Question

d. Assume that the aggregate production function is given by: where Y is aggregate output, K is capital, L is the number of workers in the economy and E is the state of technology. Further assume that capital depreciates at a rate of δ, the rate of technological progress is g, the population is growing at a rate of n and the saving rate is s. I5 marks] i. Determine the scale of production? Suppose capital is increased by a factor of 8, while effective labour is held constant. What would be the effect on output? What does this imply about returns to capital? ii. [6 marks] iii. What is the investment per effective worker in this economy? [6 marks PLEASE TURN OVER

Answer 1

solution:

(d) Given

aggregate function is

Y = $\sqrt{}$ K(EL)²    ie Y = square root of K(EL)²

(i) The scale of production is:

Y = [K x (EL)²]^0.5

Let us increase Capital (K) by a factor of N. So, now K = N x K. New output will be

Y* = [N x K x (EL)²]^0.5 = (N)^0.5 x [K x (EL)²]^0.5 = (N)^0.5 x Y

Y* / Y = (N)^0.5, which is less than N.

So, as K increases N times, output increases by less than N times. There is decreasing returns to scale.

(ii) Suppose capital is increased by a factor of 8, while effective labour is held constant, the effect on output and this imply about returns to capital is:

Y = [K x (EL)²]^0.5

Capital (K) is increased by factor of 8. So, now K = 8K. New output will be

Y* = [8K x (EL)²]^0.5 = (8)^0.5 x [K x (EL)²]^0.5 = 2.83 x Y

Y* / Y = 2.83, which is less than 8.

So, as K increases 8 times, output increases by less than 8 times. There is decreasing returns to scale, implying that as K increases, there is decreasing returns to capital.

(iii) The investment per effective worker in this economy is:

Aggregate function = Y = √K(EL)2

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(EtLt)2

Yt / LtEt = Kt (LtEt)2^1/2 = Kt;

Therefore yt = kt

Ct = (1 – s)Yt

= Ct / LtEt

= (1 – s)Yt / LtEt

= ct

= (1 – s)

It = sYt = It / LtEt   = sYt / LtEt  = it = syt = skt
  

Therefore, Investment per effective worker = skt