Problem on production function

Question

Consider a model economy with a production function

Y = K^0.2(EL)^0.8,

where K is capital stock, L is labor input, and Y is output. The savings rate (s), which is defined as s = S/Y (where S is aggregate savings), is a constant. The aggregate savings finance aggregate investment (thus It = St). The population growth rate (n), growth rate of labor efficiency level (g), and depreciation rate of capital (δ) are all constants.

(a) Show that this production function indicates constant return to scale.

(b) Show that this production function indicates decreasing marginal product of labor (MPL).

(c) Define capital per efficiency unit worker (k=K/EL) and output per efficiency unit worker (y=Y/EL). Express y as a function of k.

(d) Find steady state levels of k and y (k* and y*). Note that steady state is defined as a state where k does not change over time. Thus, the economy is in steady state at period t if and only if we have k_t+1 = k_t (= k*).

(e) Suppose there are two countries, the developed North (N) and the developing South (S). The North has 48% savings rate (s=0.48) and 0% population growth rate (n=0). The South has 9% savings rate (s=0.09) and 6% population growth rate (n=0.06). Both share the growth rate of efficiency level of 1% (g=0.01) and depreciation rate of 2% (δ=0.02). What are the steady state level of y in the North and the South (y_N* and y_S*)?

Answer 1

(a) Given, the production function is Y = K^0.2 (EL)^0.8

In order to prove that this indicates constant returns to scale, the output in the production function, Y has to increase by the same proportion, which is used to increase all the inputs. In our case, if K and L increase by m, the output Y has to increase by m.

Suppose L and K increases by m, the new production function will be

Y’ = (mK)^0.2 (mEL)^0.8  = m^0.2+0.8 K^0.2 (EL)^0.8 = m K^0.2 (EL)^0.8 = m*Y

Hence the output also increases by m. Thus this production function indicates constant returns to scale.

(b) From the production function, Y = K^0.2 (EL)^0.8

The marginal product of labor can be derived as ΔY/ΔL = 0.8 K^0.2 (EL)^-0.2 = 0.8(K/EL)^0.2

From this derived equation, as L increases, the marginal product of labor will fall (since L is in the denominator). As more workers are hired, the extra output obtained from each additional new worker will fall as L increases and marginal product of labor will fall. Thus the production function indicates a decreasing marginal product of labor.

(c) As we defined k = K/EL and y = Y/EL, and we include our production function into it,

y = Y/EL = (K^0.2 (EL)^0.8)/EL = K^0.2/EL^0.2  = (K/EL)^0.2 = k^0.2
y = k^0.2

Thus y is expressed as a function of k

(d) Labor, L grows at the rate of n (population growth rate), efficiency of labor, E grows at the rate of g (growth rate of labor efficiency level and Capital stock, K is depreciating at the level of δ (depreciation rate of capital).  Since k = K / L *E, we can see how k changes over time:

dk = dK/EL – (K/EL²) dL - (K/LE²) dE
dk = (K/EL) dK/K – (K/EL) dL/L – (K/EL) dE/E
dk = kδ – kn – kg

Here the sign of kδ is also negative, since capital is consumed by depreciation (dK/K < 0).

In the steady state condition, Δk = 0

We also know that Δk = s*f(k) – δk
In our case, Δk = s*f(k) – (δ+g+n)*k
Since Δk = 0, s*f(k) = (δ+g+n)*k
k*/f(k) = s/ (δ+g+n)
k/k^0.2 = s/ (δ+g+n)
k^0.8 = s/ (δ+g+n)

This is the steady state level for k. Since we already know y = k^0.2 (from (c)), at steady state, y* = (k*)^0.2
Thus y* and k* are determined.

(e) All details given for North and South, they are as such substituted in k* and y*.

k_N^0.8 = 0.48/(0.01+0.02+0) = 0.48/0.03 = 16
k_N* = 32
y_N* = 2
k_S^0.8 = 0.09/(0.01+0.02+0.06) = 0.09/0.09 = 1
k_S* = 1
y_S* = 1

The steady state level of y in the North and the South are 2 and 1 respectively.