me_firm.ppt

1

Chapter 7

PRODUCTION FUNCTIONS

2

Objectives• So far, we have studied consumer’s

behaviour• In this lecture, we will study the supply

side of the market (firms…)• It will be easier because many concepts

used when studying the firm have a clear counterpart in the previous chapters dedicated to the consumer

3

To have in mind…

Utility function Production function

Goods Inputs

Indifference curve Isoquant

Marginal rate of substitution

Marginal rate of technical substitution

Max utility Max profits

Min expenditure Minimize costs

4

Production Function• A firm produces a particular good (q)

using combinations of inputs• The most common used inputs are

capital (k) and labor (l)• One could think in introducing different

inputs: skilled labour, unskilled labour, raw materials, intermediate products, technology…

5

Production Function

• The firm’s production function for a particular good (q) shows the maximum amount of the good that can be produced using alternative combinations of inputs (for example, capital (k) and labor (l))

q = f(k,l)

6

Marginal Productivy• (equivalent to marginal utility). The marginal

productivity (also called marginal physical product) is the additional output that can be produced by employing one more unit of that input while holding other inputs constant.

• Given by the first derivative of the production function with respect the input under consideration

kk fk

qMP

capital of product physical marginal

ll lf

qMP

labor of product physical marginal

7

Diminishing Marginal Productivity

• The marginal physical product of an input depends on how much of the other inputs is being used

• In general, we assume diminishing marginal productivity

0112

2

ff

k

f

k

MPkk

k 0222

2

ff

fMPll

l

llNotice that we also assume that the second derivative of theUtility function with respect a good is negative

8

Diminishing Marginal Productivity

• Notice, that the Marginal Productivity generally depends on all the inputs

• For instance, the marginal productivity of labor also depend on changes in other inputs such as capital– we need to consider flk which is often > 0

• An additional unit of labor will increase the production of the good more if there is more capital available

9

Average Physical Product• Another concept is average productivity of a certain input• This is the ratio of the production and the amount of input used• For instance, average labour productivity is:

l

l

ll

),(

input labor

output kfqAP

• Note that APl also depends on the amount of capital employed

10

Isoquant Maps• To illustrate the possible substitution of

one input for another, we use an isoquant map

• An isoquant shows those combinations of k and l that can produce a given level of output (q0) (rings a bell… indifference curves)

f(k,l) = q0

11

Isoquant Map

l per period

k per period

• Each isoquant represents a different level of output– output rises as we move northeast

q = 30

q = 20

12

Marginal Rate of Technical Substitution (RTS)

l per period

k per period

q = 20

- slope = marginal rate of technical substitution (RTS)

• The slope of an isoquant shows the rate at which l can be substituted for k

lA

kA

kB

lB

A

B

RTS > 0 and is diminishing forincreasing inputs of labor

13

Marginal Rate of Technical Substitution (RTS)

• The marginal rate of technical substitution (RTS) shows the rate at which labor can be substituted for capital while holding output constant along an isoquant

0

) for ( qqd

dkkRTS

ll

14

RTS and Marginal Productivities• Take the total differential of the production

function:

dkMPdMPdkk

fd

fdq k

lll l

• Along an isoquant dq = 0, so

dkMPdMP k ll

kqq MP

MP

d

dkkRTS l

ll

0

) for (

15

Decreasing RTS

• Because MPl and MPk will both be nonnegative, RTS will be positive (or zero)

• When will we have a diminishing RTS? • If fkl is positive, then the RTS is diminishing

(sufficient condition), which translates in isoquants being convex (math proof in the book)

• This mirrors utility theory…

16

Returns to Scale

• So far, we have studied how the production changes when we change the amount of one input, leaving the rest constant (marginal productivity)

• However…How does output respond to increases in all inputs together?– suppose that all inputs are doubled, would output

double?

• Returns to scale have been of interest to economists since the days of Adam Smith

17

Returns to Scale

• Smith identified two forces that come into operation as inputs are doubled– greater division of labor and specialization

of function– loss in efficiency because management

may become more difficult given the larger scale of the firm

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Returns to Scale• If the production function is given by q =

f(k,l) and all inputs are multiplied by the same positive constant (t >1), then

Effect on Output Returns to Scale

f(tk,tl) = tf(k,l) Constant

f(tk,tl) < tf(k,l) Decreasing

f(tk,tl) > tf(k,l) Increasing

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Returns to Scale• Whether there are constant, decreasing

or increasing returns to scale depend on the production function…

• It is possible for a production function to exhibit constant returns to scale for some levels of input usage and increasing or decreasing returns for other levels

20

Production Function…

• Linear…q = f(k,l) = ak + bl

• Fixed proportions:

q = min (ak,bl) a,b > 0

• Cobb Douglas:

q = f(k,l) = Akalb A,a,b > 0

• CES, see the book

21

The Linear Production Function

l per period

k per period

q1q2 q3

Capital and labor are perfect substitutes

22

Fixed Proportions

l per period

k per period

q1

q2

q3

No substitution between labor and capital is possible. They must always be used in a fixed ratio

= 0

k/l is fixed at b/a

q3/b

q3/a

23

Cobb-Douglas Production Function

• Suppose that the production function isq = f(k,l) = Akalb A,a,b > 0

• This production function can exhibit any returns to scale

f(tk,tl) = A(tk)a(tl)b = Ata+b kalb = ta+bf(k,l)– if a + b = 1 constant returns to scale– if a + b > 1 increasing returns to scale– if a + b < 1 decreasing returns to scale

24

Cobb-Douglas Production Function

• The Cobb-Douglas production function is linear in logarithms

ln q = ln A + a ln k + b ln l– a is the elasticity of output with respect to k– b is the elasticity of output with respect to l

25

Technical Progress

• Methods of production change over time

• Following the development of superior production techniques, the same level of output can be produced with fewer inputs– the isoquant shifts in

26

Technical Progress• Suppose that the production function is

q = A(t)f(k,l)

where A(t) represents technology that changes with time– changes in A over time represent technical

progress• A is shown as a function of time (t)• dA/dt > 0

• We will not look more into this… just that you know what it is…

27

Chapter 8

COST FUNCTIONS

Copyright ©2005 by South-western, a division of Thomson learning. All rights reserved.

28

Definitions of Costs• It is important to differentiate between

accounting cost and economic cost– the accountant’s view of cost stresses out-of-

pocket expenses, historical costs, depreciation, and other bookkeeping entries

– economists focus more on opportunity cost– Opportunity costs are what could be obtained by

using the input in its best alternative use

29

Definitions of Costs

• Labour Costs– To both economist and accountants, labour

costs are very much the same thing: labour costs of production (hourly wage)

30

Definitions of Costs• Capital Costs (accountants and

economists differ…)– accountants use the historical price of the

capital and apply some depreciation rule to determine current costs

– the cost of the capital is what someone else would be willing to pay for its use (and this is what the firm is forgoing by using the machine)

• we will use v to denote the rental rate for capital

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Definitions of Costs• Costs of Entrepreneurial Services

– accountants believe that the owner of a firm is entitled to all profits

• revenues or losses left over after paying all input costs

– economists consider the opportunity costs of time and funds that owners devote to the operation of their firms

32

Example…• IT programmer, she does a new software on her free

time and sells it by £5000. • Accounting profits=£5000. This seems like a good

project• Economist profits=£5000 minus what she could have

earned working for a firm in her time. Might not seem such a good project any more…

• part of accounting profits would be considered as entrepreneurial costs by economists

33

Economic Cost

• The economic cost of any input is the payment required to keep that input in its present employment– the remuneration the input would receive in

its best alternative employment

34

Another example

• A shop owner

• If her accounting profits are smaller than the rental price of the physical shop, it means that she is having losses as she could obtain more money by no running the shop but renting it

35

Two Simplifying Assumptions• There are only two inputs

– homogeneous labor (l), measured in labor-hours

– homogeneous capital (k), measured in machine-hours

• entrepreneurial costs are included in capital costs

• Firms cannot influence the input prices, they are given… (they do not depend on firms decisions on the inputs to be used…)

36

Economic Profits• Total costs for the firm are given by

total costs = C = wl + vk

• Total revenue for the firm is given bytotal revenue = pq = pf(k,l)

• Economic profits () are equal to = total revenue - total cost

= pq - wl - vk

= pf(k,l) - wl - vk

37

Economic Profits• Economic profits are a function of the

amount of capital and labor employed– we could examine how a firm would choose

k and l to maximize profit– We will do later on…

38

Minimizing costs…• For now…

– we will assume that the firm has already chosen its output level (q0) and wants to minimize its costs

– We will examine the inputs that the firm will choose in order to minimize costs but produce q0 (kind of a compensated demand…)

39

q0

We fix the isoquant of output q0.

C1

C2

C3

Costs are represented by parallel lines with a slope of -w/v

Cost-Minimizing Input Choices

l per period

k per period

C1 < C2 < C3

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C1

C2

C3

q0

The minimum cost of producing q0 is C2


l per period

k per period

k*

l*

The optimal choice is l*, k*

This occurs at the tangency between the isoquant and the total cost curve

41

Cost-Minimizing Input Choices• Mathematically, we seek to minimize

total costs given q = f(k,l) = q0

• Setting up the Lagrangian:

L = wl + vk + [q0 - f(k,l)]

• First order conditions are

L/l = w - (f/l) = 0

L/k = v - (f/k) = 0

L/ = q0 - f(k,l) = 0

42


• Dividing the first two conditions we get

) for ( /

/kRTS

kf

f

v

wl

l

• The cost-minimizing firm should equate the RTS for the two inputs to the ratio of their prices

• This is subject to the same reservations as with utility (if RTS is strictly decreasing, no corner solutions…)

43

Cost-Minimizing Input Choices• Cross-multiplying, we get

w

f

v

fk l

• For costs to be minimized, the marginal productivity per dollar spent should be the same for all inputs

• The solution to this minimization problem for an arbitrary level of output q0 give us the contingent demand functions for inputs:

lc=lc(w,v,q0); kc=kc(w,v,q0)

44

Contingent Demand for Inputs• The contingent demand for input would be analogous

to the compensated demand function in consumer theory

• Notice that the contingent demand for input is based on the level of firm’s output. So, it is a derived demand.

45

Total Cost Function

• The total cost function gives the minimum cost incurred by the firm to produce any output level with given input prices. C = C(v,w,q)

• We can compute it as:• C = C(v,w,q)=w*lc(v,w,q)+ v*kc(v,w,q)

46

Average Cost Function

• The average cost function (AC) is found by computing total costs per unit of output

q

qwvCqwvAC

),,(),,( cost average

47

Marginal Cost Function

• The marginal cost function (MC) is found by computing the change in total costs for a change in output produced

q

qwvCqwvMC

),,(

),,( cost marginal

48

Graphical Analysis ofTotal Costs

• Suppose that k1 units of capital and l1 units of labor input are required to produce one unit of output

C(q=1) = vk1 + wl1

• To produce m units of output (assuming constant returns to scale)

C(q=m) = vmk1 + wml1 = m(vk1 + wl1)

C(q=m) = m C(q=1)

49


Output

Totalcosts

C

With constant returns to scale, total costsare proportional to output

AC = MC

Both AC andMC will beconstant

50


• Suppose instead that total costs start out as concave and then becomes convex as output increases– one possible explanation for this is that

there is a third factor of production that is fixed as capital and labor usage expands

– total costs begin rising rapidly after diminishing returns set in

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Output

Totalcosts

C

Total costs risedramatically asoutput increasesafter diminishingreturns set in

52


Output

Average and

marginalcosts

MC

MC is the slope of the C curve

AC

If AC > MC, AC must befalling

If AC < MC, AC must berising

min AC

53

Shifts in Cost Curves

• The cost curves are drawn under the assumption that input prices and the level of technology are held constant– any change in these factors will cause the

cost curves to shift

54

Some Illustrative Cost Functions

• See Example 8.2 in the book

55

Properties of Cost Functions

• If the input prices are multiplied by an amount t, the total cost is multiplied by the same amount

• cost minimization requires that the ratio of input prices be set equal to RTS, a doubling of all input prices will not change the levels of inputs purchased

• Nondecreasing in q, v, and w

56

C(v,w,q1)

Concavity of Cost Function

w

Costs

The cost function C(v,w1,q1) is concave in input prices. Why? Next slide…

C(v,w1,q1)

w1

57

Properties of Cost Functions

• Concave in input prices– The cost function increases less than

proportionally when one input price increases because the firm can substitute it by other inputs

– As the expenditure function in consumer theory

58

Reaction to input prices increases• If the price of an input increases, the cost will increase• The increase in costs will be largely influenced by the

relative significance of the input in the production process

• If firms can easily substitute another input for the one that has risen in price, there may be little increase in costs

• It is important to measure the substitution of inputs in order to predict how much costs will be affected by an increase in the price of an input (possibly due to a tax increase)

59

Input Substitution

• A change in the price of an input will cause the firm to alter its input mix

• We wish to see how k/l changes in response to a change in w/v, while holding q constant

vw

kl

60

Input Substitution• Rather than the derivative, we will use the elasticity:

l

l

/

/

)/(

)/(

k

vw

vw

ks

gives an alternative definition of the elasticity of substitution– in the two-input case, s must be nonnegative– large values of s indicate that firms change their

input mix significantly if input prices change. Hence, costs will not change so much

– It can me estimated using econometrics

61

Shephard’s Lemma

• Shephard’s lemma (a trick to obtain the contingent demand functions from the cost function)

• the contingent demand function for any input is given by the partial derivative of the total-cost function with respect to that input’s price

• See example 8.4 in the book• As we obtained the compensated demand from

the expenditure function in consumer theory

62

Short-Run, Long-Run Distinction

• Economic actors might not be completely free to change the amount of inputs

• Economist usually assume that it takes time to change capital levels, while labor can be changed quickly

• Economist say that the “short run” is the period of time in which some of the inputs cannot be changed

• “Long-run” is when the period of time when all the inputs can be changed

63

Short-Run, Long-Run Distinction

• Assume that the capital input is held constant at k1 and the firm is free to vary only its labor input

• The production function in the short run becomes

q = f(k1,l)

64

Short-Run Total Costs

• Short-run total cost for the firm is

SC = vk1 + wl

• There are two types of short-run costs:– short-run fixed costs are costs associated

with fixed inputs (vk1)

– short-run variable costs are costs associated with variable inputs (wl)

65


• In the Short-run:– Cannot decide the amount of fixed inputs. The firm

does not have the flexibility of input choice– to vary its output in the short run, the firm must

use nonoptimal input combinations– the RTS will not be equal to the ratio of input

prices– Consequently, short-run costs will be equal to or

larger than long-run costs

66


l per period

k per period

q0

q1

q2

k1

l1 l2 l3

Because capital is fixed at k1,the firm cannot equate RTSwith the ratio of input prices.Notice that short run costs will beEqual or larger than the long run cost

67

Short-Run and Long-Run costs

• The short run cost depend on the amount of fixed capital available

• There is no a unique short run cost curve. There will be as many as possible levels of capitals are

• The short run cost of producing q1 will be equal to the long run cost when the available capital in the short run is the same as the optimal level of capital for the long run (see q1 and k1 in the previous graph).

68

Short-Run Marginal and Average Costs

• Remember that short run costs are given by:

– SC = vk1 + wl

• The short-run average total cost (SAC) function is

SAC = total costs/total output = SC/q

• The short-run marginal cost (SMC) function is

SMC = change in SC/change in output = SC/q

69

Relationship between Short-Run and Long-Run Costs

Output

Total costs

SC (k0)

SC (k1)

SC (k2)

The long-runC curve is the minimum ofShort-run ones

q0 q1 q2

C

70

Chapter 9

PROFIT MAXIMIZATION

Copyright ©2005 by South-Western, a division of Thomson Learning. All rights reserved.

71

The Nature of Firms

• A firm is an association of individuals who have organized themselves for the purpose of turning inputs into outputs

• Each individual will have different objectives...

• Modeling the relation among all types of workers, managers, shareholders can be very complicated

72

Modeling Firms’ Behavior

• The simplest approach in economics is the following:– the decisions are made by a single dictatorial

manager who rationally pursues some goal• usually profit-maximization

– The dictatorial manager can monitor perfectly that everyone is working according to her guidelines (this assumption will be relaxed later in the course)

73

Profit Maximization

• A profit-maximizing firm chooses both its inputs and its outputs with the sole goal of achieving maximum economic profits– seeks to maximize the difference between

total revenue and total economic costs

74

Market Demand Curve• In the previous lectures, we studied the

demand curve for the individual consumer• If we add up the demand curve of all the

consumers of the market, then we will obtain the Market demand curve

• Its derivative with respect to price will be negative (less price, more quantity demanded)

• We will denote the market demand curve by D(p) or q(p)

75

Output Choice

• Total revenue for a firm is given by

R(q) = p(q)q

• In the production of q, certain economic costs are incurred [C(q)]->cost function

• Economic profits () are the difference between total revenue and total costs

(q) = R(q) – C(q) = p(q)q –C(q)

76

Output Choice• The necessary condition for choosing the level of q

that maximizes profits can be found by setting the derivative of the function with respect to q equal to zero

(q) = R(q) – C(q) = p(q)q –C(q)

0)('

dq

dC

dq

dRq

dq

d

dq

dC

dq

dR

77

Output Choice

• To maximize economic profits, the firm should choose the output for which marginal revenue is equal to marginal cost

MCdq

dC

dq

dRMR

78

Second-Order Conditions

• MR = MC is only a necessary condition for profit maximization

• For sufficiency, it is also required that

0)('

**

2

2

qqqqdq

qd

dq

d

• This is standard maths… at the max the second derivative must be negative

79

Profit Maximization

output

revenues & costs

RC

q*

Profits are maximized when the slope ofthe revenue function is equal to the slope of the cost function

The second-ordercondition prevents usfrom mistaking q0 asa maximum

q0

80

Marginal Revenue

• R(q) = p(q)q

dq

dpqp

dq

qqpd

dq

dRqMR

])([)( revenue marginal

The second term (dp/dq) is negative for firms that face a downward-sloping curve. Typically these are large firms so that prices decreases when they produce more.

The second term (dp/dq) is zero for competitive or price taking firms. Typically these are small firms so that the price does not depend on the quantity that the firm produces. In this case we have that the marginal revenue will be equal to the price.

81

Marginal Revenue• Suppose that the demand curve for a sub

sandwich isq = 100 – 10p

• Solving for price, we getp = -q/10 + 10

• This means that total revenue isR = pq = -q2/10 + 10q

• Marginal revenue will be given byMR = dR/dq = -q/5 + 10

82

Profit Maximization• To determine the profit-maximizing output, we

must know the firm’s costs• If subs can be produced at a constant a

marginal cost of $4, thenMR = MC

-q/5 + 10 = 4

q = 30

We should also check that the second derivative of the profit function in this point is negative

83

Marginal Revenue Curve

• The marginal revenue curve shows the extra revenue provided by the last unit sold

• For a firm that faces a downward-sloping demand curve, the marginal revenue curve will lie below the demand curve (see the formula before)

84


output

price

D (p)

MR

q1

p1

As output increases from 0 to q1, totalrevenue increases so MR > 0

As output increases beyond q1, totalrevenue decreases so MR < 0

85


• When the demand curve shifts, its associated marginal revenue curve shifts as well– a marginal revenue curve cannot be

calculated without referring to a specific demand curve

86

Supply Curve for a price taking firm

• It is the curve that tell us how much a price taking firm will produce for each possible price

• In the next few slides, we will study how to determine it

87

Analysis of the supply of a price taking firm

• Notice that if a firm is price taker, the second derivative of the profit function is equal to minus the derivative of the marginal cost

• So, the condition that the second derivative of the profit function is negative is equivalent to increasing the marginal cost must be increasing)

• A price taker firm will only maximize profits when marginal costs are increasing

88

Supply by a Price-Taking Firm

output

price SMC

SAC

SAVC

p* = MR

q*

Maximum profitoccurs wherep = SMC

89


output

price SMC

SAC

SAVC

p* = MR

q*

Since p > SAC,profit > 0

Show in the whiteboard a situation with losses !!!!

90


output

price SMC

SAC

SAVC

p* = MR

q*

If the price risesto p**, the firmwill produce q**and > 0

q**

p**

91


• We can see from previous graphs that the positively-sloped portion marginal cost function is very useful to determine how much the firm will produce because the firm maximizes profits using the rule p=MC

• So there will be a strong connection between the positively-sloped portion of the Marginal Cost curve and the Supply curve of the firm

92

Supply by a Price-Taking Firm• However, it could happen that the firm will have very

large losses even if p=MC (maximizing profits is not equivalent to have large positive profits) and hence the firm might decide not to produce at that price

• Consequently, not all the positively-sloped portion of the marginal cost curve is part of the firm supply curve

• To be part of the firm supply curve we must ensure that the firm is better of producing at that level than not producing at all!!!

• This will be different depending on whether we are in the short run or in the long run !!! So far, we will study the short-run (the long run one will be studied in the next set of slides)

93

Short-Run Supply by a Price-Taking Firm

• In the short run, we have “fixed costs” that the firm cannot eliminate them even if it decides not to produce.

• That is, in the short-run, the firm will be ready to produce and have losses, as long as the losses are not larger than the fixed costs.

• Consequently, there will be production even if price is below the minimum of the average cost in the short run

• Firms will only produce in the short run as long as the price is larger or equal to the minimum short run variable average cost.

94


– If p > SAVC, the firm operates because the difference between p and SAVC contributes towards covering fixed costs

– If p< SACV, the firm has the fixed costs PLUS a cost of (SAVC-p) for each unit produced… It is better not to produce and only have the fixed costs

• Thus, the price-taking firm’s short-run supply curve is the positively-sloped portion of the firm’s short-run marginal cost curve above the point of minimum average variable cost– for prices below this level, the firm’s profit-

maximizing decision is to shut down and produce no output.

95


output

price SMC

SAC

SAVC

The firm’s short-run supply curve is the SMC curve that is above SAVC

96

Short-Run Supply• Study example 9.3 in the book

97

Profit Functions for price taking firms

• So far, we have studied the optimal level of output of the firm.

• For this, we have formulated the firms problem of maximizing profits in terms of input prices, output prices, and level of output.

• However, there are other ways of studying the behaviour of the firm

98


• A firm’s economic profit can be expressed as a function of inputs

= pq - C(q) = pf(k,l) - vk - wl

• Only the variables k and l are under the firm’s control– the firm chooses levels of these inputs in

order to maximize profits• treats p, v, and w as fixed parameters in its

decisions

99

• The firm solves the following problem:

, ,( , ) [ ( , ) ]

k kMax k Max pf k vk w

l ll l l

• The first-order conditions for a maximum are

/k = p[f/k] – v = 0

/l = p[f/l] – w = 0

• A profit-maximizing firm should hire any input up to the point at which its marginal contribution to revenues is equal to the marginal cost of hiring the input

100

Profit Maximization and Input Demand

• These first-order conditions for profit maximization also imply cost minimization– they imply that RTS = w/v

101

Profit Maximization and Input Demand

• To ensure a true maximum, second-order conditions require that

kk = fkk < 0

ll = fll < 0

kk ll - kl2 = fkkfll – fkl

2 > 0

– capital and labor must exhibit sufficiently diminishing marginal productivities so that marginal costs rise as output expands

102

Input Demand Functions• The first-order conditions can be solved

to yield input demand functionsCapital Demand = k(p,v,w)

Labor Demand = l(p,v,w)

• These demand functions are unconditional– they implicitly allow the firm to adjust its

output to changing prices

103

Substitution and Output effect• Now we are equipped to study how

input choices changes with input prices…

• When w falls, two effects occur– substitution effect

• if output is held constant, there will be a tendency for the firm to want to substitute l for k in the production process

– output effect• the firm’s cost curves will shift and a different

output level will be chosen

104

Substitution Effect

q0

l per period

k per period

If output is held constant at q0 and w falls, the firm will substitute l for k in the production process

Because of diminishing RTS along an isoquant, the substitution effect will always be negative

105

Output Effect

Output

Price

A decline in w will lower the firm’s MC

MCMC’

Consequently, the firm will choose a new level of output that is higher

P

q0 q1

106

Output Effect

q0

l per period

k per periodThus, the output effect also implies a negative relationship between l and w

Output will rise to q1

q1

107

Cross-Price Effects

• No definite statement can be made about how capital usage responds to a wage change– a fall in the wage will lead the firm to

substitute away from capital– the output effect will cause more capital to

be demanded as the firm expands production

108

Substitution and Output Effects

• We have two concepts of demand for any input– the conditional demand for labor, lc(v,w,q)– the unconditional demand for labor, l(p,v,w)

• At the profit-maximizing level of output

lc(v,w,q) = l(p,v,w)

109

Substitution and Output Effects

• Differentiation with respect to w yields

w

q

q

qwv

w

qwv

w

wvp cc

),,(),,(),,( lll

substitution effect

output effect

total effect

110


• A firm’s profit function shows its maximal profits as a function of the prices that the firm faces

, ,( , , ) ( , ) [ ( , ) ]

( ( , , ), ( , , )) ( , , ) ( , , )k k

p v w Max k Max pf k vk w

pf k p v w p w v vk p w v w p w v

l l

l l l

l l

We use the unconditional input demands to obtain the profit function

111

Properties of the Profit Function

• If output and input prices change in the same percentage, the profits change in the same percentage– with pure inflation, a firm will not change its

production plans and its level of profits will keep up with that inflation

112


• Nondecreasing in output price

• Nonincreasing in input prices

113


• Convex in output prices– the profits obtainable by averaging those

from two different output prices will be at least as large as those obtainable from the average of the two prices

wvppwvpwvp

,,22

),,(),,( 2121

114

Graphical analysis of profits

• Because the profit function is non-decreasing in output prices, we know that if p2 > p1

(p2,…) (p1,…)

• The welfare gain to the firm of this price increase can be measured by

welfare gain = (p2,…) - (p1,…)

115


output

price SMC

p1

q1

If the market priceis p1, the firm will produce q1

If the market pricerises to p2, the firm will produce q2

p2

q2

A

B

Increase in Total Revenue= p2Aq2q1Bp1 !!!!!!!!!!!!!!!

116

• The increase in costs is the area: Aq2q1B

• Remember that:

2 2

1 1

2 1( ) ( ) ( )

area below the marginal cost curve

q q

q q

dCC q C q dq SMC q dq

dq

117

The firm’s change in profits rise by the shaded area


output

price SMC

p1

q1

p2

q2

118

Producer Surplus in the Short Run• Let’s measure how much the firm values producing at

the prevailing price relative to a situation where it would produce no output

• This is given by:

1 0

1

0

( ,...) ( ,...),

current price

shut down price

p p

p

p

This difference in profits is known as the producer surplus

119

Producer surplus at a market price of p1 is the shaded area

Producer Surplus in the Short Run

output

price SMC

p1

q1

p0

Shut down price

120


• Producer surplus is the extra return that producers make by making transactions at the market price over and above what they would earn if nothing was produced– the area below the market price and above

the supply curve

121


• Because the firm produces no output at the shutdown price, (p0,…) = -vk1

– profits at the shutdown price are equal to the firm’s fixed costs

• This implies thatproducer surplus = (p1,…) - (p0,…)

= (p1,…) – (-vk1) = (p1,…) + vk1

– producer surplus is equal to current profits plus short-run fixed costs

122

Important Points to Note:• The rule: MR=MC• The firm will never produce in the downward

sloping part of the marginal cost curve• The supply curve of the firm has to take into

account that the firm might be better of not producing if the losses at the level of output that maximizes profits are larger than the fixed costs.

123

Important Points to Note:

• Short-run changes in market price result in changes in the firm’s short-run profitability– these can be measured graphically by

changes in the size of producer surplus– the profit function can also be used to

calculate changes in producer surplus

124

Important Points to Note:

• Profit maximization provides a theory of the firm’s derived demand for inputs– the firm will hire any input up to the point

at which the value of its marginal product is just equal to its per-unit market price

– increases in the price of an input will induce substitution and output effects that cause the firm to reduce hiring of that input

me_firm.ppt

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