--%>

Problem on production function

Consider a model economy with a production function

Y = K0.2(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 kt+1 = kt (= 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 (yN* and yS*)?

E

Expert

Verified

(a) Given, the production function is Y = K0.2 (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  = m0.2+0.8 K0.2 (EL)0.8 = m K0.2 (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 = K0.2 (EL)0.8

The marginal product of labor can be derived as ΔY/ΔL = 0.8 K0.2 (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 = (K0.2 (EL)0.8)/EL = K0.2/EL0.2  = (K/EL)0.2 = k0.2
y = k0.2

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/k0.2 = s/ (δ+g+n)
k0.8 = s/ (δ+g+n)

This is the steady state level for k. Since we already know y = k0.2 (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*.

kN0.8 = 0.48/(0.01+0.02+0) = 0.48/0.03 = 16
kN* = 32
yN* = 2
kS0.8 = 0.09/(0.01+0.02+0.06) = 0.09/0.09 = 1
kS* = 1
yS* = 1

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

   Related Questions in Macroeconomics

  • Q : National income how to calculate

    how to calculate national income under value added method

  • Q : Type of market when people cannot buy

    Whenever people can’t purchase all of a good they are willing and capable to pay for at present market price, there is surely a market: (1) Price ceiling. (2) Price floor. (3) Shortage. (4) Anomaly.  (5) Surplus. Please

  • Q : Balance of trade IN which situation,

    IN which situation, there is a deficit in the balance of trade.

  • Q : Ideas in which organization is involved

    Ideas in which organization is involved: Talking about the growth of any company. There are basically three type of broad ideas in which management of any organization is involved. These are: 1. Corporate Strategy<

  • Q : Net revenue when price increases Net

    Net revenue for Macho Man fake mustaches increases after the price raised from $5 to $7, pointing that demand faced by Macho Man was: (i) Relatively elastic. (ii) Relatively inelastic. (iii) Unitarily elastic. (iv) Perfectly inelastic. (v) Perfectly e

  • Q : Define revenue receipts Define revenue

    Define revenue receipts. Write the groups in which they are categorized. Answer: Any receipts that do not either make a liability or lead to reduction in assets is

  • Q : How central bank reduce the

    Describe any two measures by which a Central Bank can attempt to decrease the gap. Answer: Central bank can decrease this gap by adopting two measures illustrated b

  • Q : Paradox of Value problem I have a

    I have a problem in economics on Paradox of Value problem. Please help me in the following question. The Diamond Water Paradox occurs from the difficulties in differentiating between: (i) Consumer surplus and the total utility. (ii) Total utility and

  • Q : Calculating exchange rate for USA dollar

    If $9 is required to buy £2, what is the exchange rate for USA dollar? Answer: £1 = 9/2 = $4.5, i.e., £1 = $4.5.

  • Q : Define Depreciation Depreciation of a

    Depreciation of a currency signifies fall in value of domestic currency in terms of foreign currency. Illustration: When value of rupee in terms of US dollars falls, state from Rs. 45 to Rs. 50 per dollar, it will be a condition of depreciation of Ind