Question

It costs a company $12 to purchase an hour of labor and $15 to purchase a unit of capital. If L hours of labor and K units of capital are available, then 0.05L 2/3K 1/3 machines can be produced. Suppose the company has $100,000 to purchase labor and capital. What is the maximum number of machines it can produce?

Note: before running Solver use some positive trial values for the decision variables; labor and capital inputs cannot be fractional and must be integers.

How does production vary with labor? Develop a data table. Let labor change from 1000 to 9000 in increments of 500; then insert a scatter plot according to your data table.

Background: It costs a company $12 to purchase an hour of labor and $15 to purchase a unit of capital. If L hours of labor and K units of capital are available, then 0.05L 2/3K /3 machines can be produced. Suppose the company has $100,000 to purchase labor and capital. What is the maximum number of machines it can produce? Note: before running Solver use some positive trial values for the decision variables; labor and capital inputs cannont be fractional and must be integers. Unit costs Labor Capital $12 $15 Production function: Production Constant *L3*Ko, where: b Constant а 2/3 1/3 0.05 Сapital Labor Inputs purchased Budget constraint Money Spent Budget $100,000 Amount of machines produced

Answer 1

Answer:

Given, and

Q=0.5LK

The cost of labor, w=$12 and cost of capital, r=$15,

Thus, C=wL+rK where C is the total cost and it is $100000 (given)

Therefore, 12L+15K=100000

Now we know that, Cost minimization condition of production is

MP, MPK ш

Where

تا انت == 1-10: تا انت

Similarly,

MK L = () *0.51$K$-1 ==+

Thus,

MPL w MIPS 历

Therefore,

22 K = 5

Now plugging the value of L in the cost function, we get

12L + 15K = 100000 => 12L + 15* (2/5) L = 100000

=> L = 5555,55556 5556

Thus,

K = 5556 * (2/3) 2222

Hence,

Q=0.5 * (5556)$(2222) = 0.5 * 313.69 * 13.05 – 2047

With $100000 the firm can produce 2047 units.

Again, given that

Q=0.5LK} = 0.5L (2L/5)

=>Q=0.5* L * * (2/5)3 * (L)} = 0.3684 * L

This the short run production function and labor and output variation is shown in the following table,

Short run Production Function 3500 3000 2500 2000 Output (Number of Machines) 500 2000 8000 10000 4000 6000 Quanlity of Labor (L)