Suppose that the economy\'s production function is: Y = The saving rate is equal

Question

Suppose that the economy's production function is: Y = The saving rate is equal to s, the capital depreciates at rate delta, population grows at rate n. Find the dynamics of capital per worker (Write k t+1 - kt as a function of output per worker yt). What is the steady state capital per worker k*? Output per worker y *? Illustrate your answer graphically. What is the growth rate of aggregate capital in the long run in this economy? Of aggregate output? The golden rule capital per worker k GR is the level of capital per worker that maximizes consumption per worker at the steady state. Write the expression of the steady state consumption per worker c* as a function of k and derive the golden rule capital per worker. Suppose that the economy's production function is now given by: Y = with productivity A growing at rate gamma. What it the growth rate of aggregate output in this economy? Of output per worker?

Explanation / Answer

Y=K•N
a=3 (most tasks have 0a1)


a)
Constant returns to scale means for z>1 ; zY[K;N]=Y[zK;zN]
So: H° is zY[K;N]=Y[zK;zN]
and H¹ is zY[K;N]Y[zK;zN]

zY[K;N]~Y[zK;zN]
lhs ~ rhs H°: lhs=rhs & H¹: lhsrhs
z(K³/N²) ~ (zK)³/(zN)²
z ~ z³/z²
z ~ z
z = z
H°=True
H¹=False
Thus given function really represents constant returns to scale.

b)
K>0 ; N>0
Decreasing returns to factor means zY[K;N]>Y[zK;N] OR zY[K;N]>Y[K;zN]
MPK = Y/K = 3•K²/N² > 0
MPK/K = ²Y/K² = 6•K/N² > 0
Y/K> 0 & ²Y/K²>0 thus there are increasing returns to capital
MPN = Y/N = -2•K³/N³ < 0
Y/N<0 thus there are decreasing returns to labor (no need to check for ²Y/N²)

Or alternatively (z>1):
zY[K;N] ~ Y[zK;N]
z•K³/N² ~ (z•K)³/N²
z ~ z³
z < z³ increasing returns to capital

zY[K;N] ~ Y[K;zN]
z•K³/N² ~ K³/(z•N)²
z ~ 1/z²
z > 1/z² decreasing returns to labor

c)
y = Y/N
k = K/N
y = (K³/N²)/N = K³/N³ = k³

d)
k=0
k=sy-k
sy-k=0
sy=k
sk³=k
k²=/s
k=(/s)
y = k³ = (/s)^1.5

e)
=0.08
s=0.32
k=(0.08/0.32)=0.25=0.5
y=(/s)^1.5 = (0.08/0.32)^1.5 = 0.25^1.5 = 0.125

s=0.16
k=(0.08/0.16)=0.5 0.707107
y=(/s)^1.5 = (0.08/0.16)^1.5 = 0.5^1.5 0.353553

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