1

Chapter 8, Part C

Long Run Production Cost

Recall that in the SR, a competitive firm can adjust only the variable input, say labor. But in the long run, a firm can change the plant size or capital input.

Plant size = K

In agriculture, some farms are larger than others. It must be true for other sectors. 

Right. In many retail businesses, some firms are larger than others.

How does one choose an optimal plant size? 

I hear some firms are too aggressive and build big plants, which turn out to be too large. 

For instance, many small hospitals bought MRI scanners, but due to lack of patients, they were underutilized. As hospitals get larger, they could include cardiac care or cancer treatment services, which raise production costs.

Consider the Discrete case where the firm has only three options:

K1 = small

K2 = medium

K3 = large

The three plants will have their own average cost curves.

Ken!

Then depending on the output level, we can choose the optimal plant for the chosen output level.

Ken. That is how we generate the Long Run Average Cost Curve = minimum of SAC curves. 

2

In agriculture, plant choice seems to be simpler than hospital, which faces a question: to add a cancer section or not.

Right.

Let us consider the continuous case, where optimal plant size can be finetuned, depending on the output level.

In the continuous Case there may be hundreds or thousands of short run AC curves. Maybe the optimal plant size can be tailored to the desired output level.

In that case, we can generate LAC

LAC curve = the envelope of all SAC curves.

What do you mean by envelope? Imagine that each SAC curve is an egg shell. Basically, you put all eggs in one bag, and the contour of the bag (envelope) is the Long Run Average Cost (LRAC) curve.

I thought LRAC would be obtained Oxi! (Greek)

3

by connecting the minimum point of each SRAC curve.

The professor who derived the LRAC curve asked his secretary to perform precisely that task.

He found out that such a curve exists, but lies above the envelope, and hence cannot be the LRAC.

4

Long Run Profit Maximization

Problem

In the long run, firms can vary those inputs that are fixed in the short run. For instance, the plant size K as well as the labor input L can be varied in the long run.

I know how to do this!

If both K and L are decision variables, the firm will maximize

pQ wL rK− −

Subject to ( , ) 0.Q f K L− =

For any given output Q, the firm must minimize production cost.

I recall that the least cost technique requires

,L

K

f wf r

=

where /if f i= ∂ ∂ is partial derivative of f with respect to input i.

Exactly. 

The solution to the cost minization problem yields input demand function,

 

( , , ),L L Q w r= and ( , , ).K K Q w r= (2)

Substituting these input demand functions into the cost function yields

( , , ) ( , , ) ( , , ).C Q w r wL Q w r rK Q w r= +

5

The cost function looks complicated, because it depends on input prices, w and r, but for competitive firms, those are given. 

An excellent observation!  

Sometimes we ignore the input prices.  

But since we are dealing with long run decisions, which need to incorporate input prices. 

Thus the firm’s LR problem is to choose its LR equilibrium output Q to maximize

( , , ).pQ C Q w rπ = −  (3) 

OK, I know how to do this.  

First order condition is:

'( , , ) 0,p C Q w r− =

where '( , , ) /C Q w r dC dQ= and factor prices (w,r) are suppressed. (Otherwise, ∂C/∂Q =CQ should be used.)

The second order condition is

" 0,C− <

which is satisfied as long as the cost function is convex.

The solution to FOC is written as

* *( , , ).Q Q p w r= (6)

And the optimal plant size is obtained by substituting (6) into (2):

* ( *, , ).K K Q w r= (7)

However, your FOC (p = LMC) is not enough.

What do you mean? P = LMC is not enough?

In the SR, it is OK for a firm to lose money, but one cannot sustain loss indefinitely. Thus, LR equilibrium of a competitive firm requires that profit must not be negative, ( , , ),pq C Q w r≥ or

6

.CpQ

≥ (8)

Now, I can summarize the rule for LR profit maximization

If P ≥ min LAC, then choose K, L, and q such that

(1) P = LMC (2) LMC is increasing (SOC).

 

Jawohl!

LONG RUN SHUTDOWN DECISION

If P < min LAC, exit the market.

What if market price falls below LAC? Should the firm exit?

Of course.

LAC is known. Although the model assumes perfect information, in the real world, firms do not know the market price in advance.

If P falls below AC, the firm will incur a loss but continue to produce some output. However, if such a low price persists for several periods, the firm will eventually have to exit the market.

For instance, the oil price dropped from $140 per barrel to about $40 about year ago. Oil companies lost money but continued to produce oil, because they knew the oil price decline was temporary. The following illustrates LR plant decision.

7

Figure 11a.

In the above diagram, no units of measurement are specified. So we cannot tell whether LAC is U-shaped or not.

In general, there are three Common Shapes of LAC curves.

1. U shaped as above

2. Horizontal

3. Horizontal up to a point and then rising

As the firm expands output, how AC changes depend on the shape of the LAC curves.

Right.

LAC ⇓ , increasing returns to scale. ∃ economies of scale.

LAC ⇒ , constant returns to scale.

8

LAC ⇑ decreasing returns to scale, ∃ diseconomies of scale.

When is the LAC curve horizontal? A firm has constant cost if its long run marginal cost and average cost are the same for all levels of output.

Then a firm has increasing cost if its long run marginal cost and average cost rise with increases in output.

Right!

Likewise, a firm has decreasing cost if LR marginal and average costs fall as output increases?

No!

You should recall the relationship between AC and MC. If AC falls, it only means MC lies below AC. MC could be rising or falling!

In many industries, as the firm gets bigger, AC seems to decline at first.

Right.

That has been the case since the industrial revolution in the 1850s.

• Employees in a large firm can specialize more in the activities at which they have comparative advantage.

What if a firm continues to expand? Will AC eventually rise or fall?

Except some industries where natural monopolies prevail, AC curve generally rises beyond a certain point.

Why? • Complex problems or organizing and coordinating the activities of employees

• Increased difficulty to insure raw material inputs

• Increased difficulty in managing internal information flows

In some industries, AC curve may

9

have a flat bottom. E.g., agriculture.

In such industries, both large ans small firms can coexist.

Minimum Efficient Scale Empirical studies for a number of industries suggest that constant returns to scale occur over a significant range of output. The minimum output level at which average cost is minimized is called the minimum efficient scale.

10

Industry MES as a percentage of US demand

diesel engines 21-30

electronic computers 15

refrigerators 14.1

cigarettes 6.6

beer brewing 3.4

bicycles 2.1

petroleum refining 1.9

flour mills 1.4

shoes 0.2 (source: lost)

11

Long Run Equilibrium

The persistence of economic profits (P > AC) or economic losses (P < AC) is not a stable or equilibrium situation in the long run for a competitive industry.

If losses continue to be sustained by the representative firm in the industry, no firms enter the market. Moreover, the incumbent firms do not replace the worn out capital, and eventually leave the industry. Thus, π < 0 implies N ⇓ .

If economic profits continue to be made by the representative firm, then potential competitors begin to enter the market, hoping to get a share of economic profits. Thus, π > 0 implies N ⇑.

Figure

(c615)

A market in long run equilibrium provides no incentive for new firms to enter or old firms to exit. In long run equilibrium, π = 0.

12

Industry long run equilibrium occurs for the competitive industry when economic profits are zero and long run average costs of the representative firm are minimized.

p = LAC (free entry)

p = LMC (long run optimality)

LAC = LMC ⇒ p = min LAC.

SAC = SMC ⇒ p = min SAC.

13

Long run Industry Equilibrium

Figure

Therefore, p = min LAC = min SAC = LMC = LMC.

q* = long run equilibrium output of firm,

N* = Q/q*.

14

ENTRY AND EXIT Entry is relatively difficult in some manufacturing, mining and government-regulated industries. This is due to large capital cost in these industries.

Entry is easier in agriculture, construction, wholesale and retail trade, and service industries.

In these industries, some firms enter and others exit the market. These firms can make different decisions depending on their abilities. Here we are considering the net entry or exit.

LONG RUN ADJUSTMENT TO AN INCREASE IN DEMAND

15

Demand curve shifts to the right from D to D'.

Price rises. Profits are positive. More firms enter the market.

This entry shifts the supply curve from S to S'.

Entry stops when price is equal to the minimum of LAC.

Constant Cost industry is based on the assumption that as N increases input prices stay put.

SR: p increases, q increases, π increases.

LR: N increases, other things return to equilibrium.

16

SHOULD GENERAL MOTORS SPLIT ITSELF IN TWO? (BUSINESSWEEK) HTTP://WWW.BUSINESSWEEK.COM/MAGAZINE/CONTENT/09_14/B4125072696638.HTM Under one bankruptcy scenario, the automaker would create a "good GM" and a "bad GM," with Hummer and Saturn part of the "bad" company

By David Welch April 6, 2009