Question

I've already posted the question 1 and got answers. Now I need to solve question 2 that might be related to Q 1.

1. Tom is a farmer. Tom grows wheat. The quantity that he produces is determined by the following production function:

Y=10I^1/2

Where “I” stands for investment and is the quantity of wheat that he plants and Y stands for output and is the total amount of wheat he harvests. (Recall that the ½ exponent means square root.)

a. Now imagine that in year 1 Tom plants 1 unit of wheat and then each year after that Tom plants half of the wheat that he produced in the previous year. Fill in the following table giving the amount of wheat he plants and harvests in each year. (I got you started.)

Year                         Investment (Planted)                            Output (Harvested)

1                                                1                                                                 10           

2                                                5

3

4

5

6

7

b. Plot these points on a graph with Year on the horizontal axis and Output (Y) on the vertical axis. Make your vertical axis go from 0 to 100. Draw a smooth curve between these points.

2. Now imagine that everything is the same except Tom plants 100 units in the first year. Fill in the following Table.

Year                         Input                        Output

1                               100

2

3

4

5

6

7

b. Plot these points on your graph above and notice what is happening to output over time in both cases.

Answer 1

The production function is

$Y=10I^{\frac{1}{2}}$

The table below gives the output for each level of I if I=100 to starts with,

Time	Input	Output
1	100	100
2	50	70.71068
3	35.35534	59.46036
4	29.73018	54.52539
5	27.26269	52.21369
6	26.10684	51.09486
7	25.54743	50.54446

In this case, it is evident that the output of the firm decreases overtime and asymptotically approaches 50 overtime. The figure below shows the time output plot

In these cases, the output approaches the steady state when 50% of the output is reinvested in production. This means no matter what input the firm starts with it will always end up producing 50 units in the long run.