Question

Consider an economy with only one good, wheat, and no money. Wheat can be planted in one period in order to produce more wheat in the next period. You want to grow some wheat but you have none, so you need to borrow some to get started. The quantity of wheat that you will produce in the next period (output) as a function of the quantity that you plant this period (investment) is as follows.

Investment            Output

1                               3

2                               5.5

3                               7.7

4                               9.7

5                               11.5

6                               13.1

7                               14.5

8                               15.8

9                               17

10                             18.1

11                             19.1

12                             20

a. What is happening to the marginal product of investment as you invest more?

b. Let’s say that you only care about how much wheat you have in the next period. If you can borrow wheat at zero interest (for each unit borrowed this period, you must repay one unit next period), how much should you borrow and invest in order to maximize next-period wheat? [Hint: if it is not immediately apparent to you how to solve this, simply figure out the quantity of wheat you would have next period after you repay the loan for each level of investment.]

c. Now imagine that the (real) interest rate is 0.1 (10%) so that for each unit that you borrow today you must repay 1.1 units next period. How much should you invest to maximize next period’s wheat? What if the interest rate is .2? .3?

d. Your answers to parts b and c above amount to a function (technically a schedule) giving the quantity of wheat you are willing to borrow for different levels of the interest rate. What is this called? Show it on a graph.

Answer 1

Answer--

This is the solution.All parts are solved.

a) if, we invest more and more marginal product of investment is declining , which can be computed as

output(t+1) - output(t) , for some time period t

b) investment output real output(output - n) n=(1,12)

1 3 3-1=2

2 5.5 5.5-2 = 3.3

3 7.7 4.7

4 9.7 5.7

5 11.5 6.5

6 13.1 7.1

7 14.5 7.5

8 15.8 7.8

now he shoud borrow only 7 units , as output is maximum there . As he borrows 8 units , real output generated is 7.8 units which is less than 8 units..

c) r=0.1 n=(1 ,12)

borrow output real output(after paying n*1.1 )

1 3 3-1.1=1.9

2 5.5 5.5 - 2.2=3.3

3 7.7 4.4

4 9.7 5.3

5 11.5 6

6 13.1 6.5

7 14.5 6.8

now note that real output is less than. investment/borrowings.implying negative returns.

hence at maximum ,he should invest 6 units.

d) when interest rate = 0 , person invests in 7 units

when r= 1 , person invests , in 6 units

we can conclude , as interest rate increases, person is willing to borrow less,hence negative relationship of investment and interest rate is confirmed.