March 17, 2003
Lecture #7
Reading: Chapter 5, pp. 159-191. Next time, Chapter 6.
Homework -For next time: Ch. 5.: 1, 4, 5, 6, 7,8, 9, 10, 11, 14, 15, 16, 17
Comments on problems:
On problem 6, add the following: Illustrate ATC, AVC and MC in a graph.
- Graphically, what is the relationship between ATC and AVC? Why?
- Graphically, what is the relationship between MC, ATC and AVC? Why?
Also: Hint : notice that qty increases in increments of 100.
On problem 15 notice that the restaurateur could have simply sold the license back to the state for $65,000. Instead she tried to sell it for $70,000
Review
III. Chapter 3: Quantitative Demand Analysis: Part 3, Demand Estimation
H. Forecasting and the MSE
I. Diagnostics.
1. Multicolinearity
2. Serial Corrleation
3. Heteroskedasticity
IV. Costs and the Production Process.
- Overview: Our interests are in (1) deciding on optimal resource use, and (2) getting resources to perform optimally.
- The production process
1.Marginal, Average and Total Product
2. Demand for inputs
3. Optimal use of a single input.
Preview
4. Long Run Production
- Algebraic Production Functions
- Isocosts and Isoquants
- Optimal use of multiple inputs
C. Costs.
1. The relationship of production functions to cost functions.
2. Short run costs.
a. Cost curves
b. Sunk vs. Variable Costs
3. Long-Run Costs
a. Long Run Average Costs
b. Economics of Scale
4. Multiple Output Cost functions
a. Economies of Scope
b. Cost complementarities
Lecture
4. Long Run (Production) Decisions. Now consider the problem of optimal resource use in an environment where all resources are variable. In terms of our two-factor production function, this means that both K and L may vary.
a. Algebraic Forms of Production Functions. The inter-relationships between these factors hinges on the nature of the production function. So prior to considering this problem, we consider some possible production function specifications.
Linear Production Function. One simple specification is the case where output may be produced by either capital or labor, perhaps at different rates. In general this would be written:
Q=F(K,L)=aK+bL
Example. Suppose that an electricity generating facility can produce 1 KW of electricity using either 4 cubic meters of natural gas, or one ton of coal. Thus,
Q=F(G,C)=4G+C
Leontief Production Function. Another specification useful in some circumstances is the case where output may be produced only with a fixed combination of inputs. This is also call a fixed-proportions production function.
Q=F(K,L)=min{bK, cL}
Example. Suppose that typed documents are produced by secretaries working at word-processors. One secretary can work at exactly one word-processor. An extra word-processor without a secretary will not increase productivity. Similarly, an extra secretary without a word processor will not increase output. (In this case b = c). Suppose a secretary with a word processor can produce 1 paper an hour. How many papers could 6 secretaries and 8 word processors produce? How would this differ from the output of 8 secretaries and 6 word-processors.
Cobb-Douglas Production Function. A third general specification is the case where different combinations of output may be used to produce a given input, but where the inputs are imperfect substitutes. For example consider the problem of using labor and cloth to cut clothes patterns. One could economize on labor by having workers not worry about waste cloth. On the other hand, one could get a similar output, with far less waste, but with far higher labor costs by having labor cut with extreme care. One way to characterize this type of relationship as follows:
Q=F(K,L)=KaLb
Example. Suppose that a = b= 1/2. Then
Q=F(K,L)=K1/2L1/2
Algebraic Measures of Productivity. Notice that TP and AP relationships may be readily calculated from the algebraic production function. For instance, using the above Cobb-Douglas function, if the firm used K=9 and L = 4, then
Q=F(K,L)=(9)1/2(4)1/2
=(3)(2)=6
The average productivity of Labor and Capital in this circumstance are
APL= 6/4=1.5and APK= 6/9=.67
Marginal productivities can be calculated as derivatives (for the linear and Cobb-Douglas cases.
Marginal Product for a Linear Function. If
Q=aK+bL
Then the marginal productivities of K and L are,
Q=aandQ= b
KL
Marginal Product for a Cobb-Douglas Function. If
Q=KaLb
Then the marginal productivities of K and L are,
Q=a Ka-1LbandQ= b KaLb-1
KL
Comments:
- I will not ask you to do such calculations in your homework. However, I do want for you to understand how marginal productivities are derived from a production function.
b. Long Run Production. Isocosts and Isoquants. To consider the optimal mix of multiple inputs, it is necessary to evaluate the productivity of different input combinations in terms of their costs. For the case of two inputs, this may be done graphically, with the development of isoquant and isocost curves.
1. Isoquants The collection of all input combinations that can be used to produce a given level of output. The shape of the an isoquant depends on the nature of production
K / Q1 / K / Q1 / K / Q1L / L / L
Linear Production Leontiff Production Cobb-Douglas Prod.
(Perfect Subs.) (Fixed Proportions) (Imperfect Subs.)
a) Notice that to produce a higher output, more inputs must be used, so the curves radiate out as Q expands.
b) The slope of the isoquant is easily developed from the linear case.
Q = aK + bL
Solving for Slope-intercept form:
K = Q/a - (b/a)L
Recalling that
MPK = a and MPL = b, it follows that
Slope equals-MPL/MPKThis ratio is called the Marginal Technical Rate of Substitution
c) The case illustrated by the Cobb-Douglas function is perhaps the most general. This case illustrations the notion of a diminishing MTRS.
2. Isocost Curve: A curve indicating all input combinations that can be purchased for a given budget
a) Most typically, inputs can be purchased for fixed prices. Thus, algebriacally:
I = rK + wL
b) Solving for slope intercept form
K = I/r - (w/r)L
c) Graphically
K / -(w/r)I
L
d) Notice that larger budgets permit the use of more inputs. The curves radiate outward as the budget is increased.
3. Minimum Cost Production. The least costly method of producing an output level Q is the point where the isoquant is tangent to the isocost curve closest to the origin:
a. Graphically
K7 / Qo
I
5 L
In the above example, 5 units of Labor and 7 units of capital are optimally used to product Qo.
b) Notice that at this point, -MPL = -w
MPKr
This suggests the rule for optimal resource use: Hire all resources until the Marginal product per dollars worth is equal for each input. or until
MPL=MPK
w r,
Example: Jones & Printing Co. is presently paying $10 per hour for labor, and $5 per hour for Printing Machines.
- If the Marginal Productivity of labor is 100 pages per day, and the marginal productivity of machines is 150 per day is it using a least cost combination of inputs?
- If not which machines should be used relatively more?
Example: Suppose that Randolph Tire Co. Currently uses Labor and Rubber Banding Machines to finish tires. Suppose that labor is negotiating its contract, and asks for a 33% increase in salary. Management argues that the consequence of such a wage increase will be a sharp reduction in the number of employees. Employees don’t believe management. They firmly believe that management will continue to employ the same number of workers even after a substantial wage increase.
Using isoquant/isocost curves illustrate how the difference of opinion between management and labor is an empirical one regarding the MTRS.
C. Costs and the Theory of the Firm
1. The Relationship of Production Functions to Cost Functions. To provide some context for this discussion, consider the problem of producing blueberry tarts in my house. We may have the following relationship.
(a)Inputs:
(Number of Workers) / (b)
Outputs
(Number of Blueberry Pies) / (c)
Marginal Pies / (d)
Marginal Costs Per pie (Assume that labor costs $20 per unit, and that ingredients are free)
0 / 0
1 / 5 / 5 / $20/5 =$4
2 / 15 / 10 / $20/10=$2
3 / 23 / 8 / $20/8 = $2.5
4 / 29 / 6 / $20/6 = 3.33
5 / 33 / 4 / $20/5 =$5
6 / 35 / 2 / $20/2 = $10
In the lecture on short run production, we focused on the relationship between the number of workers (column a) and the marginal productivity of those workers (column c). A closely related question pertains to the relationship between the number of pies made (column b) and the marginal costs of making pies (column d).
Observations
- After exhausting gains from specialization the relationship between units of labor and marginal productivity is inverse. This motivates the labor demand curve we developed last time in class.
- Conversely, the marginal cost curve first falls, and then, upon exhausting gains from specialization, increases.
- Intuitively, marginal costs increase as Marginal productive decreases, because the marginal productivity increases imply that more labor is “imbedded in each unit of output.
Now we focus on this latter view of the relationship between inputs and outputs. We start with the single output case, first in the short run, and then in the long run. Then we consider some aspects of costs in a multi-product environment.
2. Short Run Costs. Recall that the short run is defined as a timeframe where there are unavoidable input commitments, as well as variable inputs. The short run cost function may be used to describe this relation.
a. Cost function components.
Fixed costs: Costs associated with fixed input commitments. These costs do not change with the level of output
Variable costs: Costs associated with the variable components. These costs vary with the level of output.
b. Total cost relationships. One way to represent these costs is in terms of total expenditures. For example, consider the following table, where K is fixed at 2 and where L can vary
K / L / Q / FC / VC / TC(K*1000) / (L*400) / FC + VC
2 / 0 / 0 / 2000 / 0 / 2000
2 / 1 / 76 / 2000 / 400 / 2400
2 / 2 / 248 / 2000 / 800 / 2800
2 / 3 / 492 / 2000 / 1200 / 3200
2 / 4 / 784 / 2000 / 1600 / 3600
2 / 5 / 1100 / 2000 / 2000 / 4000
2 / 6 / 1416 / 2000 / 2400 / 4400
2 / 7 / 1708 / 2000 / 2800 / 4800
2 / 8 / 1952 / 2000 / 3200 / 5200
2 / 9 / 2124 / 2000 / 3600 / 5600
2 / 10 / 2200 / 2000 / 4000 / 6000
2 / 11 / 2156 / 2000 / 4400 / 6400
Graphically, these relationships appear as follows:
Observations
-Notice that the TVC and the TC curves both take on the shape of a “recliner”: that is, first increasing at a decreasing rate, and then increasing at an increasing rate. The difference between the two curves is TFC, which is a fixed amount.
-Notice also that the slope of the line tangent to either TVC or TC is the marginal cost. Marginal costs first decrease and then increase due to the law of diminishing returns. (This is the same logic as diminishing marginal productivity: At the outset, gains from division of labor increase. Thus marginal productivity increases, and marginal costs increase, reflecting gains from specialization. Later, as the law of diminishing returns sets in, marginal productivity falls, as marginal costs increase. Intuitively, more labor is “imbedded” in each unit of output.)
c. Average Cost Relationships. The same relationships can be generated by dividing costs by quantity, to get per unit costs. In this case:
Q / AFC / AVC / ATC / MC0
76 / 26.32 / 5.26 / 31.58 / 5.26
248 / 8.06 / 3.23 / 11.29 / 2.33
492 / 4.07 / 2.44 / 6.50 / 1.64
784 / 2.55 / 2.04 / 4.59 / 1.37
1100 / 1.82 / 1.82 / 3.64 / 1.27
1416 / 1.41 / 1.69 / 3.11 / 1.27
1708 / 1.17 / 1.64 / 2.81 / 1.37
1952 / 1.02 / 1.64 / 2.66 / 1.64
2124 / 0.94 / 1.69 / 2.64 / 2.33
AFC=TFC/Q(Average Fixed Costs)
AVC=TVC/Q(Average Variable Costs)
ATC=TC/Q(Average Total Costs
MC=TC/Q
Graphically, these curves are represented as
Observations
- ATC and AVC approach each other as quantity expands. This is because the difference between the two curves is AFC. AFC is a fixed quantity allocated over an increasing number of units.
- MC intersects ATC and AVC at their minimum points. This follows for the same reason that MP intersects AP at its peak: The marginal drives the average. The averages reflect the same information as the marginal. The marginal is more volatile, however, because it is not weighed down by the effects of any output other than the current increment.
d. Algebraic Forms of the Cost Function. As with production functions, the appropriate specification of the cost function depends on the underlying cost relationships. However, a cubic cost function is often used, because it is sufficiently general to allow any relationship.
If C(Q)f+ aQ+bQ2+cQ3
Notice that fixed costs are f.
Marginal costs are the derivative of C(Q), or
C’(Q)=a+2bQ+3cQ2
Notice that other cost relations are easily derived. For instance, average variable costs are
AVC=a+bQ+cQ2
Example: Suppose C(Q)=20+3Q2
What are marginal costs, average fixed costs, average variable costs and average total costs when Q=10? When are average total costs minimized? (To be worked in class)
e. Sunk vs. Variable Costs. A final distinction (and one we’ve made before). Fixed costs may be divided into two components: Sunk costs and recoverable costs. Sunk costs are costs forever lost after they are paid. This is an important distinction, for the opportunity costs of recoverable assets and sunk cost assets is remarkably different.
Example: Suppose you are choosing between the purchase of a Toyota Corolla (for $12,500) and a GEO Probe (for $11,000). After a year the book value on the cars will be $11,000 and $7,000. What are the sunk cost components associated with the purchase of each car? How does this difference affect your decision to undertake a 5 year loan to pay for the cars?
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.
/ ATC0ATC1 ATC3
ATC2
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. Economics of Scale
Economies of scale arise when LRAC falls as the plant expands.
Diseconomies of scale arise when LRAC increases as the plant expands.
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?
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+aQ1Q2 + Q12 + Q22
Then MC1=aQ2 + 2Q1
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, Q2)=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.
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