Question

6. A construction company has two types of employees: skilled and unskilled. A skilled employee can build 1 yard of a brick wall in one hour. An unskilled employee needs twice as much time to build the same wall. The hourly wage of a skilled employee is $15. the hourly wage of an unskilled employee is $8.

a. Write down a production function with the inputs of LS and LU, where LS is the number of hours of skilled workers and LU is the number of hours worked by unskilled employees. The output, Q is the number of yards of brick wall.

b. The firm needs to build 100 yards of a wall. On the set of axes below, sketch the isoquant that shows all combinations of skilled and unskilled labor that result in building 100 yards of the wall.

c. What is the cost-minimizing way to build 100 yards of wall? Illustrate your answer on the graph in part b.

d. What is the level of Total Cost at this equilibrium?

Answer 1

a. The production function of number of yards of brick built in term of skilled and unskilled labor will be given by,

Q = LS + 1/2LU

Since Skilled labor can build 1 yard in one hour that's why the coefficient of LS is 1. And similarly the unskilled labor can build only 1/2 yard in one hour hence the coefficient of LU is 1/2. And Q is the total yard of brick built.

b. When Q = 100 yards, putting this into our production function we get,

100 = LS + 1/2LU

And we can plot this on a graph that will be our isoquant. As the name suggests isoquant shows the different combination of two factors here skilled and unskilled labor such the production remains constant.

In the above figure we have skilled labor LS on the Y axis and unskilled labor on the X axis. Since the isoquant curve is linear in LS and LU so in order for draw the isoquant all we need to do is to find the Y and X intercept.

Let's calculate the Y intercept.

To calculate Y intercept we need to put LU=0

Putting LU = 0 in isoquant equation we get,

100 = LS + 0

LS = 100

So we get Y intercept as 100. Similarly to calculate the X intercept we need to put LS =0. Putting LS = 0 in our isoquant equation we get,

100 = 0 + 1/2LU

LU = 100×2

LU = 200 so the X intercept is equal to 200. Now we have to draw a line through Y and X intercept.

C. The cost minimization way to build the 100 yard of wall will be such that it minimizes the overall cost of production.

Now see that each skilled labor costs $15 per hour and builds 1 yard per hour. And on the other hand each unskilled labor costs $8 per hour and builds 1/2 yard per hour.

So as you can see unskilled labor will take two hours to build one yard and two hours of unskilled labor will cost 8×2 = $16 while the skilled labor costs $15. So skilled labor is building each yard at low cost, so to build 100 yards only skilled labor should be employed.

So the cost minimization way of building 100 yards is to employ 100 skilled labors.

We can see this directly in the following figure.

Since the marginal productivity of skilled and unskilled labor is constant, so to build 100 yards of brick wall either skilled labor or unskilled labor will be employed based on the lower possible cost.

As you can see employing 100 skilled labor will cost 15×100 = $1,500 and employing 200 unskilled labors will cost 200×8 = $1,600. Since $1,500 is less than $1,600 so employing 100 skilled labor is less costly tham employing 200 skilled labor.

D. As we have already calculated in part C the cost of building 100 yards of brick wall be equal to, when the wall is being built by 100 skilled labors at a wage rate of $15 per hour.

= 100 × 15

= $1,500

I hope I was able to help you, thank you