P2:  A steel producer possesses the production function

 

 y = (10)(K.25 )(L.75 )

 

 where y is the quantity of steel, in metric tons, K is the firm’s capital stock, and L is the number of labor hours it employs (measured in thousands).

 

                (a) Assume the firm’s current capital stock is fixed at K = 4096. If the firm employs L = 81  labor units, then how much steel does it produce?

 

If K = 4,096 and L = 81, then y = (10)(4,096.25 )(81.75 ) = 10(8)(27) = 2,160 tons.

 

(b) Suppose that the firm receives an order to produce 270 metric tons of steel. If its capital stock is fixed at K = 4096, how many hours of labor must it employ to satisfy the order?

 

 Here, y = 270 and K = 4,096.   L is the unknown.  According to the production function, the firm must pick L so that

                 270 = 10(4,096.25)(L.75)

 

Simplifying, L = 5.06.

 

 

(c) In the long run, the firm is free to vary both its capital stock and the number of worker hours it employs. Given this, what happens to the firm’s output if it doubles the amount of labor and capital it uses?  What happens if it quadruples the amount?  What does this result suggest in general?

 

 

Let K0 and L0 be given.  The firm then produces y0 = (10()K.25)( L.75)  units of output.

 

If it then doubles these inputs, it produces y = 10(2K.25)(2 L.75)  units of output.

 

 

Which implies doubling the inputs doubles the firm’s output. If the firm quadruples its inputs, then afer running through the same steps, we see quadrupling the inputs quadruples the output.

 

 

P3.

 

The facts are as follows: all markets are competitive; the product price is $p = $1 per unit; the wage rate is $W = $12 per hour; the firm’s production function is given by y = L∙(32 − L), where L is the level of employment; and the firm’sfi􀄗xed costs are zero.

 

(a) Write down the firm’s profits in terms of L.

(b) What is the marginal-revenue product and marginal cost of labor?

Hint: MPL = 32 − 2∙L.

(c) Use your answer to (b) to calculate the profit-maximizing level of employment, denoted L*. What are the firm’s maximum profiits?

(d) Suppose that the 􀄗rm employs L* + 1 hours. What happens to its pro􀄗ts?

 

 

(a) Π = p∙y − w∙L. But, y = L∙(32 − L), soΠ = p∙L∙(32 − L) − w∙L where, p = $1 and $W = $12.

 

(b) Under competitive conditions, the marginal cost of labor is simply the wage MCL = W.

 

MRPL = p∙MP.

 

 Using the hint, this implies MRPL = p∙(32 − 2L) = 2∙p∙(16 − L).

 

The profit-maximizing level of employment, L*, is governed by the condition MRPL = MCL.

 

W = MCL   =   MRPL = 2∙p∙(16 − L*)

 

(1/2)∙(W/p) = 16 − L*

 

From which,

 

L* = 16 − (1/2)∙(W/p)

 

In this particular problem, (W/p) = (12/1) = 12, so L* = 16 − 6 = 10.

 

Π = 1∙10∙(32 − 10) − 12∙10 = $100

 

(d) If the firm employs L = L* + 1 = 11 workers, then

 

Π = 1∙11∙(32 − 11) − 12∙11 = $99,

so its profits are lower.

 

 

P3.8  Sorry about the problems I had with this on Friday.  Right before class I “cheated” and looked at the answer key so that I would not have any problems.  Never trust an answer key.  They confused nominal and real wages and I tried so hard in class to make the answer the same as the one in the key.  Which, if they mixed nominal and real, will never happen.

 

This is what they wanted.  I will do the problem as stated in class.

 

Suppose that the firm possesses some monopsony power and that it faces the labor supply curve L = 2W.  Assume the product market is competitive and that its product price is $p = $10. Finally, suppose that its marginal product of labor schedule is given by MPL = 10 − (0.1)L.

 

(a) Evaluate its MRPL and MCL schedules.

 

The product market is competitive, so that the marginal revenue product of labor takes a standard form MRPL = p∙MPL. We are told that $p = $10 and that MPL = 10 − (0.1)L, so MRPL = 100 − L.

 

 

The firm is a monopsonist, so that its marginal cost of labor is MCL = W + WL ∙L, where WL is the slope of the wage requirement’s schedule.  Here, the trick is to observe that W(L) = (0.5)L (because L = 2W). Therefore, WL = 0.5. Using this gives, MCL = 0.5L + 0.5L = L.   ****This is where I was confused.  The problem says little w, but it should have been big W*******

 

 

(b) Characterize and calculate the optimal level of employment, L*, and the optimal wage,

$W*.

 

As we have seen many times the optimal level of employment, L* is governed by the

condition MRPL = MC

 

so            100 − L* = MRPL         =         MCL = L*

 

 L* = 100/2 = 50.

 

 W* = $25.

 

 

 

(c) What would happen to the optimal level of employment if a union imposes a wage 􀄚oor

of $40?

 

 Suppose that a union imposes a wage 􀄚oor of W0 = $40. In this case, the monopsonist can hire up to L = 2∙40 = 80 workers at the constant wage of $40 per person. It follows that the marginal cost of labor in this range is then $40. The marginal revenue product of labor is unchanged, so the optimal level of employment, denoted L*, is governed by

 

100 − L*0 = MRPL          =           MCL = 40.

 

It follows that, L* = 60.

 

In this case, the imposition of the wage floor raises the monopsonistic firm’s optimal level of employment!

 

2.  The U-shape of each indifference curve tells us that the individual prefers more consumption to less for any given level of leisure.

 

As for the individual’s optimal choice, this is governed by the standard tangency condition between the

budget line and the highest attainable indifference curve.

 

The only effect of the U-shape is that the individual is always predicted to be a labor-force participant. The reason is that a non-participant is somebody for whom, at the maximum, the slope of his indifference curve exceeds the slope of the budget line (in absolute value). This cannot happen with U-shaped indifference curves.

 

4.  A thorough analysis of this issue requires a sophisticated dynamic analysis. Nevertheless, it is possible to go far with little. Intuitively, the loss in home equity values represents a decline in individual wealth or non-labor income .   This results in a parallel shift of the budget line. Given that leisure is a normal good, the prediction is that individuals will work harder following the decline in their wealth.

 

8.  The simplest way of tackling this problem is as follows. Once again, suppose that an individual’s utility depends on his consumption and leisure according to u = u(c, ℓ). His budget constraint is c = A0 + W∙h, where A0 is initial wealth, and h denotes the number of work hours.

 

Once commuting time, Q , enters the picture, we have T = ℓ + h + Q , which, in turn, implies h = T − Q − ℓ. this fact allows us to write the budget constraint as c = A0 + W∙(T − Q − ℓ) = {A0 − W∙Q} − W∙(T − ℓ).

 

This is essentially the same as the standard budget constraint described in the text but for the appearance of the shi􀄛 term − W∙Q (notice that the slope of the line is still − W).  Hence an increase in commuting time, Q , is qualitatively similar to a decrease in wealth.  Therefore the individual is predicted to reduce his leisure as Q rises. The effect on total hours worked, h, depends on the strength of the income effect.

 

 

10.  The kink in the budget line occurs to the left of the worker’s current leisure choice.  Therefore, the overtime payment either has no effect (because the worker is better off remaining with his initial choice), or it encourages him to reduce his leisure (because the new tangency is located to the le􀄛 of his current choice, on the new steep part of the budget line). Before, the overtime payment can never reduce the number of hours this worker chooses to work.