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Econ 7040 Macroeconomic Analysis Tutorial 3
发布时间:2022-04-02
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Macroeconomic Analysis (Econ 7040)
Tutorial 2: The Augmented Solow
1. Suppose that the augmented Solow model represents the world well enough. Assume that Yt = AK (ZtNt)1
.
(a) Find the new transitional dynamics equations of the model, in eciency units terms [TD2].
(Backward engineering: given that the model displays long-run equilibrium in ef- ciency units, we divide everything by ZtNt)
Kt+1 = sAK(ZtNt)1
+(1
δ)Kt ,
divide it by ZtNt (eciency units of labor) to get
Kt+1 = Kt+1 Zt+1Nt+1 = sAK(ZtNt)
+(1
δ) Kt
t+1(1+z)(1+n) = sA
+(1
δ)
t ,
t+1 = (1+z)(1+n) hsA
+(1
δ)
ti [TD2]
(b) Find an expression for the steady state value of =
and k =
. How do we know that
exists?
Impose that the long-run equilibrium is reached (t =
t+1 =
).
t+1 = (1+z)(1+n) hsA
+(1
δ)
ti
= (1+z)(1+n) hsA
+(1
δ)
i
(1+z)(1+n)
(1
δ)
= sA
[(1+z)(1+n)
(1
δ)] = sA
(1
) = sA
=
(1
) .
The existence of
is guaranteed by the properties of F(K,ZN), diminishing returns and INADA conditions, which by CRS apply also to f(
). Now, we also now that kt = Zt
t. Therefore, at steady state:
k = Zt
= Zt
(1
)
(c) Find an expression for the steady state value of yˆ = and y =
yˆ = A
yˆ = A
.
To nd y
use the fact that yt = Ztyˆt .
(d) Find an expression for the steady state value of =
and c =
= (1
s)yˆ = (1
s)A
= (1
s)A
.
To nd c
use the fact that ct = Zt
t .
(e) Use the macrohistory data set JSTdatasetR4.dta to compute the growth rate of population in the U.S for the period 1860-2016. Plot a graph, clearly labeling the axis. Do you observe any trend or structural change in n?
Code and data attached. There is a clear downward trend in population growth. A permanent decline in n. In 1940-1960 n 1.7%, while between 2010-2016 n
0.75%.
(f) What are the model’s implications on y,k, and r from the observed changes in n in the U.S?
In the short-run we have
t+1 = (1+z)(1+n) hsA
+(1
δ)
ti
yˆt = A ,
yt = Ztyˆt ,
and in the long-run we have
yˆ = A
.
y = Ztyˆ
The decline in n implies an increased marginal productivity of workers in t+ k (k 1), all else equal (diminishing returns of inputs). But also the fact that capital accumulation also speeds up (see TD equation) increases output and labor productivity even more. While marginal productivity of capital should decrease, all else equal, the fact that there is less and less people to feed and that the productivity of workers increases (due to lower N but also higher K) implies a positive long-run e
ect on
and yˆ
. The implications for r are determined by
r = MPK = Aα
1 .
In the short-run, faster capital accumulation would decrease r. In the long-run, as is expected to increase, r is expected to fall (more supply of capital).
(g) Is the observed evolution of y,c and r consistent with these predictions? Discuss how the model can (or cannot) make sense of the evolution of these variables more broadly.
We observe that real GDP, as expected, appears to negatively comove with n. The OLS regression between real GDP or log GDP (as dependent variable) on n (in- dependent variable) displays a negative and statistically signicant coe
cient. To study the relationship between n and the real interest rate (r) we proxy de
ne using the Fisher equation: r = i
π, where i is the nominal rate of return on cap- ital and π is the in
ation rate (annual growth rate of the CPI
consumer price index). The relationship between n and r in the data appears positive, as ex- pected. However, during 1930’s and 2010-2016 they appear to comove negatively. While, overall, the model appears to be good in explaining the facts, there are sev- eral weaknesses of our analysis. First, analysing the evolution of non-stationary variables could lead to spurious correlations. Do they share the same trend (un- derlying mechanism)? A cointegration analysis is required to test this.1 Second, we have an important omitted variable problem. What happened to the saving rate s = I/Y and to total factor productivity A in the US during 1870-2016? The data shows important
uctuations in s = I/K that will certainly drive the evolution of y,r, as well. The database does not provide information on A. Besides, there are other elements outside the model (for now) that could a
ect the evolution of y,r (evolution of credit conditions, uncertainty,
scal and monetary policy, etc.).