Due Date: Thursday, September 8Th (At the Beginning of Class)

Problem Set 5 FE411 Spring 2007 Rahman

Due Date: Thursday, April 12th (at the beginning of class)

INSTRUCTIONS: Use your own paper to answer the questions. Please turn in your problem sets with your name clearly marked on the front page and all pages stapled together. You are encouraged to work together, but you must hand in your own work. You must show your work for credit and answer in complete sentences when appropriate (such as when the question asks you to “describe” or “explain”).

1) From Chapter 10 of Weil, do problems 2 and 3.

Chapter 10, Problem 2)

A 1 The level of productivity in Country X, relative to the United States is given as X  . AY 2 T The level of technology in Country X, relative to the United States is X  (1 g) G , TU .S. where g is the growth rate of technological progress and G is the number of years that Country X lags behind the United States. Given values of g = 1% and G = 20, we substitute and solve for the ratio of technology differences.

T X  (1 0.01)20  0.82 TU .S.

Since productivity is defined as T*E, we can divide the productivity level of Country X by the productivity level of the United States to obtain the ration of efficiency in both countries. E 0.5  X *0.82 EU .S.

Solving yields a value of 0.61 for the raion of efficiency in Country X relative to the United States. Stated differently, Country X has 61% of the efficiency of the United States.

of 5 Problem Set 5 FE411 Spring 2007 Rahman

E A Assuming that India  1, India  0.35 , and that the growth rate of technology is 0.81%, EU .S. AU .S. we write the functional productivity from of each country in ration form and substitute in the necessary values to obtain:

A E T T India  India * India  0.35  1* India  1*(1 g) G  (1.0081) G AU .S. EU .S. TU .S. TU .S.

Hence, we have an equation that we can solve for to find the magnitude of the technology gap in years between India and the United States. Taking the log of both sides and rearranging to solve yields,

 ln(0.35) ln(0.35)  (G)ln(1.0081), G   130.13 ln(1.0081)

The level of technology in India is, thus, approximately 130 years behind that of the United States.

2) Consider a country in which there are two sectors, called Sector 1 and Sector 2. The production functions in the two sectors are:

1/ 2 Y1  L1

1/ 2 Y2  L2

where L1 is the number of workers employed in Sector 1 and L2 is the number of workers employed in Sector 2. The total number of workers in the economy is L = L1 + L2 = 8. Workers can move freely between sectors. a. If workers gets paid their marginal products, calculate the number of workers employed in each sector (L1 and L2), and the wage that each worker earns (w1 and w2).

First note that because workers can migrate across sectors, wages will be the same. Further, since production in each sector is exactly the same, employment will be the same, so that L1 = L2 = 4. Take labor in sector 1:

1/ 2 1/ 2 w1  (1/ 2) *(L1 )  w1  (1/ 2) *(4)  1/ 4 . Thus workers in each sector earn ¼.

b. Now assume that sector 1 imposes a minimum wage of w1 = ½. This will be what workers in sector 1 earn. Now calculate the number of workers employed in each sector (L1 and L2), and the wage that workers in sector 2 earn (w2).

First calculate the labor that will be employed in sector 1:

1/ 2 1/ 2 . Thus we see that L1 = 1, which w1  (1/ 2) *(L1 )  1/ 2  (1/ 2) *(L1 ) automatically means that everyone else must work in sector two, or L2 = 7.

1/ 2 1/ 2 w2  (1/ 2) *(L2 )  w2  (1/ 2) *(7)  0.189 c. Use a diagram such as Figure 10.4 in the Weil text to analyze part b graphically. Be sure to highlight the amount of the inefficiency. In words, why does this create an inefficient allocation of resources?

Not shown (see Figure and discussion in text).

of 5 Problem Set 5 FE411 Spring 2007 Rahman d. Go back to the original set-up. Production is still described as above, there are no minimum wage constraints, and L = L1 + L2 = 8. However, now workers in sector 2 are paid their average products, while workers in sector 1 are still paid their marginal products. Calculate how many workers will work in each sector. In words, why does this create an inefficient allocation of resources?

1/ 2 Y2 L2 1/ 2 Average Product =   L2  w2 L2 L2

Again, workers are free to move between sectors. So in equilibrium, the wage of both sectors must be equal. Thus we set w1=w2 to solve for the number of workers in each sector.

1/ 2  1  1/ 2 w2  L2   L1  w1  L  4L  2  2 1

The umber of workers in sector 2 will be 4 times greater than the number of workers sector 1. Thus, with L = 8, L1 = 1.6, and L2 = 6.4.

3) From Chapter 11 of Weil, do problems 4 and 5.

In an economy perfectly open to the world capital market, the steady-state level of output per capita is given by equation 11.2 in the Weil text. Let’s call that yold. Given that the world rental price of capital doubles, or goes from rw to 2rw, we can write the new steady- state level of output per worker:

    1 1   1   1  1 1  1 1 1   1   ynew  A    A        yold  2rw   rw   2   2 

And with a value of ½ for α,

1/ 2  1 11/ 2  1  ynew    yold   yold  2   2 

Thus the new steady-state level of income per capita falls by half when the world rental rate of capital doubles in an economy open to the world capital market.

In a closed economy where one slice of cheese is consumed with one slice of bread, we can write a production function for each sector as Yc  Lc and Yb  (1/ 2)Lb , where c denotes the cheese sector, and b denotes the bread sector. In equilibrium, each must be produced equally. This, combined with the fact that Lc + Lb = 60, gives us Lb = 40, and Lc = 20. Therefore, the allocation of labor hours towards the bread sector will be 40, resulting in an output of 20 slices of bread, and the allocation of labor hours towards the cheese sector will be 20, resulting in an output of 20 slices of cheese.

With the economy opened up to trade where the price of bread equals the price of cheese on the world market, the economy is strictly better off when they specialize in the good that they have a comparative advantage in and trade for the other. Because the economy can produce two slices of cheese for every slice of bread, the economy can specialize in cheese production. 60 labor hours devoted to cheese production would lead to 60 slices of cheese and a dormant bread sector. In turn, 30 of those slices of cheese can be traded for 30 slices of bread on the world market. The economy can now consume 30 slices of cheese and 30 slices of bread, a consumption level strictly greater than in the closed scenario. Yeah!