**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 labor. The inputs are the number of hours of skilled workers,***L*, the number of hours worked by unskilled employees,_{S}*L*, and the output is the number of yards of brick wall,_{U}*Q*.**b) The firm needs to build 100 yards of a wall. 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 a wall? Illustrate your answer on the graph in part (b).**

- a) The production function is
*Q*=*L*+ ½_{S}*L*where_{U}*L*denotes hours worked by skilled workers and_{S}*L*denotes hours worked by unskilled workers. Both types of labor are perfect substitutes._{U}

- b) The isoquant is a straight line.
- c)
*MP*;_{Ls}/w_{s}= 1/15*MP*. Thus, the “bang for the buck” is higher for skiled labor, and the firm will use only skilled labor._{Lu}/w_{u}= 0.5/8 = 1/16

Note that the total cost of building 100 yards with skilled labor is (100 hours)($15/ hour) = $1500.

The total cost of building 100 yards with unskilled labor is (200 hours)($8/ hour) = $1600.The isocost line representing a $1500 expenditure is drawn as a dotted line in the graph in (b). The isocost line is more steeply sloped than the isoquant in the graph because the marginal rate of technical substitution of unskilled labor for unskilled labor is equal to ½, while the ratio of input prices is equal to 8/15.

**A paint manufacturing company has a production function***Q***=***K***+ √***L***. The firm faces a price of labor***w*that equals $1 per unit and a price of capital services*r*that equals $50 per unit.**a) Verify that the firm’s cost-minimizing input combination to produce***Q***=****10 involves no use of capital.****b) What must the price of capital fall to in order for the firm to use a positive amount of capital, keeping***Q*at 10 and*w*at 1?**c) What must***Q*increase to for the firm to use a positive amount of capital, keeping*w*at 1 and*r*at 50?

- a) First, note that this production function has diminishing
*MRS*. The tangency condition would imply that or_{L,K}*L*= 625. Substituting this back into the production function we see that*K*= 10 – 25 = –15. Since the firm cannot use a negative amount of capital, the tangency condition is not valid in this case.

Looking at the corner with *K = *0, since *Q = *10 the firm requires *L = Q*^{2} = 100 units of labor. At this point, *MP _{L}* /

*w*= (1/20)/1 = 0.05 >

*MP*1/50 = 0.02. Since the marginal product per dollar is higher for labor, the firm will use only labor and no capital.

_{K}/ r =

- b) The firm will use a positive amount of capital when , or Thus
*L =*25*r*^{2}. From the production constraint*K = =*10 – 0.5*r*.*K >*0 then we must have 10 – 0.5*r >*0, or*r*< 20.

- c) Again, using the tangency condition we must have Therefore, since
*r*= 50,*L*= 625. From the production constraint, the input demand for capital is*K = =**Q –*So if*K >*0 then we must have*Q >*25.

**firm’s production function is Q = min(K , 2L), where Q is the number of units of output produced using K units of capital and L units of labor. The factor prices are w = 4 (for labor) and r = 1 (for capital). On an optimal choice diagram with L on the horizontal axis and K on the vertical axis, draw the isoquant for Q = 12, indicate the optimal choices of K and L on that isoquant, and calculate the total cost.**

The isoquant Q = 12 is shown for this Leontief technology. To produce Q = 12, the firm will need at least K = 12 and L = 6. This will cost the firm C = wL + rK = 4(6) + 1(12) = 36. The isocost line representing an expenditure of 36 is drawn. The optimal basket of inputs is A.

**Suppose a production function is given by***Q***=****10***K***+****2***L*. The factor price of labor is 1. Draw the demand curve for capital when the firm is required to produce*Q***=****80.**

With this production function the firm views *K* and *L* as perfect substitutes.

The firm will be at a corner point with *K* = 0 when *MP _{K}/r* <

*MP*, or when 10/

_{L}/w*r*<2/1, or when

*r*> 5.

The firm will be at a corner point with

*L*= 0 when

*MP*>

_{K}/r*MP*, or when 10/

_{L}/w*r*>2/1, or when

*r*< 5. When the firm needs to produce

*Q*= 80, how much capital will it need? The production function shows that 80 = 10

*K*, or

*K*= 8 units.

When r = 5, the firm might use any combination of

*K*and

*L*along the isoquant 80 = 10

*K*+ 2

*L*. The firm might therefore use any

*K*such that 0

__<__

*K*

__<__8.

The graph of the demand for labor is as shown.

**Consider the production function***Q***=***K***+ √***L***. Derive the input demand curves for***L*and*K*, as a function of the input prices*w*(price of labor services) and*r*(price of capital services). Show that at an interior optimum (with*K >*0 and*L >*0) the amount of*L*demanded does not depend on*Q*. What does this imply about the expansion path?

The tangency condition implies that . Clearly the demand curve for *L* is not a function of the level of output, *Q*. Therefore, as the level of output changes, the amount of labor is constant. Therefore, if we were to graph isoquants with labor on the horizontal axis, the expansion path for labor would just be a straight, vertical line.

The demand curve for capital can be derived by substituting the demand curve for labor into the production function. That is, .