Question

Consider twocompetitive economies that have the same quantities of labor (L= 400) and capital (K= 400), and the same technology (A= 100). The economies of the countries are described by the following Cobb-Douglas production functions:North Economy:Y= A L0.3K0.7South Economy:Y= A L0.7K0.3a.Which economy has the larger total production? Explain.b.In which economy is the marginal product of labor larger? Explain.c.In which economy is the real wage larger? Explain.d.In which economy is labor's share of income larger? Expla

Answer 1

1. A) Both economies have the same total production. Because the amount of capital and labour is the same, and the total production is calculated by replacing K=400 , L=400 and A=100 in the corresponding equations for each country, both countries will produce
   `Y=100*(400)^(0.3)*(400)^(0.7)=40,000`.
2. B)The south economy will have the larger marginal product of labor. This is because in this economy, the relative weight of labor in the production function is larger (0.7 in the south against 0.3 in the north economy). If you calculate the marginal derivative of product in terms of labour (wich mathematically express the concept of marginal product) in the south
   `Y_L^S`you will get
   `Y_L^S=0.7((K)/(L) )^(0.3)=0.7((400)/(400))^(0.3)=0.7` while in the north, the marginal product of labor
   `Y^N_L` will be
   `Y^(N)_L=0.3((K)/(L) )^(0.7)=0.3((400)/(400))^(0.7)=0.3`.
3. C) The real wage is expected to be larger in the economy where the productivity of labor (or marginal product of labor) is larger, which is the south economy. This is because in equilibria conditions, each factor of production's payment equals its marginal production. Then, if the marginal production of labor is greater in the south, it is expected that its payment (which would be equal to marginal production) is greater than in the countries with lower marginal production.
4. D) Total production or income (Y) would be used to pay factors of production's services. This can be expressed with the following equation:
   `Y=w*{L}+r*{K}` which means that product should be enough to pay the market price of the units of labor and capital used in the process. The market price is the wage ("w") and the return rate of capital ("r"). As mentioned before, in equilibria, factors are paid at their productivity rate (or marginal product), which in this case means that wage would be paid at
   `w_S=0.7` in the south and at
   `w_N=0.3` in the north. Then, from total production (which was calculated in point 1), labor would have a larger share on income in the south
   `s_L^S=\frac{L*{w}}{Y} =100*(400*0.7)/(40,000) =0.7`, because the marginal product of labor (hence its payment) is greater there, while in the north this share on income for labor would be
   `s_L^N=\frac{L*{w}}{Y} =100*(400*0.3)/(40,000) =0.3`.