Production

6 6.1

Firms Why Do Firms Exist?

Why do …rms exist? The fact that a …rm exists suggests that its particular form of organization is superior to some other alternative institution. Airlines are not mom-and-pop operations while most used book stores are. The alternative to …rms is the market. Markets cause people to specialize and cooperate, albeit usually unconsciously Firms, on the other hand, consciously force specialization and cooperation. In the words of Dennis Robertson, …rms are "islands of conscious power in the ocean of unconscious co-operation like lumps of butter coagulating in a pail of buttermilk." While workers have relative labor mobility to move from …rm to …rm, long-term contracts between individuals and …rms turn unconscious decision-making into conscious decision-making where workers are required to do what their employer says as long as they are in their employ. To illustrate the distinction, consider an example o¤ered by McCloskey (1985, The Applied Theory of Price). Why do grocery stores exist as …rms? Why don’t grocery stores operate like marketplaces, like the farmers market that operates in downtown Beloit on Saturday during the summers and fall. The butcher, the baker, the stockboys, the manager of the dairy section, etc., the checkout people, could all be independent contractors. Why do we not see this form of organization? As Nobel Laureate Ronald Coase put it, the fact that this situation does not exist suggests that "the operation of a market costs something and by forming and organization and allowing some authority (an ’entrepreneur’) to direct the resources, certain marketing costs are saved." In today’s terms, we would say that …rms exist to reduce the transaction costs of bargaining (and writing) contracts in ongoing markets. It might be costly (and nearly impossible) to specify the right contract between the butcher and the checkout person, since the checkout person is also working for the baker, and the manager of the dairy section, etc. By changing the form of organization to allow one …rm owner/manager to hire people she can order around the store, these transactions costs are eliminated in a way that leads to economic progress.

6.2

The Pro…t Motive

The theory of the …rm asks why …rms choose a certain scale of activity. Why does Beloit have 1250 students? Why does a particular farmer plant and harvest 200 acres of corn instead of 250? To most people, this question might appear stupid. However, as enlightened individuals with some economic training you realize that everyone does not do the most that they can do in every endeavor. Sometimes it is optimal to do a half-assed job. In the same manner, it is wrong to say that they farmer planted 200 acres because it was the most he could do. Surely if it was pro…table to do so he could hire additional help, purchase more land, work even longer hours to produce more. So the answer depends on the motiviation of the …rm’s owners. The farmer

64

might like only working a 60 hour week, or like not having to oversee any employees. He could also like the money that comes with planting 200 acres and realize he would lose money if he planted more land. When it comes to most business, we feel comfortable assuming that the primary motivator is pro…t. Alchian (1950) in his article "Uncertainty, Evolution, and Economic Theory" makes the argument that we should employ the pro…t maximization hypothesis not because it is what …rms actually do, but rather because …rms that do not approximate pro…t maximization will not survive. Thus as a description of …rm behavior in the long run the assumption of pro…t maximization will be correct.

6.3

The Uses of the Hypothesis of Pro…t Maximization

Pro…t-making is not the only motivation in society of course. However, even if you work only to provide for your family, making more money allows you to provide more for your family. What aobut areas of life where the pro…t motive is absent? Can we say something about these areas because of the lack of a pro…t motive? Well for one thing, we might observe managers or workes engage other tastes besides their desire for pro…ts. After all, if there is no residual claimant, people cannot maximize their utility by maximizing pro…ts. Instead individuals can indulge their tastes for 1) a quite life, 2) large o¢ ces, 3) socially responsible behavior, 4) socially irresponsible behavior, etc. For example, George Borjas (1978, JLE) found that the Department of Health, Education, and Welfare discriminated against women at a rate higher than the colleges and universities it was charged with investigating for discrimination. Why did we need laws to segregate buses in the South? Because some "unscrupulous" bus companies caring only about pro…ts, didn’t want to segregate their buses. Even if the majority white voters (and riders) wanted segregated buses, that was not enough to get the bus companies to segregate the buses. They had to use the power of the law to force the bus companies to do so.

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6.4

How Do Firms Seek To Do What They Want To Do (Make Money)

Just like individuals do, they do an activity until MB (marginal bene…t) = MC (marginal cost). MR, MC MC

Loss MR Profit

MC MR

Q*

Q^

Q

At Q*, MR=MC. At Q^, MC>MR, so at that point we are losing money on each sale. 6.4.1

Digression on the Derivatives and MR=MC Condition

Suppose we had a pro…t function that looks like this

66

Π

Π* Π2 Π1

Q1

Q*

Q

Q2

Mathematically, what can we use to …nd Q*, the point where the pro…t function is maximized? First of all, let us think about why we want to …nd this out. What we really want to know is what is the …rm’s pro…t-maximizing level of output (i.e., the output consistent with MB=MC). So we will want to keep producing as long as the is increasing when Q is positive.That is, when: Q

>0

(289)

Conversely, we want to scale back production when increasing production causes pro…ts to fall <0 Q Logically then, the pro…t maximizing level of output is when Q

=0

(290)

(291)

Well, all the total derivative tells us is the rate of change along the function when there are only two variables for very small changes in q. In fact derivates are just the change in one variable as an other variable changes by as the amount by which it changes (which we can denote by h) approaches zero. df lim f (q + h) d = = dQ dQ h!0 h 67

f (q)

(292)

With respect to pro…t maximization, Here we can think of the pro…t function as being pro…t as being a function only of how many units are sold. Mathematically, we can write this as = f (q) (293) So looking at a particular point, Q1 for example, we can see whether the pro…t function is increasing or decreasing at that point by looking at the derivative of the function at that point. d >0 (294) dQj Q=Q1 Which is obvious from the picture since pro…ts are increasing at point Q1 if we increase Q. What about at Q2? Well, as output increases pro…ts fall, so d <0 dQj Q=Q1

(295)

So to …nd the pro…t maximizing level of output (Q ) we just …nd

If

d =0 dQj Q=Q

(296)

d df = =0 dQ dQ

(297)

(q) = f (q), then

Mathematically, we can also think of cost as only being dependent upon the level of output we produce (not that unrealistic, think of an activity with zero …xed costs). We then can write the cost function as c (q) Revenue also can depend only on the amount sold (and usually price, but here we’ll ignore price for simplicity), making the revenue function R(q). The pro…t function is just whatever is left when costs are subtracted from revenues, or (q) = R(q)

c (q)

(298)

So …nding the point where the pro…t function is maximized is found by taking the total derivative of the pro…t function and setting it equal to zero. d dR = dq dq

dc =0 dq

Rearranging terms on the RHS, we see that dc dR = dq dq

d dq

(299)

= 0 (is optimized) when (300)

Which is our MR=MC condition, since dR = MR dq 68

(301)

and

dc = MC dq

(302)

Here is an example of a more realistic pro…t function with …xed costs (and thus losses). Profits

q*

q

Losses

6.5

Firm Supply

Firm supply comes from the production function, which speci…es the technical tradeo¤ that …rms can make in the production function between capital (K) and labor (L). A generic production function thus looks like this y = f (L; K) A production function with a speci…c functional form looks like this p y = LK

(303)

(304)

The production function de…nes our isoquant curves. Each isoquant gives you all the combinations of K and KL that give you the same quantity of Y. (Note: "iso" means "the same", hence "the same quantity"). So thinking of y = 1 we can …gure out what combinations of K and L achieve that level out output given the nature of the production function.

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L 1 1 2

2

K 1 2 1 2

Y 1 1 1

We can graph these points to give us our isoquant for y = 1 K

2

1

1/2

Y=1

1/2

2

1

L

We can do the same thing with y = 2, y = 3, and so on. Just as there are an in…nite number of indi¤erence curves radiating out from the origin, there are in…nite number of isoquants, limited only by the "lumpiness" of the production process.

70

6.6

ISOCOST Curves

Just as individuals are constrained by their budget, we can think of …rms being constrained by costs. So let’s say that the …rm has $100 to spend on K and L. If the price of K is r and the price of L is w, then wL + rK = c

(305)

Let c = $100. then solving for K 100 w L (306) r r What is the slope of the isocost curve? Taking the derivative of K with respect to L yields dK w = (307) dL r thus the isocost curve is a straight line with a slope of wr and a vertical intercept of 100 r . This is our isocost line for total costs equal $100. Isocost lines are all the combinations of K and L that cost the same. K=

K

100 r

Slope = - w r

100 w

6.7

L

Marginal Physical Product

De…nition 1 The Marginal Physical Product of an inupt is the additional output that can be produced by employing one more unit of that input while holding all other inputs constant. 71

There are generally two types of marginal physical products employed, they are: 1. Marginal Physical Product of Capital (often shortened to Marginal Product of Capital or M PK : 2. Marginal Physical Product of Labor (often shortened to Marginal Product of Labor, or M PL : We’ll use both of the shortened terms for these. How do we measure the marginal change of one variable on another holding all other variables constant? That’s right, using partial derivatives. We’ll …nd the following: @f (L; K) (308) @K @f (L; K) (309) M PK = @K An Example: Bob Tollision is currently using 5 units of labor and 10 units of capital. Given the following production function M PK =

y = f (L; K) = 600L2 K 2

K 3 L3

(310)

…nd his marginal product of labor. The …rst thing we do is plug in 10 for K, since we can only evaluate the marginal productivity of labor from a speci…c reference point (and holding all other factors constant). Thus, at K=10, the production function is f (L; K) = 60; 000L2

1000L3

(311)

Taking the partial derivative with respect to L and setting equal to zero yields @f (L; K) = 120; 000L @L

3000L2 = 0

(312)

Thus, moving from L=5 to L=6 we can expect output to increase by 120000 6 720000

3000(6)2

108000 = 612; 000

(313) (314)

So adding one more worker would while holding the level of capital constant would increase output by 612,000. Now if we solve the MPL equation for L we can …nd the point where output reaches its maximum value: L2 = 40L (315) L = 40

(316)

Thus L=40 is the point where output reaches its maximum value. That is, given K=10, the addition to output from the 40th worker is zero. So hiring a 41st 72

worker would actually lower output (think about people getting in each other’s way, etc.) What about K? Setting L=5 f (L; K) = 15; 000K 2 125K 3 (317) @f (L; K) = 30; 000K @K So moving from K=10 to K=11 yields: 30000(11)

375K 2 = 0

375(11)2 = 284625

(318)

(319)

Thus adding one more unit of capital from (5,10) increases output by 284,625 units (whatever they are). Note that this doesn’t tell us whether this is a good idea or not, because we haven’t factored in what a unit of labor or a unit of capital costs. This just tells us what the value of adding an additional unit of homogenous capital will give us. Solving the MPK equation for K gives us the point where output reaches its maximum value. 375K 2 = 30; 000K (320) 30000 375 K = 80

K=

(321) (322)

Thus K=80 is the point where output reaches its maximum value. The marginal product of labor is just the slope of the production function holding other other factors (K) constant.

73

Y Slope = MPL

L (# of workers)

7

Diminishing Marginal Productivity

The marginal physical product of a input depends on how much of it is used. We cannot keep adding fertilizer to a …eld, for example, before the grass starts getting greener and begins to die. The same goes with labor. At some point, more the productivity of each unit of labor begins to decline. Mathematically, diminishing marginal productivity of labor can be found by utilizing the second derivative. The second derivative is found by taking the partial derivative of the …rst partial derivative. Using, for example, our M PL equation from above: @f (L; K) = 120000L @L

3000L2

(323)

@ 2 f (L; K) = 120000 6000L (324) @L2 Has diminishing marginal productivity of labor set in yet? Let’s evaluate at L=5 @ 2 f (L; K) = 120000 6000 (6) = 90000 (325) @L2 L=5 Nope, since the second derivative is still positive. What about L=21? @ 2 f (L; K) @L2

= 120000 L=21

74

6000 (21) =

6000

(326)

Note that this is not telling us that we should not hire the 21st worker. What it is telling us is that the contribution to output of the 21st worker is 6000 fewer than the 20th worker. Note that if we took the second order condition and set it equal to zero we could get the point where diminishing marginal returns set in @ 2 f (L; K) = 120000 @L2 120000 = 6000L L = 20

6000L = 0

(327) (328) (329)

So at K=10, any workers beyond 20 add less to output than the preceding workers. What about capital? @f (L; K) = 30000K @K

375K 2

(330)

@ 2 f (L; K) = 30000 @K 2

750K

(331)

@ 2 f (L; K) @K 2

= 30000

750(10) = 22500

(332)

K=10

So diminshing marginal productivity of capital has not set in yet either. At what point will it decline? Let’s try K=41 @ 2 f (L; K) @K 2

= 30000

750 41 =

750

(333)

K=41

It would appear at K=40 diminishing marginal returns begin to set in. Note that this does not mean that additional units of capital do not increase ouput. That point is K=80. Diminishing marginal productivity is why the production function ‡attens out as more and more variable inputs are added (holding all others constant).

75

7.1

Technical Rate of Substitution (TRS)

De…nition 2 The Technical Rate of Substitution shows the rate at which labor can be substituted for capital while holding output constant along an isoquant. Mathematically, dK dL

T RSLK =

(334) y=y

We are going to prove that T RSLK =

dK dL

= y=y

M PL @f (L; K)=@L = M PK @f (L; K)=@K

(335)

Let us de…ne the a representative isoquant f (L; K) = y

(336)

Totally di¤erentiating the production function: @f (L; K) @f (L; K) dL + dK = dy @L @K

(337)

However, note that dy = 0 since we are treating it like a constant. Thus @f (L; K) @f (L; K) dL + dK = 0 @L @K

(338)

Rearranging: @f (L; K) @f (L; K) dL = dK @L @K

(339)

Rearranging some more: dK = dL

@f (L;K) @K @f (L;K) @L

(340)

M PL M PK

(341)

Which proves that T RSLK =

7.2

Pro…t Maximization (Generic Case)

What do …rms want to do? Maximize pro…ts. The …rm’s problem is to maximize output (y) subject to total costs (TC). Let denote pro…ts. What are pro…ts? They are what are left from revenues once all costs have been paid. Total revenues thus are equal to the amount sold (y) times the price at which they are sold. (p). So the …rm’s problem is to maximize y subject to TC. TR = p y (342)

76

Recall that the prduction function is y = f (L; K)

(343)

T R = pf (L; K)

(344)

So total revenues equal Then recall that a typical cost function looks like T C = wL + rK

(345)

Just as in the consumer theory, we can set up the producer’s optimization problem using the Lagrangian. L = pf (L; K) + [T C

wL + rK]

(346)

Taking the partial derivatives of the pro…t function with respect to L and K and ;we get our …rst order conditions (FOCs). @f (L; K) @ =p @L @L @ @f (L; K) =p @K @K

w=0

(347)

r=0

(348)

@ = T C wL + rK = 0 (349) @ Here I want to focus attention just on the …rst two FOCs. Let’s isolate w and r and then divide one by the other, yielding p @f (L;K) w @L = @f (L;K) r p @K

(350)

Prices cancel out and we are left with w = r

@f (L;K) @L @f (L;K) @K

(351)

Which is equal to w M PL = (352) r M PK This equation tells us that in order to maximize output, we must use inputs in a combination so that the ratio of their marginal products equals the ratio of their relative prices. Note the similarity to the fact that consumers must equalize the marginal utility of their consumption of a good relative to it’s price level across all goods consumed. Perhaps the relationship is clearer if we rewrite the above equation this way: M PL M PK = (353) r w The economic intuition behind this is if you are using one good that has a low M PK ratio relative to the MwPL ratio, you would be dumb not to shift your r production to utilize more of the product the input that gives you higher ouput per cost. 77

7.3

Cost Minimization (Generic example)

Most often in the real world entrepreneurs has a certain level of output to produce and wants to …nd the least costly way of doing so. For example, a developer might have space to build 10 homes and wants to miminize the costs of producting those 10 homes. So again we have a production function y = f (L; K)

(354)

and the …rms has to pay w in wages per worker and r to rent machinery so the total cost function is: T C = wL + rK (355) If the developer wants to minimize costs subject to a …xed level of output (y), then we can set this up as an cost minimization problem using the Lagrangian. L = wL + rK + [y

f (L; K)]

(356)

FOCs:

@L @f (L; K) =w =0 (357) @L @L @L @f (L; K) =r =0 (358) @K @K @L (359) = y f (L; K) = 0 @ Since both of the …rst 2 FOCS are equal to 0, we can set them equal to each other @f (L; K) @f (L; K) w =r (360) @L @K Simplifying @f (L;K) w @L = @f (L;K) (361) r @K

Note that this is the same optimality condition that we found from the pro…tmaximizing approach. Thus the optimality conditions to maximize output subject to a cost constraint are exactly the same as the optimality conditions when we minimize cost subject to an output constraint. Again, this is know as the principle of duality. We encountered this previously with the consumer optimization problem, but the principle of duality basically says that any constrained maximization problem has associated with it a dual problem in constrained minimization. Both approaches should yield the same answer, but which approach to use will depend on the circumstances at hand.

78

7.4

Pro…t Maximization (Numerical Example) y = f (L; K) = L0:5 K 0:4

(362)

108; 000 = 4000L + 3000K

(363)

L = L0:5 K 0:4 + [108; 000 4000L 3000K] @L = 0:5L 0:5 K 0:4 4000 = 0 @L @L = 0:4L0:5 K 0:6 3000 = 0 @K @L = 108; 000 4000L 3000K = 0 @L Dividing …rst two FOCs equal to each other: 0:5L 0:5 K 0:4 4000 = 0:4L0:5 K 0:6 3000

(364) (365) (366) (367)

(368)

Simplifying K = 1:33 L K 1:33 = L 1:25 K = 1:064L

1:25

(369) (370) (371)

Substituting into the 3 FOC: 108; 000

4000L

108; 000

3000 (1:064L) = 0

4000L

108; 000

3192L = 0

7192L = 0

7192L = 108; 000 108000 L= 7192 L = 15 108; 000

4000(15)

3000K = 0

(372) (373) (374) (375) (376) (377) (378)

3000K = 48000

(379)

K = 16

(380)

Let’s also calculate the value of . Using the …rst FOC: 0:5(15)

0:5

(16)0:4

4000 = 0

4000 = 0:39136 0:39136 = 4000 = 0:00009784

(381) (382) (383) (384)

Here = 0:00009784 says that if we relax our TC by one unit ($1), output will change by 0.00009784. Or if we increased our budget by $10,000 output would increase nearly one unit (0.9784) 79

7.5

Cost Minimization (Numerical Example)

Let’s approach this problem from the cost minimization side to illustrate the principle of duality. y = f (L; K) = (15)0:5 (16)0:4 (385) Gives us y = 11:741.So we can say what levels of K and L minimizes cost for that level of production. Setting up the Lagrangian: L = 4000L + 3000K +

L0:5 K 0:4

11:741

@L = 4000 0:5 L 0:5 K 0:4 @L @L = 3000 0:4 L0:5 K 0:6 @K @L = 11:741 L0:5 K 0:4 @ From the …rst two FOCs: 0:5 L 0:5 K 0:4 4000 = 3000 0:4 L0:5 K 0:6 K L K = 1:064L

1:3333 = 1:25

(386) (387) (388) (389)

(390) (391) (392)

Plugging into the 3rd FOC: 0:4

11:741

L0:5 (1:064L)

=0

(393)

11:741

L0:5 1:0251L0:4 = 0

(394)

0:9

1:0251L

= 11:741

(395)

L0:9 = 11:454

(396)

L = 15

(397)

K = 1:064(15)

(398)

K = 16

(399)

Solving for 4000

0:5 (15)

0:5

= 10; 221

(16)0:4

(400) (401)

Here lamda measures the change in the objective function (Total Cost) when we change production by one unit. To be more precise, it is the …rm’s marginal cost of producing one more unit of output.

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