Question

Consider a closed (no trade) economy "I" with a fixed labor force equal to 1000 and a fixed capital stock equal to 100 (L=1000, K=100). There is a representative firm with a Cobb-Douglas production function that rents capital and hires labor to produce. Assume that TFP parameter equals one (A=1) , we have Y=K^1/3 L^2/3. Markets are competitive.

1. graph the following: plot output per capita on the Y axis and capital per capita on the x axis. and show with an X the point that characterizes the equilibrium. In this plot, output per capita (Y) and capital (k) are you variables, while all other are constant and equal to their assumed values.

2. graph 2: plot wages on the Y axis and capital per capita on the x axis. show with an X the point that characterizes the equilibrium. Plot wages w and k capital as your variables.

Consider an economy "II" with labor equal L=500 and capital to K=20. Assume that this economy has the paramater that equal ones (A=1), Y=K^1/3 L^2/3. First, assume each of these economies are in autarky, so capital cannot flow across countries.

1. show in your previous graph with a circle where the equilibium of economy II is.

2. assume now that these economies are open to capital flows and capital can move freely across them. which direction would you expect capital to flow? why?

Answer 1

Consider the given problem here the production function of economy1 is given by, “Y = K^1/3*L^2/3”. So, we can write it in per worker form.

=> Y = K^1/3*L^2/3, => Y/L = K^1/3*L^2/3/L = K^1/3*L^(-1/3), => y = k^1/3.

Now, given the production function, => Y = K^1/3*L^2/3, => MPL = dY/dL = (2/3)*L^(2/3-1)*K^1/3.

=> MPL= (2/3)*L^(-1/3)*K^1/3 = (2/3)*(K/L)^1/3 = (2/3)*k^1/3, => MPL= (2/3)*k^1/3.

Consider the following fig.

Now, for the “1^st country” the capital is “100” and the “labor” is “1000”, => “K/L=100/1000 = 0.1”, => the corresponding output per worker is given by, “y = k^1/3 = 0.46”. So, in the above fig at “k=k1=0.1” the corresponding output per worker is “y1=0.46” correspond the position of “country 1”.

So, here the real wage must be equal to the “MPL”.

=> W/P = MPL, => W/P = (2/3)*k^1/3, => w = (2/3)*k^1/3. Consider the following fig.

So, here “k = k1 = 0.1”, => the corresponding wage is given by, “w1 = 0.31” represent the “country1”.                 

Consider the above fig. Now, for the “2^nd country” the capital is “20” and the “labor” is “500”, => “K/L = 20/500 = 0.04”, => the corresponding output per worker is given by, “y = k^1/3 = 0.34”. So, in the above fig at “k = k2 = 0.04” the corresponding output per worker is “y2 = 0.34” correspond the position of “country 2”.

Consider the above fig. So, here “k = k2 = 0.04”, => the corresponding wage is given by, “w2 = 0.03” represent the “country2”.

Now, here we can see that “country1” has higher “capital per worker” compare to “country2”, => under free capital movement capital of “country1” will move to “country2” and will continue until perfect equality will established, => until capital per worker will same in both the country.