Question

2. Consider two countries: Mahaliaville and UWIville. Both countries have the same production: Y = K Neither country experiences population growth nor technological progress and both countries have a depreciation rate of 10%. Ma- haliaville saves 10 percent of ouput each year and UWIville saves 30% of output each year. (a) Find the steady-state levels of capital per worker, income per worker and consumption per worker for each country. (b) If both countries start with a capital stock per worker of 1, how long will it take before consumption per worker in UWIville is higher than consumption per worker in Mahaliaville? (Be sure to show your calculations in a table).

Answer 1

Answer:-

Given That:-

Mahaliaville and UWIville both countries have the same production.

$Y = K^{\frac{3}{4}} L^{\frac{1}{4}}$

Each countries have the same production,also depreciation rate is same as 10%. Saving rate of Mahaliaville is 10 percent and Saving rate of UWIville is 30%.

For country Mahaliaville and UWIville

$Y = K^{\frac{3}{4}} L^{\frac{1}{4}}$

Y KL

$\frac{Y}{L} = \frac{K^{\frac{3}{4}}}{L^{\frac{3}{4}}}$

y = (R)

y $\rightarrow$ output per worker

capital per worker

(R) +

Also,

$\delta$ = 10% = 0.1 in both the countries

S_Mahaliaville = 10% of Y

S_M = 0.1 Y

SM _ 0.1 - 1 1

S_M = 0.1 y -(ii)

S_{M $\rightarrow$ Saving per worker in Mahaliaville}

_Also

S_UWIville = 30% of Y

S_U = 0.3 Y

$\frac{S_{U}}{L} = \frac{0.3 Y}{L}$

S_U = 0.3 y -{iii)

S_{U $\rightarrow$ saving per worker in UWIville}

(a)

At Steady state:-

dR = 0 [Change in capital stock per worker]

$i - \delta R = 0$

$\Rightarrow i = \delta R$

Also, I = A [In closed Economy]

I/L = S/L

i = s [Investment per worker = Saving per worker]

Now,

In Mahaliaville :-

_{At Steady state:-}

_{$i = \delta R$}

_{$s_{M} = \delta R$}

_{$0.1 y = \delta R$}

_{$0.1 R^{\frac{3}{4}} = 0.1 R$ [From (i) and (ii)]}

_{$R^{\frac{3}{4}} = R$}

_{$\frac{R}{R^\frac{3}{4}} = 1$}

_{$R^{\frac{1}{4}} = 1$}

_{R = 1}

_{So capital per worker = 1}

_{$\rightarrow$ Income per worker = $y = R^{\frac{3}{4}}$}

_{= $y = (1)^{\frac{3}{4}} = 1$}

_{y = 1}

_{$\rightarrow$ Consumption per worker}

_{Y = C + I}

  _{$\frac{Y}{L} = \frac{C}{L} + \frac{I}{L}$}

_{y = c + i}

_{c = y - i}

_{s = saving rate}

_{So, c = y(1 - 0.1) = 0.9y = 0.9 $\times$ 1 = 0.9}

_{In UWIville}

_{At Steady State}

_{$i = \delta R$}

_{$s_{U} = \delta R$}

_{0.3 y = 0.1 R}

_{$0.3 R^{\frac{3}{4}} = 0.1 R$}

_{$\frac{R}{R^{\frac{3}{4}}} = \frac{0.3}{0.1}$}

_{$R^{\frac{1}{4}} = 3$}

$R^{*}_{U} = 3^{4} = 81 \rightarrow$ Capital per worker

Also, $y = R^{\frac{3}{4}}$

$y = 81^{\frac{3}{4}} = 27\rightarrow$ Income per worker

s = saving rate

c = (1 - 0.3)y = (1 - 0.3)27 = 18.9

(b)

Consuming in Mahaliaville

C_M = (1 - s)y

s = saving rate

y = output per labor

C_M = consumption per labor

$\because$ Saving Rate in Mahaliaville at the end of each year is 10% of y

So, C_M = (1 - 0.1)y = 0.9 y = 0.9 R^3/4

consumption in UWIville

C_U = (1 - s)y

s = saving rate

y = output per labor

C_U = consumption per labor

$\because$ Saving Rate in UWIville is 30% of y

C_U = (1 - 0.3)y = 0.7y = 0.7 R^3/4

Since, the production output/income in both the countries are same.

So, at any level of output/worker or capital per worker, consumption in UWIville is always less than consumption in Mahaliaville as UWIville has a higher saving rate than  Mahaliaville

$C_{M}(0.9 R^{\frac{3}{4}})$

R = 1, C_M = 0.9(1)^3/4 = 0.9

R = 2, C_M = 0.9(2)^3/4 = 1.681 $\times$ 0.9 = 1.513

R = 3, C_M = 0.9(3)^3/4 = 0.9 $\times$ 2.28 = 2.052 .........

$C_{U}(0.7R^{\frac{3}{4}})$

R = 1, C_U = 0.7(1)^3/4 = 0.7

R = 2, C_U = 0.7(2)^3/4 = 0.7 $\times$ 1.681 = 1.177

R = 3, C_U = 0.7(3)^3/4 = 1.595 .......

At every level capital per worker consumption per worker in UWIville will be leass.