Question

Kaldor (1961) documented a set of stylized facts on the growth process of industrialized countries. We discussed these facts in lecture 2. Explain if and how the Solow-Swan model can rationalize each of these facts. You are allowed to focus on the steady state equilibrium of the model. Every statement needs to be supported with a mathematical equation. You can refer to the notes of lecture 5 and 6 for this.

a. Output per worker Yt/Nt grows at a rate that does not diminish.

b. Physical capital per worker Kt/Nt grows over time at a constant rate.

c. The ratio of physical capital to output Kt/Yt is constant. In the following exercise, you are allowed to assume that the production function F(K, AN) satisfies the Inada conditions. A firm chooses capital and labor to maximize its profits: max Kt,Lt F(Kt , AtNt) − wtNt − r K t Kt where wt and r K t denote the wage and rental rate of capital, respectively.

d. Derive the firm’s first order conditions with respect to capital and labor. Interpret these equations.

e. Under constant returns to scale, we can write F1(Kt , AtNt) = F1 ( Kt AtNt , 1) and F2(Kt , AtNt) = F2 ( Kt AtNt , 1). 1 Show that the rate of return to capital r K t is constant in the steady state equilibrium

Answer 1

The Solow growth model is constructed around 3 building blocks:
1. The aggregate production function: Y (t) = AF (K (t) , N (t)), which it is assumed to satisfy the following series of technical conditions:
a)FK > 0, FN > 0
b)FKK < 0, FNN < 0
c) AF (λK, λN) = λAF (K, N)
d)lim_K→0FK = lim_N→0 FN = +∞.
For example, one production function that satisfies these properties is the Cobb-Douglas production function :
  F = AK (t)α N (t)1−α ........................... (1)
The constant A captures T F P and should be interpreted in a broad sense: it incorporates the effects of the organization of markets and production on the efficiency with which factors are utilized.

2. The law of motion for the stock of capital. In discrete time we have: Kt+1 = It + (1 − δ) Kt
K (t + Δ) = I (t) Δ − (1 − δΔ) K (t)
where the the flow-variables variables are the ones multiplied by the length of the interval. Dividing both sides by Δ and taking the limit as Δ → 0, we arrive at·
K(t) = I (t) − δK(t) .................................(2)
3. The savings/investment function. It is assumed to be of a Keynesian nature, i.e. savings (and investment in a closed economy) equals a constant fraction s of total income Y (t) , or
S (t) = I (t) = s Y(t) .................................(3)

Let’s also assume that population grows at rate n, i.e. N (t) = N0e^nt ........(4)
Taking logs of both sides to obtain ln N (t) = lnN0 + nt
and take the derivative with respect to time of both sides which yields
N(t)^{^}/N (t) = n
In growth theory we are interested in the determination of income per capita (or per worker), as a measure of welfare of society. Thus, it is convenient to express all the variables of interest in per capita terms. Let’s use small letters to denote per capita (or per worker) variables.
We start transforming the model from income per capita:
y (t) =Y (t)/N (t)=AK (t)^α N (t)^1−α/N (t)= [AK (t)/N (t)]^α= Ak (t)^α
Next, the law of motion for capital K(t)^{^}/N (t)=I (t)/N (t) − δK(t)/N (t) ⇒K(t)^{^}/N (t)= i (t) − δk (t)..................(5)
Now,
k(t)^{^} =∂k (t)/∂t =∂ (K (t) /N (t))∂t=[K(t)^{^} N (t) −·N(t)^{^} K (t)] /N (t)² =K (t)^{^}N (t) −·N(t)^{^}/N (t)*K (t)/N (t)...............(6)
k(t)^{^} =K(t)^{^}/N (t) − nk (t) ⇒K(t)^{^}/N (t)= k (t)^{^} + nk (t)
which substituted into the law of motion for capital (5) gives
k(t) = i (t) − (δ + n) k (t) ...................(7)
Hence the growth rate of the labor force N (t) works like an additional source of depreciation.The reason is for a given stock of capital K (t), the larger the population, the lower the stock of capital per worker.
This is a nonlinear first order differential equation in the variable k (t) that describes the process of capital accumulation in the economy: capital accumulation is the engine of growth, since output per capita depends on the per-capita capital stock k (t) . According to Solow, to understand the growth process, we need to understand the reasons for capital accumulation.