Question

1. Consider the Solow growth model with population growth and growth in the efficiency of labor. Suppose that 2 countries A and B have the same production function given by Yt = Ktα(LtEt)1−α, the same rate of growth of E (g), the same depreciation rate of physical capital (δ) and the same saving rate s. The initial level of E, E0, is lower in country A than in country B.

   1. (a) Compare the steady-state levels of output per effective worker of these two coun- tries.

   2. (b) Assume now that at time 0 both countries are above the steady state and that k0A = k0B (i.e., the level of capital per effective worker at time 0 is the same in both countries) and continue to assume that the initial level of E is higher in country B than in country A. Draw log of output per capita for these two countries as a function of time (have time on the x-axis) starting in period 0 and show how they both converge to their respective balanced growth paths.

   3. (c) Do these countries converge to each other in output per capita? Defend your answer.

Answer 1

Given: Yt = Kt^α(LtEt)^1−α

Et+1 = (1 + g)Et

depreciation rate of physical capital = δ)

saving rate = s.

assuming growth in labor, n = 0 (since not given in the question)

Output per effective worker: yt ≡ Yt/LtEt ,

Capital per effective worker: kt ≡ Kt/LtEt ,

Consumption per effective worker: ct =Ct/LtEt

Investment per effective worker: it ≡ It/LtEt .

Yt = Kt^α(LtEt)^1−α, therefore,  Yt/LtEt = ( Kt/LtEt )^α ⇒ yt = kt^α

Ct = (1 − s)Yt

Ct/LtEt = (1 − s) Yt/LtEt,

ct = (1 − s)yt = (1 − s)kt^α

It = sYt

It/LtEt = s Yt/LtEt

it = syt = skt^α (Capital gain)

Capital loss = (n+g +δ)kt =  (g +δ)kt (since n=0)

Law of motion states that: change in capital stock (net capital stock), ∆kt = capital gain - capital loss = kt+1 − kt

∆kt = it − (g + δ)kt = skt^α − (n + g + δ)kt

If, it < (g + δ)kt ⇒ ∆kt < 0

it = (g + δ)kt ⇒ ∆kt = 0 (steady-state)

it > (g + δ)kt ⇒ ∆kt > 0

Now, at steady state: it = (n + g + δ)kt or sk_ss^α = (g + δ)k_ss

or solving for k_ss = [ s / (g + δ) ] ^1/1−α, where k_ss = steady-state capital per effective worker.

Since, s, g, δ and α is same for both countries, and steady-state level depends only on these factors, therefore, both countries will have same level of steady-state capital per effective worker and output per effective worker.

b) yt = Yt/EtLt or yt = [(Yt/Lt)/Et]

taking log both sides, we get,

logyt = log (Yt/Lt) - Log(Et)

differentiating w.r.t. to time t, we get

(1/yt)dyt/dt = d(log (Yt/Lt)/dt - (1/Et)dEt/dt

since at steady-state, LHS = 0, and (1/Et)dEt/dt = g, we get

d(log (Yt/Lt)/dt = g, if an economy is at steady state, than the output per workers grows at rate g.

Given at time 0, both countries are above the steady state and that k_0A = k_0B such that k_0A > k_ss and k_0B > k_ss

now, k_0A = k_0B, this implies, [(K/L)/E]_0A = [(K/L)/E]_0B

If E_A < E_B then (K/L)_0A < (K/L)_0B to maintain the equality.

therefore, Country B will have higher output per worker than Country A at steady state.

the transition graph will look like following

c) The countries will converge to same output per effective worker but not to same output per capita.

yt = Yt/(LtEt) or Yt/Lt = yt*E. We know that y_ss will be the same in both countries, but that E_B>E_A.

Therefore y_ss*E_B>y_ss E_A

This implies that (Y/L)_B > (Y/L)_A.

Thus, Country B will have higher output per worker than Country A at steady state as explained in part (b).