is growing at the rate of 5% (g = 0.05), that the size of the household (Lt/H) is growing at the rate of 3% (n = 0.03), and that there is no depreciation. All the other assumptions of the model hold.
a. Write the expressions for c•t and k•t in terms of the model's parameters. Find the values c∗ and k∗ at which c•t = 0 and k•t = 0, and at which the conditions k∗ ≥ 0 and lims→∞e-Rse(n+g)sk∗ = 0 are satisfied as well. Show all your work.
b. Assume that the economy is operating at the steady state that you found in point a. Denote the steady state values of consumption per worker and capital per worker c∗old and k∗old. Assume that coefficient of relative risk aversion θ suddenly increases to θ = 0.8. Find the new values c∗new and k∗new at which c•t = 0 and k•t = 0, and at which the conditions k∗ ≥ 0 and lims→∞e-Rse(n+g)sk∗ = 0 are satisfied as well. Draw the phase diagram showing the locus at which c•t = 0 and the locus at which k•t = 0. Show the points c∗old, k∗old, c∗new and k∗new in the phase diagram.
c. i. Consider the instantaneous utility function: u(ct) = ct1-θ/1-θ. How is the marginal utility of consumption affected by the change in θ? Is it going to increase or decrease at any given level of ct? Explain your answer an show your work. Recall that the marginal utility of consumption is u'(ct).
ii. Now consider the dynamics of the model in the moment in which θ increases to θ = 0.8, but consumption per worker and capital per worker are still ct = c∗old and kt = k∗old. Write the expressions for c•t and k•t in terms of the model's parameters. Will ct be increasing or decreasing? Will kt be increasing or decreasing? Explain your answer and show your work.
2. Consider the basic Solow model, and assume that output is produced according to the following Cobb-Douglas production function:
Yt = Ktα(AtLt)(1-α) (2)
where Yt is aggregate output, Kt is aggregate capital, and AtLt is aggregate effective labor. You know from the data that α = 1/3.
a. In class, we wrote the expression:
f'(kt)(kt/f(kt)) ≈ (ln(Y2/L2) - ln(Y1/L1)/ln(K2/L2) - ln(K1/L1)) (3)
i. Use (2) to write f(k), where Kt/AtLt = kt, and show that in this case f'(k)(k/f(k)) = α.
ii. Show that (∂(Yt/Lt)/∂(Kt/Lt))(Kt/Lt)/(Yt/Lt) = f'(kt)(kt/f(kt)) (which is the reason why we are able to write the expression in (3)).
Use the fact that Yt/Lt = Atyt, and that ∂(Yt/Lt)/∂(Kt/Lt) = (∂(Yt/Lt)/∂kt )(∂kt/∂(Kt/Lt)) = (∂(Yt/Lt)/∂kt) 1/At.
iii. Let t = 1 be the first quarter of 2010, and t = 2 be first quarter of 2015. Go to the FRED (Federal Reserve Bank of St. Louis) webpage, and find the data on Y and L for these two quarters (use Y = GDPC1 and L = PAYEMS; the PAYEMS variable is measured monthly, use the month of February as an approximation of the number of workers for the quarter). Write Y2/L2 = X (Y1/L1), and compute the value of X.
iv. In class we said that (3) implies that:
[ln(K2/L2) - ln(K1/L1)] α 
[ln (Y2/L2) - ln (Y1/L1)]
so if Y2/L2 = X(Y1/L1):
[ln (K2/L2) - ln (K1/L1)] α ln (X)
Use the value of X that you obtained in a.iii. and the fact that α = 1/3 to compute: (K2/L2)/(K1/L1) = X1/α. Use the values of L1 and L2 that you obtained from the data to compute the value of the ratio K2/K1 that you should observe if the predictions of the Solow model were correct. Go to the FRED website and look at the variable K = RKNANPUSA666NRUG (this variable is measured annually, so use K1 = 2010 and K2 = 2015). Are the numbers that you obtained consistent with the actual data? Explain your answer and show your work.
b. Consider the real rate of return to capital (R/P)t, and recall that in the Neoclassical Theory we have that: (R/P)t = f'(kt), where f(kt) is the function that you obtained in point a.i.
i. Show that if the production function is (2), then (R/P)t = f'(kt) implies that (R/P)t = αyt(α-1)/α.
ii. Show that (∂(R/P)t/∂(Yt/Lt))(Yt/Lt)/(R/P)t = - (α-1)/α. Use the fact that ∂(R/P)t/∂(Yt/Lt) = ∂(R/P)t/∂yt (∂yt/∂(Yt/Lt)) = (∂(R/P)t/∂yt)1/At.
iii. Notice that we can write: (∂(R/P)t/∂(Yt/Lt))(Yt/Lt)/(R/P)t (ln((R/P)B)-ln((R/P)A)/ln(YB/LB)-ln(YA/LA)). Show that if YB/LB = Z(YA/LA) and α = 1/3, then we can write: -2 ln(Z) = [ln((R/P)B) - ln((R/P)A)].
iv. Consider now two countries, Country A and Country B, and assume that YB/LB = 5(YA/LA). Write the expression for (R/P)B/(R/P)A implied by your answer to point b.iii., and use this expression to compute the value of the ratio (R/P)B/(R/P)A. Interpret the numbers that you obtained. According to the Solow model, a country with lower output per worker (Country A) should have higher or lower real rate of return to capital relative to a country with higher output per worker? Explain your answer and show your work.
c. Consider again the Solow model with the Cobb-Douglas production function (2) and α = 1/3. You obtained data on the growth rate of output per worker in Country C, and you want to use this data to study whether this country is on a path of conditional convergence to its own steady state.
Use the Cobb-Douglas production function (2) to write the expression for ((Yt/Lt)/(Yt/Lt))• as a function of (kt/kt)• and g.
Use the approximation kt• ≈ -λ(kt - k∗), where λ is a constant that does not depend on the level of kt, to write ((Yt/Lt)/(Yt/Lt))• as a function of (kt - k∗). Use this expression to discuss the behavior of ((Yt/Lt)/(Yt/Lt))• that you expect to observe in the data if this country is converging to its steady state value k∗. Show all your work.
d. Consider now the following linear regression model:
ln [(Y/L)i] = a + b ln(si) + c ln(ni + gi + δi) + ∈i (4)
where g is the growth rate of At, n is the growth rate of Lt, δ is the depreciation rate, and s is the saving rate. The subscript i denotes country i. For each country, we can assume that the error term ∈i is not correlated with the independent variables in the regression.
You obtained data on Y/L, s, n, g, and δ for 120 different countries, i = {1, . . . , 120}, and you used this data to estimate the linear regression model (4). Your estimate for b is b^ = -0.09, and your estimate for c is c^ = -0.24; both parameters are statistically different from zero.
i. If the production function is (2), are the signs of b^ and c^ consistent with the Solow model?
ii. Assume that I decide to use the estimated model to test the following hypothesis: b + c = 0. If the production function is (2), would this hypothesis be consistent with the Solow model?
iii. If the production function is (2) and α = 1/3, what should the true values of b and c be? Explain all your answers.