29 March, 2006
Reading: Ch 5. pp. 186-192
Problem #11: Due Friday
Lecture 26
REVIEW___________________________________________________:
IV. Chapter 5. The Production Process and Costs
C. Costs
2. Short Run Costs.
c. Algebraic Forms of the Cost Function.
d. Sunk costs vs. Variable Costs
Preview__________________________________________________________
3. Long Run Costs
a. Long run average costs
b. Economies of Scale, Diseconomies of Scale and Long Run
Survivability
4. Multi-Output Cost Functions
Lecture____________________________________________________________
C. Costs and the Theory of the Firm
3. Long-Run Costs. In the Long run, all costs are variable. As I indicated at the outset of
this chapter, the long run may be viewed as the planning horizon, since the project for the
firm is to pick the optimal plant size. Information about the Long-Run Average Cost
curve is very useful for determining the structure of an industry.
a. Long Run Average Costs. The long run average cost curve (LRAC) is the
envelope of all short run costs curves. That is, the LRAC is the tangency of all
efficient production points on for each plant size.
ATC0
ATC1
ATC2
ATC3
Economics of
Diseconomies of Scale
Scale
Efficient Plant Size
Q
In the above chart, the bold line is the LRAC. Note that the efficient scale of operation is
not at the point of minimum marginal costs unless the firm is at an optimal plant size.
b. Economies of Scale
Economies of Scale arise when LRAC falls as the plant expands.
Reasons for Economies of Scale:
- Physical Relationships (double a pipe diameter, and flow through
expands quadratically)
- Gains from specialization and division of labor
Diseconomies of scale arise when LRAC increases as the plant expands.
Reasons for Diseconomies of Scale
Transportation Costs
“Crowding” with respect to managerial efficiency
Minimum Efficient Scale (MES): The first point where LRAC is at a minimum
If a range exists where costs neither increase nor decrease, there exist
constant returns to scale.
Application: The shape of the LRAC can determine how many firms can survive in an
industry.
- Suppose that MES is 10,000 units, and that market demand, at a competitive
price is 40,000 units. How many efficient firms can survive in the industry?
- Suppose that MES is 10,000 units, but that diseconomies of scale set in very
soon after achieving the optimal plant size. Can a single firm efficiently service
the industry?
- Suppose an industry is characterized by continuous diminishing returns to scale.
What is the optimal industry structure?
c. Measuring Returns to Scale. The notion of scale economies and diseconomies
is related to production, and can be measured with a production function.
A firm is said to enjoy increasing returns to scale if a constant increase in all
inputs causes a more than proportionate increase in output. A firm that enjoys
economies of scale does because it realizes increasing returns to scale
The firm enjoys constant returns to scale if a constant increase in all inputs causes
a constant increase in outputs. A firm with a range of plant choices at MES will
enjoy constant returns to scale over this entire range.
The firm has decreasing returns to scale if a constant increase in outputs causes a
decreasing increment to outputs. Firms with diseconomies of scale may suffer
decreasing returns to scale.
Increasing, constant or decreasing returns to scale can be determined analytically.
Here we introduce this by focusing on a specific form of production function
Q
=
F(K,L)
=
(KL)a
Consider a production function consisting of both Capital and Labor, Q = F(K,L)
= K x L. When one unit of each input is used Q = 1. Doubling both inputs
increases outputs to 2x2 = 4. This firm has increasing returns to scale.
A firm with the production function Q = F(K,L) = (KL).5 has constant returns to
scale. To see this, observe that when K = 1, and L = 1, Q =1. When K=2, L = 2,
Q =2, etc.
A firm with the production function Q = (K,L).25 has decreasing returns to scale
When K=1 and L=1, Q =1. When K=2, L=4, Q = 4.25 =1.41
RULE: For any production function of the form
Q = F(K,L) = (KL)a
The firm exhibits increasing returns to scale if a>.5
The firm exhibits decreasing returns to scale if a<.5
The firm exhibits constant returns to scale if a=.5
Example: Suppose that the production function for Grecko’s Smoothies is
F(K, L) = (KL).25
Grecko currently produces with K=4 and L=4.
- What is output?
- Would Grecko enjoy increasing, decreasing or constant returns to scale if it doubled
output to K = 8, L = 8?
- How would your answer change if Greko’s production function wast
F(K, L) = (KL)1
4. Multiple Output Cost Functions. All of the insights pertaining to single product output
apply to a multi-product firm. There are, however some interesting additional cost issues
that arise. In finishing this chapter, consider two points: The notion of economies of
scope, and economies of scale.
To illustrate, we will consider cost conditions for a firm that produces just two
products: Q1 and Q2. Denote the cost function for this firm as C(Q1, Q2).
a. Economies of Scope: Exist when the joint production of two goods is less
expensive that the production of both goods separately. Mathematically, if
C(Q1, 0) + C(0, Q2) > C(Q1, Q2)
Economies of scope are an important reason why firms produce multiple
products. For example, it may be more efficient to produce both cars and light
trucks in a single plant than to produce both good separately, the two products
may share many parts of the same assembly (such as the chassis) and producing
the products separately would require considerable duplicative construction.
b. Cost complementarities: These exist in the marginal cost of producing one
good increases when the output of another product is increased. Mathematically,
when:
MC1(Q1, Q2)< 0
Q2
This often arises when one product is a by-product of another. For example there
are cost complementarities in the production of in the production of Flouride and
Aluminum Ingot from Alumina.
Cost complementarities are an important reason for economies of scope.
These notions are conveniently expressed algebraically with at quadratic cost
function:
C(Q1, Q2)
=
f
+
Then MC1
=
aQ2 + 2Q1
aQ1Q2 + Q12 + Q22
Cost complementarities exist whenever a < 0.
Economies of scope exist whenever
C(Q1, 0) + C(0, Q2) > C(Q1, Q2)
C(Q1, 0)
=
f
+ Q12
C(0, Q 2)
=
f
+ Q22
Thus, economics of scope exist if
2f+ + Q12+ Q22 > f+ aQ1Q2 +Q12+Q22
Or if
f - aQ1Q2 > 0.
Comparing the two conditions, it is seen that cost complementarities are a
stronger condition than economies of scope. Given a<0, cost complementarities
always exist. However, economics of scope may also even without cost
complementarities, if the costs of paying fixed set-up costs twice are sufficiently
high.
Example: Suppose
C
=
100 - .5Q1Q2 + Q12 + Q22
Do cost complementarities exist?
Do economies of scope exist? What about the case where
C
=
100 + .5Q1Q2 + Q12 + Q22 ?
Notice the existence of economies of scope and cost complementarities have a lot
to do with the effectiveness of mergers and the sales of subsidiaries. Sales of an
unprofitable subsidiary may not reduce losses much, due to cost
complementarities. Similarly, due to economies of scope, it may be the case that
multi-product mergers are efficient.