Theory of the Firm
• In last lecture, we focused on the demand side of the market—preferences and behavior of consumers
• Focus is now on supply side of the market
• Theory of the firm describes how a firm makes cost-minimizing production decisions and how the firm’s resulting cost varies with its output.
• We will also see that there are strong similarities between optimizing decisions made by firms and those made by consumers.
Production decisions are best understood in three distinct steps
• Production technology: – We need a way of describing how inputs (such as labor,
capital, raw materials) can be transformed into outputs (such as cars and televisions)
– A firm can produce a particular level of output by using different combinations of inputs
• Cost constraints: – Firms must take into account the prices of inputs – A firm will want to produce in a way that minimizes its
total production costs for a given level of output
• Input choices: – Given its production technology and the price of labor,
capital, and other inputs, the firm must choose how much of each input to use in producing its output.
Overview
I. Production Analysis
– Total Product, Marginal Product, Average Product.
– Isoquants.
– Isocosts.
– Cost Minimization
II. Cost Analysis
– Total Cost, Variable Cost, Fixed Costs.
– Cubic Cost Function.
– Cost Relations.
Production Function
• Production Function – The maximum amount of output that can be produced with
K units of capital and L units of labor. – Q = F(K,L)
• Q is quantity of output produced. • K is capital input. • L is labor input. • F is a functional form relating the inputs to output.
• Important to bear in mind that inputs and outputs are flows. • Production functions describe what is technically feasible
when the firm operates efficiently – Presumption that production is always more technically efficient need
not always hold, but it is reasonable to expect that profit-seeking firms will not waste resources.
• Production function applies to a given technology—that is a given state of knowledge about various methods that might be used to transform inputs into outputs. Technology may change over time.
Production Function Algebraic Forms
• Linear production function: inputs are perfect substitutes.
Q = F (K , L) = ak + bL
• Leontief production function: inputs are used in fixed proportions.
Q = F (K , L) = min{bK , cL}
• Cobb-Douglas production function: inputs have a degree of substitutability.
Q = F (K , L) =
K a Lb
Production Decisions: Short- vs. Long-Run
• Production decisions by firms can be very different in the short-run compared to those made in the long-run
– For example, in short-run, firms vary the intensity with which they utilize a given plant and machinery; in long-run, they may vary the size of the plant.
• Short-Run vs. Long-Run Decisions – Short-Run: Period of time in which quantities of one or more
production factors cannot be changed
– Long-Run: Amount of time needed to make all production inputs variable
• Fixed vs. Variable Inputs – Fixed input: a production factor that cannot adjust (in the short-
run) – Variable factors can adjust to alter production
Productivity Measures: Total Product
• Total Product (TP): maximum output produced with given amounts of inputs.
• Example: Cobb-Douglas Production Function:
Q = F(K,L) = K.5 L.5
– K is fixed at 16 units.
– Short run Cobb-Douglass production function:
Q = (16).5 L.5 = 4 L.5
– Total Product when 100 units of labor are used?
Q = 4 (100).5 = 4(10) = 40 units
Productivity Measures: Average Product of an Input
• Average Product of an Input: measure of output produced per unit of input. – Average Product of Labor: APL = Q /L.
• Measures the output of an “average” worker.
• Example: Q = F(K,L) = K.5 L.5
– If the inputs are K = 16 and L = 16, then the average product of labor is APL = [(16) 0.5(16)0.5]/16 = 1.
– Average Product of Capital: APK = Q /K.
• Measures the output of an “average” unit of capital.
• Example: Q = F(K,L) = K.5 L.5
– If the inputs are K = 16 and L = 16, then the average product of capital is APK = [(16)0.5(16)0.5]/16 = 1.
Productivity Measures: Marginal Product of an Input
• Marginal Product of an Input: change in total output attributable to the last unit of an input. – Marginal Product of Labor: MPL = ∆Q /∆L
• Measures the output produced by the last worker.
• Slope of the short-run production function (with respect to labor).
– Marginal Product of Capital: MPK = ∆Q /∆K
• Measures the output produced by the last unit of capital.
• When capital is allowed to vary in the long run, MPK is the slope of the production function (with respect to capital).
Q
Increasing, Diminishing and Negative Marginal Returns
Increasing
Marginal
Returns
Diminishing
Marginal
Returns
Negative
Marginal
Returns
Q=F(K,L)
AP
L MP
• It can never be profitable to use additional amounts of a costly input to product less output, i.e. where MP<0
• MP > 0 as long as output is increasing.
• AP and MP curves are closely related:
MP>AP if AP is increasing MP<AP is AP is decreasing
• Why should we expect the MP curve to rise and then fall?
• Principle of diminishing marginal returns says that as the use of an input increases with other inputs fixed, the resulting additions to output will eventually decrease.
Assume here that (in the short-run) capital is
fixed and only labor can vary.
Guiding the Production Process
• The manager’s role in guiding the production process is twofold:
1. Ensure that firm operates on the production function • Difficult to achieve in practice • For example, for the case of labor this means aligning
incentives to induce maximum worker effort
2. Employing the “right” level of inputs • When labor or capital vary in the short run, to maximize
profit a manager will hire: – labor until the value of marginal product of labor equals the wage:
VMPL = w, where VMPL = P x MPL. – capital until the value of marginal product of capital equals the rental
rate: VMPK = r, where VMPK = P x MPK .
Isoquant
• Curve showing all combinations of inputs that yield the producer the same level of output.
• The shape of an isoquant reflects the ease with which a producer can substitute among inputs while maintaining the same level of output.
• As we will see below, by taking into account this flexibility in the production process, managers can choose input combinations that minimize cost and maximize profits.
Marginal Rate of Technical Substitution (MRTS)
• The rate at which two inputs are substituted while maintaining the same output level:
MRTS = - ∆K/∆L (for a fixed level of output)
• MRTS is closely related to marginal products of inputs. To see this, consider the following:
Additional output from increase use of labor = MPL ∆L
Additional output from increase use of capital = MPK
∆K
Moving along an isoquant, the change in output is constant:
MPL ∆L+ MPK ∆K = 0
Rearranging we get: MPL / MPK = - ∆K / ∆L = MRTS
• This is analogous to Marginal Rate of Substitution (MRS) in consumer theory.
Linear Isoquants
• Different production functions imply different MRTS. K
• Here: linear production function.
• Capital and labor are perfect substitutes – Q = aK + bL
– MRTSKL = b/a
– Linear isoquants imply that inputs are substituted at a constant rate, independent of the input levels employed.
Increasing
Output
Q1 Q2 Q3
L
Leontief (Fixed-Proportions) Isoquants
• Capital and labor are perfect complements. K
• Capital and labor are used in fixed-proportions.
• Q = min {bK, cL}
• Since capital and labor are consumed in fixed proportions there is no input substitution along isoquants (hence, no MRTSKL).
Q3
Q2
Q1 Increasing
Output
L
2
1
Cobb-Douglas Isoquants
• Inputs are not perfectly substitutable.
• Diminishing marginal rate of K technical substitution. Q3
– As less of one input is used in the production process, increasingly more of Q the other input must be employed to produce the same output level. Q
– Isoquants are convex (bowed inward).
• Diminishing MRTS tells us that productivity of any one input is limited.
– As more labor (capital) is added in place of capital (labor), the productivity of labor (capital) falls.
– “Production needs a balanced mix of both inputs.”
• Q = KaLb
• MRTSKL = MPL/MPK
Increasing
Output
L
Returns to Scale
• Analysis so far focused on happens when a firm substitutes one input for another while keeping output constant.
• In long-run, with all inputs variable, firms must also consider the best way to increase output.
• One way is to change scale of operation by changing all of the inputs to production in proportion.
• Returns to scale: rate at which output increases as inputs are increased proportionately. – Increasing (decreasing) returns to scale: output more (less)
than doubles when all inputs are doubled – Constant returns to scale: output doubles when all inputs
are doubled.
NOW
Production decisions are best understood in three distinct steps
• Production technology: – We need a way of describing how inputs (such as labor,
capital, raw materials) can be transformed into outputs (such as cars and televisions)
– A firm can produce a particular level of output by using different combinations of inputs
• Cost constraints: – Firms must take into account the prices of inputs – A firm will want to produce in a way that minimizes its
total production costs for a given level of output
• Input choices: – Given its production technology and the price of labor,
capital, and other inputs, the firm must choose how much of each input to use in producing its output.
Isocost • The combinations of inputs K
that produce a given level of
New Isocost Line
associated with higher
output at the same cost:
wL + rK = C
• Rearranging,
C1/r
C0/r
costs (C0 < C1).
K= C/r - (w/r)L C0
C /w C1
C /w L
• For given input prices, isocosts farther from the origin are associated with higher costs.
• Changes in input prices change the slope of the isocost line.
K
C/r
0 1
New Isocost Line for
a decrease in the
wage (price of labor:
w0 > w1).
L C/w0
C/w1
Cost Minimization
• Marginal product per dollar spent should be equal for all inputs:
MPL = w
• But, this is just
MPK ⇔ r
MPL = w
MPK r
MRTSKL
= w
= MPL
r MPK
Cost Minimization
• At the optimal input mix for K and L:
K − Slope of isocost = Slope of isoquant
− MRTS = w/r Cost-minimizing − MP
/w= MP /r input combination •
L K
If condition did not hold, technical rate at which producer could substitute between K and L would differ from market rate
− Suppose: MPL/w > MPK/r − Then, at the margin, firm should
use less capital and more labor to
Q minimize costs. If firm reduced its expenditure on capital by 1 CHF, it could produce the same level of output if it increased its
L expenditures on labor by less than 1 CHF.
A
B
Q0
Optimal Input Substitution
• A firm initially produces Q0 by employing the combination of inputs represented by point A at a cost of C0.
• Suppose w0 falls to w1. – The isocost curve rotates
counterclockwise; which represents the same cost level prior to the wage change.
– To produce the same level of output, Q0, the firm will produce on a lower isocost line (C1) at a point B.
– The slope of the new isocost line represents the lower wage relative to the rental rate of capital.
K
K0
K1
0 L0 L1
C0/w0 C1/w1
C0/w1 L
Cost Analysis
• Types of Costs – Short-Run
• Fixed costs (FC)
• Sunk costs
• Short-run variable costs (VC)
• Short-run total costs (TC)
– Long-Run
• All costs are variable
• No fixed costs
Cost Curves
• We have so far described cost-minimizing behavior of a firm.
• Here we continue that investigation through the use of an important geometric construction, the cost curve.
• Cost curves can be used to depict graphically the cost of a firm and are important in studying the determination of optimal output choices.
Total and Variable Costs
C(Q): Minimum total cost of $ producing alternative levels
of output:
C(Q) = VC(Q) + FC VC(Q):
Costs that vary with
output.
FC: Costs that do not vary
with output.
C(Q) = VC + FC
VC(Q)
FC
0 Q
Fixed and Sunk Costs
FC: Costs that do not vary $ with output.
Sunk Cost: A cost that is
forever lost after it has been
paid.
Decision makers should
ignore sunk costs to
maximize profit or minimize
losses
C(Q) = VC + FC
VC(Q)
FC
111
Q
Some Definitions
Average Total Cost
ATC = AVC + AFC
ATC = C(Q)/Q $
Average Variable Cost
AVC = VC(Q)/Q
Average Fixed Cost
AFC = FC/Q
Marginal Cost
MC = dC/dQ
Why is ATC curve U-shaped in short-run?
• U-shaped ATC curve is combination of AFC
and AVC curves.
• AVC curve may initially slope down but
need not. However, it will eventually rise,
because of diminishing marginal returns (as
long as there are fixed factors that
constrain production in the short-run).
MC ATC
AVC
AFC
Q
The Shapes of Cost Curves
• AFC is downward sloping, toward zero for large output
• Shapes of remaining curves are $ determined by relationship between marginal and average cost curves: • If MC<AC, AVC curve falls • If MC>AC, AVC curve increases • Minimum of AVC is when
MC=AVC • Minimum of ATC is when
MC=ATC • Vertical distance between ATC
and AVC is decreasing in Q. • Minimum of AVC lies left to
minimum of ATC (this follows because
MC=AVC at its minimum point and MC=ATC at its minimum point. Because ATC is always greater than AVC and the marginal cost curve MC is rising, the minimum point of the ATC curve must lie above and
to the right of the minimum point of the AVC curve.)
MC ATC
AVC
AFC
Q
Fixed Cost
$
ATC AFC Fixed Cost
AVC
Q0×(ATC-AVC)
= Q0× AFC
= Q0×(FC/ Q0)
= FC
MC
ATC
AVC
Q0 Q
Variable Cost
Q0×AVC $
= Q0×[VC(Q0)/ Q0]
= VC(Q0)
MC
ATC
AVC
AVC
Variable Cost
Q0
Minimum of AVC
Q
Total Cost
Q0×ATC MC
$ = Q0
×[C(Q0
)/ Q0]
ATC
= C(Q0) AVC
ATC
Total Cost Minimum of ATC
Q0 Q
Cubic Cost Function
• C(Q) = f + a Q + b Q2 + cQ3
• Marginal Cost?
MC(Q) = dC/dQ = a + 2bQ + 3cQ2
An Example
– Total Cost: C(Q) = 10 + Q + Q2
– Variable cost function:
VC(Q) = Q + Q2
– Variable cost of producing 2 units:
VC(2) = 2 + (2)2 = 6
– Fixed costs:
FC = 10
– Marginal cost function:
MC(Q) = 1 + 2Q
– Marginal cost of producing 2 units:
MC(2) = 1 + 2(2) = 5
Long-Run vs. Short-Run Cost Curves
• We saw that short-run average cost (SRAC) curves are U-shaped (because of diminishing marginal returns).
• In the long-run, all factors of production are flexible and fixed costs are zero (in the sense that it is always possible to produce zero units of output at zero costs).
• Shape of long-run average cost (LRAC) curve is determined by relationship between the scale of the firm and the inputs required to minimize its costs.
– Constant returns to scale: AC are constant for all levels of output.
– Increasing (decreasing) returns to scale: AC falls (rises) with output because doubling of costs is associated with a more (less) than twofold increase in output.
Long-Run Average Costs (LRAC)
LRAC curve: Curve relating AC cost of production $ to output, when all inputs are variable.
LRAC
Economies
of Scale
Diseconomies
of Scale
Q* Q
Economies and Diseconomies of Scale
• Economies of scale: Situation in which output can be doubled for less than doubling cost.
• Diseconomies of scale: Situation in which a doubling of output requires more than a doubling of cost.
• Economies of scale includes increasing returns as a special case, but it is more general because it reflects input proportions that change as the firm changes its level of production.
Economies of Scale—Examples
• As output increases, the firm’s average cost of producing that output is likely to decline, at least to a point. This can happen for the following reasons:
– If the firm operates on a larger scale, workers can specialize in the activities they are most productive in.
– Scale can provide flexibility. By varying the combinations of inputs utilized to produce the firm’s output, managers can organize the production process more effectively.
– The firm may be able to acquire inputs at lower cost because it is buying them in large quantities and can therefore negotiate better prices. The mix of inputs might change with the scale of the firm’s operation if managers take advantage of lower-cost inputs.
Diseconomies of Scale—Examples
• At some point, however, it is likely that average cost of production will begin to increase with output. This can happen for the following reasons:
– At least in the short-run, limited factory space and machinery may make it more difficult for workers to do their jobs effectively.
– Managing a larger firm may become more complex and inefficient as the number of tasks increases.
– The advantages of buying in bulk may have disappeared once certain quantities are reached. At some point, available supplies of key inputs may be limited, pushing their costs up.
Relationship between Short-Run and Long-Run Cost
• Assume that a firm is uncertain about future demand for its output and is considering three possible plant sizes.
• SRAC cost curves are given by SRAC1 , SRAC2 , and SRAC3
• If firm expects to produce Q1, then it should build the smallest plant (described by SRAC1).
• In long-run, the firm can change the size of its plant it will always choose the plant that
Cost (Zmk per unit of output)
LRAC curve with discrete
plant size is envelope of the
three SRAC curves.
minimized AC. • LRAC curve is given by envelope of the three
SRAC curves (if it can choose only between the three plant sizes) − Points A and B are on the LRAC curve in this case.
• LRAC curve is U-shaped and is given by the red curve if plants of any size could be built. − Point B is not on the LRAC curve in this case. − With economies and diseconomies of scale in the
long-run, the minimum points of the SAC curves do not lie on the LRAC (except for the point where SRAC curve is tangent to minimum of LRAC curve; as shown for Q2 here).
SRAC1
A B
SRAC2
SRAC3
LRAC
Q1 Q2 Q3 Q
Conclusion
• To maximize profits (minimize costs) managers must use inputs such that the value of marginal of each input reflects price the firm must pay to employ the input.
• The optimal mix of inputs is achieved when the marginal rate of technical substitution equals the ratio of input prices.
