topic 1.3

ECON 377/477

Topic 1.3

Cost functions

Outline

• Introduction• An example• Properties• Deriving conditional input demand functions• Short-run cost functions• Marginal and average costs• Economies of scale and scope• Revenue functions• Profit functions

3ECON377/477 Topic 1.3

Introduction

• The cost-minimisation problem for a firm that is a price taker can be written as:

c(w,q) = min w׳x such that T(q,x) = 0

where w = (w1, w2, …, wN) is a vector of input prices

• The RHS entails a search over technically feasible input-output combinations to find the input quantities that minimise the cost of producing the output vector q

• This minimum cost varies as w and q vary


An example

• Refer to CROB, pages 22-23, for an example of a cost function:

• The cost minimization problem can be written:

such that

1 21 2 1 1 2 2,

( , , ) min x x

c w w q w x w x

1.25 2.52 10.177 0x x q

0.5 0.41 22q x x


An example

• Or, substituting for x2:

• Take the first derivative with respect to x1 and set it to zero:

• Solving for x1 we obtain the conditional input demand function

• That is, input demands are conditional on the value of output

1

1.25 2.51 2 1 1 2 1( , , ) min 0.177 .

xc w w q w x w x q

2.25 2.51 2 10.221 0.w w x q


An example

• The cost function is:

• An interesting property of this cost function is that it has the same functional form as the production function, namely Cobb-Douglas

• This property is shared by all Cobb-Douglas production and cost functions

• Such functions are said to be self-dual• The algebra quickly becomes unmanageable with many

inputs and outputs and/or a less tractable functional form than the Cobb-Douglas

0.556 0.444 1.1111 2 1 1 1 2 2 2 1 2 1 2( , , ) ( , , ) ( , , ) 0.92 .c w w q w x w w q w x w w q w w q


An example

• We can gain some insights into the properties of the cost function by computing the cost-minimising input demands (and associated minimum costs) at different values of the right-hand-side variables

• Several aspects of this numerical example are depicted graphically in Figure 2.6 in CROB


Properties

• The cost function satisfies the following properties:

C.1 Nonnegativity: Costs can never be negative

C.2 Nondecreasing in w: An increase in input prices will not decrease costs

C.3 Nondecreasing in q: It costs more to produce more output

C.4 Homogeneity: Multiplying all input prices by an amount k > 0 will cause a k-fold increase in costs

C.5 Concave in w: Input demand functions cannot slope upwards


Properties

• These properties of the cost function are used in three important ways:1. Evidence that one or more properties are violated can be

regarded as evidence that a firm is not minimising costs

2. The properties can be used to establish qualitative results concerning changes in market structure or government policy

3. They can be used to obtain better econometric estimates of cost and conditional input demand functions

10ECON377/477 Topic 1.3

Deriving conditional input demand functions

• Shephard’s Lemma says that:

• This result has an important practical implication• Once a well-behaved cost function has been specified or

estimated econometrically, we can use Shephard’s Lemma to obtain the conditional input demand equations quickly and easily

• Refer to the example in CROB, page 25

( , )( , ) .n

n

cx

w

w q

w q

11ECON377/477 Topic 1.3


• The approach where Shephard’s Lemma is used to derive input demand equations is known as the dual approach

• The approach involving constrained minimisation of the cost function is known as the primal approach

• In practice, the dual approach is used much more widely than the primal approach, partly because it is easier but also because (estimated) cost functions are often closer to hand than production functions

12ECON377/477 Topic 1.3


• If the cost function is twice-continuously differentiable and satisfies the five cost function properties (C.1 to C.5), Shephard’s Lemma can be used to show that conditional input demand functions have the properties of:

D.1 Nonnegativity

D.2 Nonincreasing in w

D.3 Nondecreasing in q

D.4 Homogeneity

D.5 Symmetry

13ECON377/477 Topic 1.3

Short-run cost functions

• Until now, we have assumed that all inputs are variable, which they would be in the long run

• For this reason, the cost function is sometimes known as a variable or long-run cost function

• A useful variant of this function is obtained by assuming that a subset of inputs are fixed, as some inputs would be in the short run

• The resulting cost function is known as a restricted or short-run cost function

( , )c qw

14ECON377/477 Topic 1.3


• Let the input vector x be partitioned as

where xf and xv are subvectors containing fixed and variable inputs, respectively

• And let the input price vector w be similarly partitioned as

• Then the short-run cost minimisation problem can be written:

such that

( , )f vx x x

( , ).f vw w w

( , , ) min v

f v v f fc x

w q x w x w x

( ) 0.T q,x

15ECON377/477 Topic 1.3


• Note that this problem only involves searching over values of the variable inputs

• In every other respect, it is identical to the long-run cost minimisation problem

• Thus, it is not surprising that satisfies properties C.1 to C.5

• In addition, short-run costs are no less than long-run costs and the function is nondecreasing in fixed inputs

( , , )fc w q x

16ECON377/477 Topic 1.3

Marginal and average functions

• Associated with long-run and short-run cost functions are several concepts that are frequently used when discussing firm behaviour

• In the case of a single-output firm we can define

Short-run variable cost:

Short-run fixed cost:

Short-run total cost:

Short-run average variable cost:

Short-run average cost: …

SVC ( , , )v v fqw x w x

SFC f fw x

STC ( , , )v v f f fq w x w x w x( , , )

SAVC v v fq

q

w x w x

( , , )SAC fc q

q

w x

17ECON377/477 Topic 1.3


Short-run average fixed cost:

Short-run marginal cost:

Long-run total cost:

Long-run average cost:

Long-run marginal cost:

SAFC f f

q

w x

( , , )SMC fc q

q

w x

LTC ( , )c q w

( , )LAC

c q

q

w

( , )LMC

c q

q

w

18ECON377/477 Topic 1.3


• To illustrate the nature of some of these quantities, suppose the production function is

and x2 is fixed in the short run

• These equations express various types of costs as functions of up to four variables

0.5 0.41 22q x x

19ECON377/477 Topic 1.3

Marginal and average functions• We can represent the four functions graphically

by holding all but one of the right-hand side variables fixed

• We set (w1, w2, x2) = (150, 1, 100)

• The input, x1, is allowed to vary and we plot LTC, SVC, SFC and STC against output on the next slide

• Note that the STC function is the sum of the SVC and SFC functions and is convex in q

20ECON377/477 Topic 1.3


0

100

200

300

400

500

600

0 10 20 30 40

LTC

SVCSTC

SFC

11.5

Note that this was incorrectly annotated

as SVC in CROB

21ECON377/477 Topic 1.3


• This LTC function is ever-so-slightly convex in q, a property that must be satisfied if there is to be a unique solution to the long-run profit maximisation

• It is also tangent to the STC function when q = 11.5• The conditional input demand equations can be

used to verify that the values x1= 0.833 and x2 = 100 minimise the long-run cost of producing output q = 11.5 when prices are w1 = 150 and w2 = 1

22ECON377/477 Topic 1.3

Economies of scale and scope

• A measure of overall scale economies is:

• The firm will exhibit increasing, constant or decreasing returns to scale as ε is greater than, equal to or less than one

• In the multiple-output case, it is also meaningful to consider the cost savings resulting from producing different numbers of outputs

1

1

ln ( , )

ln

w qM

cm m

c

q

23ECON377/477 Topic 1.3


• Three measures of so-called economies of scope are:

where denotes the cost of producing the m-th output only and denotes the cost of producing all outputs except the m-th output

1

( , ) ( , ) 1w w qM

mm

S c q c

( , ) ( , ) ( , )

( , )

w w q w q

w qm M m

m

c q c cS

c

2 ( , )w qmn

m n

cS

q q

( , )w mc q( , )w qM mc

24ECON377/477 Topic 1.3


• The measure defined by the first equation is of global economies of scope, and gives the proportionate change in costs if all outputs are produced separately

• If S > 0, then it is best to produce all outputs as a group

• If S < 0, then it is best to produce all outputs separately

25ECON377/477 Topic 1.3


• The measure defined by the second equation is of product-specific economies of scope, and gives the proportionate change in costs if the m-th output is produced separately and all other outputs are produced as a group

• If Sm > 0, then it is best to produce all outputs as a group

• If Sm < 0, then it is best to produce the m-th output separately

26ECON377/477 Topic 1.3


• The measure defined by the third equation is another measure of product-specific economies of scope

• It gives the change in the marginal cost of producing the m-th output with respect to a change in the production of the n-th output

• The firm experiences economies of scope with respect to the n-th output if this derivative is negative

27ECON377/477 Topic 1.3

Revenue functions

• The function that gives us maximum revenue is known as a revenue function

• In this section the revenue function and its properties are discussed

• The revenue maximisation problem for a multiple-input multiple-output firm can be written:

• such that

where is a vector of output prices over which the firm has no influence

( , ) max r q

p x p q ( ) 0T q,x

1 2( , ,..., )Mp p p p

28ECON377/477 Topic 1.3

Revenue functions

• The revenue function satisfies the properties

R.1 Nonnegativity:

R.2 Nondecreasing in p: If then

R.3 Nondecreasing in x: : If then

R.4 Convex in p:

for all

R.5 Homogeneity:• These properties are analogous to the cost

function properties C.1 to C.5

( , ) 0.r p x

0 1p p 0 1( , ) ( , ).r rp x p x

0 1x x0 1( , ) ( , ).r rp x p x

0 1( (1 ) , )r p p x 0 1( , ) (1 ) ( , )r r p x p x

0 1. r(kp, x) = kr(p, x) for k > 0

29ECON377/477 Topic 1.3

Revenue functions

• Because the revenue maximisation and cost minimisation problems are conceptually similar, we can work backwards from the revenue function to the conditional output supply functions simply by differentiating with respect to output(s)

• We can also define short-run revenue functions by assuming one or more outputs are fixed

30ECON377/477 Topic 1.3

Revenue functions

• In the single-output case we can define:

Long-run total revenue (LTR): pq

Long-run average revenue (LAR): p

Long-run marginal revenue (LMR): p• When we plot LTR, LAR and LMR against q, the

equation for LTR is the equation of a straight line that passes through the origin and has slope p, while the equation for LAR = LMR is a horizontal line with intercept p

31ECON377/477 Topic 1.3

Profit functions

• In this section we look at how firms choose inputs and outputs simultaneously

• We usually assume that firms make these decisions in order to maximise profit (that is, revenue minus cost)

• We assume multiple-input multiple-output firms solve the problem

such that• Again, we have used the notation on the left-hand-side

to emphasise that maximum profit varies with p and w

,( , ) max

q xp w p q w x ( ) 0.q,xT

32ECON377/477 Topic 1.3

Profit functions

• To illustrate the solution to the profit maximisation problem and some associated ideas, CROB consider the problem of maximising profits when the production function takes the simple Cobb-Douglas form, q = x0.5

• The equation on the previous slide becomes:

such that• or, by substituting for q,

,( , ) max

q xp w pq wx

0.5 0,q x

0.5( , ) max .x

p w px wx

33ECON377/477 Topic 1.3

Profit functions

• The first-order condition,

• can be solved for the (unconditional) input demand function:

• Substituting this result back into the production function yields the output supply function,

• and eventually the profit function

2 2( , ) 0.25x p w w p

1( , ) 0.5 ,q p w w p

1 2( , ) ( , ) ( , ) 0.25 .p w pq p w wx p w w p

0.50.5 0,px w

34ECON377/477 Topic 1.3

Profit functions

• To make these ideas more concrete, consider the profit-maximising input and output levels when p = 1 and w = 4

• Rewrite the profit function in the form

• This is the equation of an isoprofit line – a straight line with intercept and slope that gives all input-output pairs capable of producing profit level p

pq wx ( / ) ( / ) .q p w p x

/ p /w p

35ECON377/477 Topic 1.3

Profit functions

• On the next slide we depict the production function, q = x0.75 and two such isoprofit lines

• The isoprofit line having slope, w/p = 4/1, is tangent to the production function at the point (x, q) = (0.016, 0.125) and intersects the vertical axis at π/p = 0.063

• The isoprofit line with slope w/p = 4/3 (for p = 3) is tangent to the production function at the point (x, q) = (0.141, 0.375) and intersects the quantity axis at π/p = 0.188

• Since p = 3, this implies π = 0.563

36ECON377/477 Topic 1.3

Profit functions

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.05 0.1 0.15 0.2 0.25

Isoprofit line with slope w/p = 4 and

intercept p/p = 0.063

q

x0.016 0.141

0.125

0.375

Isoprofit line with slope w/p = 1.33' and intercept p/p = 0.563/3 = 0.181

Production function q =

x0.75

37

Profit functions

• The profit function satisfies the following properties:

P.1 Nonnegativity:

P.2 Nondecreasing in p: if then

P.3 Nonincreasing in w: if then

P.4 Homogeneity: for k > 0

P.5 Convex in (p,w):

for all• These properties are generalisations of the

properties of cost and revenue functions

0 1 0 1 1 1( (1 ) , (1 ) ) ( , ) p p w w p w1 1(1 ) ( , ) p w 0 1.

( , ) 0 p w

0 1p p 0 1( , ) ( , ). p w p w

0 1w w 0 1( , ) ( , ). p w p w

( , ) ( , )k k k p w p w

38ECON377/477 Topic 1.3

Profit functions

• The use of Shephard’s Lemma could be generalised to the case of a profit function

• If the profit function is twice-continuously differentiable, then Hotelling’s Lemma says that

• An example of the application of Hotelling’s Lemma to derive input demand and output supply equations is described by CROB, page 37

( , )( , )n

n

xw

p w

p w( , )

( , ) .mm

qp

p w

p w

39ECON377/477 Topic 1.3

Profit functions

• If the profit function is twice-continuously differentiable and satisfies properties P.1 to P.5, then Hotelling’s Lemma can be used to establish the following properties of input demand and output supply functions:

• X.1 Nonnegativity: ∂xn(p, w) ≥ 0

• X.2 Nonincreasing in w: • X.3 Homogeneity: xn(kp, kw) = xn(p, w) for k >

0• X.4 Symmetry:

( , ) / 0.n nx w p w

( , ) / ( , ) / .n m m nx w x w p w p w

40ECON377/477 Topic 1.3

Profit functions

Q.1 Nonnegativity: qm(p, w) ≥ 0

Q.2 Nondecreasing in p:

Q.3 Homogeneity: qm(kp, kw) = qm(p,w) for k > 0

Q.4 Symmetry: • The nonnegativity properties X.1 and Q.1 stem from

the fact that the profit function is nondecreasing in p and nonincreasing in w (properties P.2 and P.3)

• The monotonicity properties X.2 and Q.2 stem from the convexity property of the profit function (property P.5)

( , ) / 0.m mq p p w

( , ) / ( , ) / .n m m nq p q p p w p w

41ECON377/477 Topic 1.3

Profit functions

• The homogeneity properties X.3 and Q.3 stem from the fact that the profit function is homogeneous of degree 1 (property P.4) and the result that the first derivative of any linearly homogeneous function is homogeneous of degree 0

• The symmetry properties X.4 and Q.4 follow from the fact that the order of differentiation is unimportant (Young’s Theorem)

• These properties can be used to explore the effects of possible changes in economic policy and obtain better estimates of economic quantities of interest, including measures of productivity and efficiency

42ECON377/477 Topic 1.3

Restricted profit functions

• The profit function treating all inputs and outputs as variable is an unrestricted or long-run profit function

• Special cases of this function are obtained by assuming that one or more inputs or outputs are fixed, as they would be in the short run

• The resulting profit function is known as a restricted or short-run profit function

43ECON377/477 Topic 1.3


• We have already considered two restricted profit functions in this chapter:1. the cost function is (the negative of) a restricted profit function

corresponding to the case where all outputs are fixed

2. the revenue function is a restricted profit function where all inputs are fixed

• Another restricted profit function is obtained by assuming only a subset of inputs are fixed

44ECON377/477 Topic 1.3


• The resulting short-run profit maximisation problem can be written as

such that

• This problem is identical to the long-run profit maximisation problem except we now only search over values of the outputs and variable inputs (that is, q and xv)

• Because we no longer search over (potentially more profitable) values of the fixed inputs, short-run profit can never be greater than long-run profit

,( , , ) max

vf

q xp w x p q w x ( ) 0.T q,x

45ECON377/477 Topic 1.3

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