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MICROECONOMICS 1 – REVEALED PREFERENCE

Our earlier discussions on the behaviour of the consumer

relied on fundamental assumptions about preferences and

the budget constraint to derive demand functions for

individual consumers.


In the case of revealed preference, we want to go about

things the other way.

That is, we want to use information about the consumer’s

demand to discover information about his/her preferences.


Why this the case? In other words why the reliance on

revealed preference?

In real life, preferences are not directly observable. We

discover them by observing their behaviour.


The notion of revealed preference was introduced into

economics by Paul Samuelson (1938; 1947) in his

investigation of the empirical content of the theory that

consumers maximize their utility.


His analysis was based on observable data, and thus attempted to

characterize the data sets that are consistent with the existence of

some utility function.

The main criticism of the ordinal approach was from a

methodological point of view, in that it used non-observable

concepts and propositions.


As Samuelson (1938) argued, one ought to analyse the

consumer’s behaviour without having recourse to the

concept of utility at all, since this did not correspond to

directly observable phenomena.


Basically the theory of revealed preference makes a virtue of

assuming nothing whatsoever about the psychological causes of

our choice behaviour.

Instead, it pays attention only to what people do. It assumes that

we already know what people choose in some situations, and uses

this data to deduce what they will choose in other situations.


A few Assumptions are in order:

Preferences are stable: they remain unchanged whilst we observe

consumers’ behaviour.

Preferences also indicate rational choices by the consumer

Rational choices indicate optimization-based approach to decision

making



Consistency, that is, the consumer’s choice behaviour must be

consistent. .

Strict Preferences: It is thus usual to write A B to mean that the

consumer likes B strictly more than A. Such a strict preference

relation is said to be consistent if it is both asymmetric and transitive.



A preference relation is transitive if a b and b c implies a c.

It is only when transitivity holds that we can describe the consumer’s

preferences by simply writing a b c. Without transitivity, this

information wouldn’t imply that a c.



A preference relation is asymmetric if we don’t allow both A B

and B A.

It represents a full set of strict preferences on a set X if we insist that

either A B or B A must always hold for any A and B in X that

aren’t equal (complete or total).


The revealed preference axiom: The consumer, by choosing a collection of

goods in any one budget situation, reveals his preference for that particular

collection. The chosen bundle is revealed to be preferred among all other

alternative bundles available under the budget constraint.

The chosen ‘basket of goods’ maximises the utility of the consumer. The revealed preference for a particular collection of goods implies the

maximisation of the utility of the consumer.

MICROECONOMICS 1 – REVEALED PREFERENCE Figure 1: Revealed Preference: the bundle (x1,y1) that the consumer chooses is

revealed preferred to the bundle (x2,y2) that he could have chosen

(x1, y1) •

• (x2,y2)

Y

X

0


In Figure 1, the consumer is faced with two bundles (x1,y1) and (x2,y2).

Both bundles are clearly affordable to the consumer, as is any bundle on

or beneath the budget line.

However, bundle (x1,y1) is the optimal bundle and thus a unique

demanded bundle (for reasons that are familiar to us).


Thus, from Figure 1 we can conclude that all other bundles on or

beneath the budget line are revealed worse than the chosen bundle

(x1,y1). This is because those other bundles are affordable and could

have therefore been chosen, but were rejected in favour of bundle

(x1,y1).


The algebra of revealed preference

With quantities (xi, yi) and prices (px, py), and a given income, m the

two bundles can be expressed algebraically as follows:

For the chosen bundle (x1,y1) this condition must be satisfied � + � = �, whilst for the other bundle this condition must be

satisfied � + � �.



Thus, putting these two together, the fact that (x2,y2) is affordable

at the budget (px, py, m) means that � + � � + � .



If the above inequality is satisfied and (x2,y2) is actually different from

(x1,y1), we say that (x1,y1) is directly revealed preferred to (x2,y2). Thus

revealed preference is a relation that holds between the bundle that is

actually demanded at some budget and the bundles that could have been

demanded at that budget.


The principle of revealed preference

Let (x1,y1) be the chosen bundle when prices are (px, py), and let

(x2,y2) be some other bundle such that � + � � +� . Then if the consumer is choosing the most preferred bundle

he/she can afford, we must have , , .


The principle of revealed preference

A point worth noting! If oat porridge is revealed preferred to maize

porridge, it doesn’t automatically mean that oat porridge is preferred to

maize porridge. This is because ‘revealed preferred’ just means that oat

porridge was chosen when maize porridge was affordable (and

available).


Indirect revealed preference

In our earlier discussion, we noted that , , . Now

suppose we know that (x2,y2) at prices (p1, p2) and that (x2,y2) is

itself revealed preferred to some other bundle (x3,y3).



That is, � + � � + � . Then we know that , , and that , , .

Thus, from the transitivity assumption we can conclude that , , .



Hence, from revealed preference and transitivity, we can conclude

that (x1, y1) is indirectly revealed preferred to (x3, y3). In Figure 2

we depict the idea of indirect revealed preference. The bundle (x1,

y1) is indirectly revealed preferred to (x3, y3).

MICROECONOMICS 1 – REVEALED PREFERENCE Figure 2: Indirect Revealed Preference

F

• (x1, y1)

•

(x2, y2)

• (x3, y3)

X

Y

0


Derivation of the Demand Curve

We have discussed the concept of revealed preference. Now we can

use this concept to derive the demand curve.

As usual we make use of the budget line and the well-known

concept of compensated budget line.



Suppose the consumer is faced with the budget constraint AB in

Figure 3 and chooses bundle Z, thus revealing his preference. Of

course everything else we learnt previously is obvious; within the

class of bundles affordable, Z is revealed preferred to all.

MICROECONOMICS 1 – REVEALED PREFERENCE Figure 3: Derivation of the Demand Curve

•

• •

Z

N

W

X1 X2 B x3 B’ C

y

x

A’

A



Now suppose the price of commodity X falls, such that the new

budget line rotates outwards, to become AC.

But first, suppose we make a ‘compensating variation of income so

that the consumer is left with just enough money to continue

purchasing bundle Z if he/she so wishes.



The compensated budget line is shown by A’B’, which passes

through bundle Z to illustrate the idea of income compensation.

Because bundle Z is still available to the consumer, he/she will not

choose any bundle to the left of Z. Why?



Thus, the consumer will continue to consume bundle Z, in which

case the substitution effect of the price fall is zero, or choose a batch

on the segment ZB’, such as bundle W (which includes larger

quantities of commodity X, for which the substitution effect is

negative).



Now if we allow the consumer to move back to the new budget

line, AC, the consumer may choose a bundle to the right of W, say

N (if commodity X is normal with a positive income effect).



The new revealed equilibrium position (N) includes a larger

quantity of commodity X (x3) resulting from the fall in its price.

Thus, the revealed preference axiom and the implied consistency of

choice lead us to a direct derivation of the demand curve: as price

falls of a commodity, more of it is purchased.


The Weak Axiom of Revealed Preference

How do we know the consumer is following the maximising

model?

What kind of observation would lead to us to conclude that the

consumer was not maximising?



The weak axiom of revealed preference can be stated as follows: if

(x1, y1) is directly revealed preferred to (x2, y2), and the bundles are

not the same, then it cannot happen that (x2, y2) is directly revealed

preferred to (x1, y1). The weak axiom can be explained using

Figures 4 and 5.

MICROECONOMICS 1 – REVEALED PREFERENCE Figure 4: Violation of the Weak Axiom of Revealed Preference

•

(x1, y1)

• (x2, y2)

Y

X



In Figure 4, we observe the consumer making choices that do not

follow from the logic of revealed preference.

This is because two conclusions can be arrived at: 1) we observe that is

one case, (x1, y1) is revealed preferred to (x2, y2); and 2) in another

instance, (x2, y2) is revealed preferred to (x1, y1).


Figure 5: Satisfying the Weak Axiom of Revealed Preference

● (x1, y1)

● (x2, y2)

Y

X

A

B



The Weak Axiom of Revealed Preference is however satisfied in

Figure 5. Here we observe that his/her choices are consistent with

the logic of revealed preference. That is, when either bundle of

goods is chosen, the other is not affordable to the consumer.



This leads us to restate the weak axiom of revealed preference

algebraically as follows:



If a bundle of goods (x1, y1) is purchased at prices (p1, p2) and

different bundle (x2, y2) is purchased at prices (p3, p4), then if

� + � � + � ,

then it must not be the case that

◦ � + � � + �



In simple words, if bundle B is not affordable when bundle A is

purchased, then when bundle B is purchased, bundle A must not be

affordable.


The Strong Axiom of Revealed Preference

The weak axiom requires that if A is directly revealed preferred to

B, then we should never observe B being directly revealed preferred

to A.

The Strong Axiom of Revealed Preference requires that the same

sort of condition hold for indirect revealed preference.



More formally, the Strong Axiom of Revealed Preference can be

stated as follows: if (x1, y1) is revealed preferred to (x2, y2) (either

directly or indirectly),and (x2, y2) is different from (x1, y1), then

(x2, y2) cannot be directly or indirectly revealed preferred to (x1, y1).



Thus, the Strong Axiom of Revealed Preference is a necessary

implication of optimizing behaviour: if a consumer is always

choosing the best things that he/she can afford, this his/her

observed behaviour must satisfy the strong axiom of revealed

preference.



Further, we can also conclude that the Strong Axiom of Revealed

Preference is a necessary condition for optimizing behaviour: if the

observed choices satisfy the strong axiom of revealed preference,

then it is always possible to find preferences for which the observed

behaviour is optimizing behaviour.

MICROECONOMICS 1 – PRODUCTION THEORY

Here our emphasis is on the behaviour of the firm; thus an

examination of the supply side of the circular flow (the concept of

the circular flow must be familiar to us).

Firms are crucial productive agents in an economy, and are

engaged in the conversion of resources (inputs) into final goods

(output).

1 LECTURE MATERIAL ON MICROECONOMICS 1: PREPARED BY DR. EMMANUEL CODJOE


In thinking of firms, we can categorise them by many factors

and aspects. These include:

Sectors

Production Scale

Ownership



Of course, it is worth noting that households engage in production, but

our analysis would not focus on household production. We mainly

concern ourselves with production by firms as is the feature of modern

economies.

Production can be interpreted very broadly to include the production of

both physical goods, such as rice, automobiles, etc., and services, such

as legal advice, medical care, financial services, etc.



Inputs and Output(s): production is the process of transforming

inputs into output(s). Many production processes require a wide

variety of inputs.

Inputs are also termed factors of production. Broadly defined,

these include: land; labour; raw materials; capital (physical and

financial); intermediate goods purchased from other firms.



Fixed and Variable Factors: in any given time period, some

inputs may be difficult to adjust. This is because the firm may have

contractual obligations to employ certain inputs at certain levels

(e.g., the lease on a building which houses the factory and offices).

A fixed factor refers to a factor of production that is in a fixed

amount for the firm under the period under examination.



A variable factor is a factor of production that can be used in

different amounts under the period under examination.

A quasi-fixed factor refers to an input that must be used in a fixed

amount, independent of the output of the firm, as long as output is

positive (e.g., electricity for lighting and water for cleaning)



The production decision by firms is usually constrained by several factors.

These include:

Customers – preferences, demand conditions, etc.

Competitors – products, prices, quality, market size, etc.

Nature – what is possible given available resources. The quality and quantity

of these resources define the different ways by which any output can be

produced.



Describing Technological Constraints: we have seen earlier that

nature imposes technological constraints on firms. In other words,

only certain combinations of inputs are feasible ways to produce a

given amount of output.

The firms is thus limited to the technologically feasible production

plans available to it.



The production set describes the set of all combinations of inputs

and outputs that comprise a technologically feasible way to

produce.

If we take the case of a single input production relation, where x is

the input and y the output, then the production set might take the

shape indicated below.


10

A Production Set illustrating the possible shape for a production

Function

Y = f(X)

X

Y

• • •

C

D

B

Production Set

Inefficient point

Production Function

LECTURE MATERIAL ON MICROECONOMICS 1: PREPARED BY DR. EMMANUEL CODJOE


The production set shows the possible technological choices facing a firm.

And as long as the inputs to the firm are costly it makes sense to limit our

analysis to the maximum possible output for a given level of input.

This limit is the boundary of the production set. The function describing the

boundary of this set is known as the production function; it measures the

maximum output that can be obtained from a given amount of input.



The concept of the production function applies equally well if

there are several inputs. In the typical case of two inputs, the

production f (x1, x2) would measure the maximum output, y, given

the inputs x1 units of factor 1, and x2 units of factor 2.

In the two input case, the production function can conveniently be

depicted as an isoquant.



An isoquant is the set of all possible combinations of inputs 1 and

2 that are just sufficient to produce a given amount of output.

Or, the locus of all the technically efficient methods (or all the

combinations of factors of production) for producing a given level

of output.



Isoquants are similar to indifference curves, but differ in one

important respect. Further, the shape of the isoquant is

defined by the nature of the production technology.

Consequently, isoquants may assume different shapes. Here

we will only consider a few cases.



Fixed Proportions Isoquants (Leontief production function):

here the production technology requires specific minimum input

combinations.

The production function is written as � � ,� = min{� , � }

E.g., in painting a wall or building, one painter would need one

paint brush.



An illustration of a fixed proportion isoquant.


Paint brush

Painter

Q = 200 m2

Q = 400 m2

Also known as the Input-

Output Isoquant, it

assumes strict

complementarity (i.e.,

zero substitutability of

inputs in the production

process.


An illustration of the perfect substitute (linear) isoquant.


Red pencils

Blue pencils

Q = 20

Q = 30

This type of production

technology assumes perfect

substitutability of inputs in the

production process


An illustration of the Kinked isoquant.


P0

P1

P2

P3

The Kinked Isoquant

assumes limited

substitutability. There are

only few processes for

producing any commodity.

Substitutability of factors is

only possible at the kinks.

19

The Convex (Smooth) Isoquant. Also known as the Cobb-Douglas Production Function

X1

X2

0

Q = 120

Q = 100

As the number of processes

increases the kinked line

looks increasingly like the

typical isoquant. The

smooth isoquant assumes

continuous substitutability

of inputs in the production

process.


The Cobb-Douglas Production Function: this production

is widely used in economics both for theoretical and

empirical work.

The usual form this production function takes is given by � � , � = ��



The magnitude of the production function does matter, so we have

to allow these parameters to take arbitrary values.

The parameter A measures the scale of production: how much

output we would get if we used one unit of each input.

The parameters a and b measure how the amount of output

responds to changes in inputs.



Given the following Cobb-Douglas production function, � � , � = �� , several important relationships can be derived.

First, we look at the marginal products of x1 and x2, then the marginal

rate of technical substitution, returns to scale, factor intensity, elasticity

of substitution.



We continue from where we left off, examining some

concepts in production theory.

As we saw earlier, the Marginal Rate of Technical

Substitution (MRTS) measures the trade off between two

inputs in production with output constant.



A formal derivation of the formula for the MRTS is given below.

Suppose our production function as generally specified is given by

Then if we consider the change in the use of factors 1 and 2 that

keeps output unchanged, then we have:


),( 21 xxfy

LECTURE MATERIAL ON MICROECONOMICS 1: PREPARED BY DR. EMMANUEL CODJOE 3


),(

),(

),(),(

,0),(),(

212

211

1

2),(

22121211

22121211

21 xxMP

xxMP

x

xMRTS

xxxMPxxxMP

xxxMPxxxMPy

xx


It is worth highlighting that the Marginal Rate of Technical

Substitution (MRTS) is also the slope of the isoquant.

That is,


211 ,

1

2

1

2 | xxyy MRTSx

x

dx

dx


Another closely related assumption about the nature of technology

embodied in the production process is that of Diminishing

Marginal Rate of Technical Substitution (MRTS).

That is, as we increase the amount of one factor, say x1, and adjust

the second factor, say x2, so as to stay on the same Isoquant, the

MRTSx1,x2 declines.



It means that the slope of the MRTS must decrease in

absolute value as we move East and must increase as we

move North

Not to be confused with the law of diminishing marginal

product (law of diminishing marginal returns).


7

MRTSL,K is high; labour is scarce

so a little more labour frees up

a lot of capital

K

L

•

•

MRTSL,K is low; labour is

abundant so a little more

labour barely affects the

need for capital

LECTURE MATERIAL ON MICROECONOMICS 1: PREPARED BY DR. EMMANUEL CODJOE



Diminishing MRTS and diminishing marginal returns are

closely related but are not exactly the same.

Diminishing marginal returns is an assumption about how

the marginal product changes as we increase the amount of

one factor, holding the other factor fixed.



But diminishing MRTS is about how the ratio of the

marginal products – the slope of the isoquant – changes as we

increase the amount of one factor and reduce the amount of

the other factor so as to stay on the same isoquant.



Elasticity of Substitution: the MRTS despite its insights has

a serious defect, in that it is dependent on the units of

measurement of the factors.

Hence a better measure of the degree of factor substitution is

the elasticity of substitution.



Elasticity of Substitution: this is defined as the percentage

change in the capital-labour ratio, divided by the percentage

change in the MRTS.

Formally, the elasticity of substitution, can be

represented by the following expression:


LECTURE MATERIAL ON MICROECONOMICS 1: PREPARED BY DR. EMMANUEL CODJOE 12



Factor Intensity: the factor intensity of any process is measured

by the slope of the line through the origin representing the

particular process.

In other words, the factor intensity is the capital-labour ratio at

the particular point of interest in the production process.




A

B

100

30

50 200 L

K

Q=12


In the figure above, what is apparent regarding points A and B are that:

That is, the upper part of the isoquant is more capital-intensive than

the lower part, which includes more labour-intensive techniques



In general therefore, a production method that uses relatively

more labour than capital is labour-intensive, while a method

that uses relatively more capital than labour is capital-

intensive.



Returns to Scale: now suppose that instead of increasing one

input whilst the other is held unchanged, we increase both inputs

in the production process.

In other words, let’s scale the amount of all inputs up by some

constant factor, e.g., use three times as much of x1 and x2



Returns to Scale describes the relationship between inputs and output

when all factors of production vary. In other words, it describes the

output response to a proportionate increase of all inputs.

In general, if we scale all inputs by some amount, t, then three

possibilities can arise: constant returns to scale, increasing returns to

scale and decreasing returns to scale.



Returns to Scale are easily defined for homogeneous production

function. A production function is homogeneous of degree k if

where k is a constant and t is any positive real number.


),(),( 2121 xxfttxtxfk


Constant Returns to Scale: this arises if we use twice as much

of each input and we get twice as much output. Thus, in the case

of two inputs,


2� �1, �2 = �(2�1, 2�2) �� 1 , �2 = �(��1 , ��2)


Why might we expect this outcome? It should be possible for the firm

to replicate what it was doing before. Thus, if the firm has twice as

much of each input, it can just set up two plants side by side and

thereby get twice as much output.

But it is perfectly possible for a production technology to exhibit

constant returns to scale and diminishing marginal product to each

factor.



Increasing Returns to Scale: this arises when we scale up all

inputs by some factor, t, we get more than t times as much

output.

Mathematically, this is given by

for all t > 1


�� 1, �2 > �(��1 , ��2)


Decreasing Returns to Scale: this arises when we scale up all

inputs by some factor, t, we get less than t times as much output.

Mathematically, this is given by

for all t > 1


�� 1, �2 < �(��1 , ��2)


Returns to Scale: in general if both inputs are increased by the

factor t, and output is increased by the factor tk , returns to scale

are increasing if k > 1, constant if k = 1, and decreasing if k < 1.



Economies of Scale Vs. Returns to Scale

A production process is said to exhibit economies (constant

economies, diseconomies) of scale over a particular range of

output per unit of time if the long-run average production costs

fall (remains unchanged, increases) as output increases.



Economies of Scale Vs. Returns to Scale

The term returns to scale refers to the effect on output of a

proportionate change in the level of use of all inputs.

It is therefore important not to confuse the two concepts.


PRODUCTION COSTS

In this section we introduce production costs into the analysis of the firm. So far, our emphasis has been on the production process without any consideration of costs.

However, production activities do involve costs – implicit and explicit.

But cost is a rather complicated concept. It is a term that is open to more interpretations. Note for example the difference between consumers’ cost and producers.

1 Econ 311 Microeconomics 1 Lecture Material Prepared by Dr. Emmanuel Codjoe

PRODUCTION COSTS

Social Costs of production, refer to costs to society when its resources are employed to make a given commodity. Since economic resources are limited, when resources are used to produce a certain product, less can be produced of some other product that can be made with those resources.

Private costs, are in contrast with social costs. Private costs are defined to be costs to the individual firm or producer. Take case of pollution!


PRODUCTION COSTS

Explicit costs include the ordinary items that

an accountant would include as the firms

Expenses, e.g., payroll, payments for raw

materials, etc.

Implicit costs (alternative costs or opportunity

cost) include opportunity costs of resources

owned and used by the firm’s owner. This type is often omitted in calculating the firm’s costs.

These costs generally come under private costs


PRODUCTION COSTS

The economic cost of any activity is the value

of the best forsaken alternative.

The firm in order to attract the resources or

“factors” necessary to engage in production, must pay resource owners amounts sufficient to

induce them to sacrifice their best alternatives

(whether employment or leisure).

But it is important to note that in production,

cost and price are different.


PRODUCTION COSTS

As we shall see later, costs are important in determining a firm’s optimum profit position.

They are also the basis for a firm’s supply curve.

Having noted all these, it is worth stating the goal of the firm. Economists usually assume that the firm maximises profit, which is defined as the difference between revenue and cost.

This section on costs assumes that the firm under analysis is a competitive or “price-taking firm”.


PRODUCTION COSTS

Distinguishing between short-run and long-run costs

This is related to the same concept in production theory.

In the short-run some costs are fixed, whilst in the long-run they become variable. This is the fundamental difference between the two.

Nevertheless, the distinction is a matter of degree.


PRODUCTION COSTS

Distinguishing between short-run and long-run costs

The longer the run contemplated, the greater the range of costs regarded as variable rather than fixed.

Hence, if a firm is not committed to any outlays, it is in the long-run. In the long-run all options are open!

The situation changes to the short-run once a commitment to some factor of production has been undertaken.


PRODUCTION COSTS

Distinguishing between short-run and

long-run costs

Thus, the short-run is characterised by

fixed and variable costs.

In the long-run, all costs are variable

since all costs depend on the volume of

output.


PRODUCTION COSTS

Total, Average and Marginal Costs

The total cost can be regarded as the sum, taken

over all resources employed, of factor prices times

factor quantities.

In other words, it is the sum of all costs incurred

by the firm to produce a given level of output.

From the total cost, two other measures emerge:

average and marginal costs.


PRODUCTION COSTS


The average cost (AC) is defined as the cost

per unit of output. Formally, this is defined as:

Where TC is total cost and Q is output

Q

TCAC


PRODUCTION COSTS


The marginal cost (AC) is defined as the

change in total cost resulting from a unit change

in output. Formally, this is defined as:

Where TC is total cost and Q is output

Q

TCMC


PRODUCTION COSTS


Some important conclusions are worth making!

We have noted earlier that in the short-run, the

firm faces both fixed and variable costs. But as the

firm alters its output, only the variable costs

change (why?).

The marginal cost that a firm experiences as it

expands output from given fixed resources are

entirely due to its variable costs.


PRODUCTION COSTS


This therefore leads to a major conclusion!

Decisions about output are based

entirely on marginal costs; fixed

costs are totally irrelevant to any

output decisions.


PRODUCTION COSTS


. But based on our understanding from

production, we know that cost is a multivariable

function, that is, it is determined by many

factors.

Thus, in the short-run,

),,,(

KPTXfC f


PRODUCTION COSTS


Where C = total costs; X = output; Pf =

prices of factors; T = technology; and

= fixed factors.

In the long-run

K

),,( fPTXfC 15 Econ 311 Microeconomics 1 Lecture Material Prepared by Dr. Emmanuel Codjoe

PRODUCTION COSTS


We have earlier made the assumption that the

firm is a competitive firm.

Thus, it seeks to be efficient in production,

aiming to produce at the minimum cost of

production for any given output level, Q, when

factor prices are Pf .

If we also assume that firms are price takers in

the factor markets, then Pf is fixed.


PRODUCTION COSTS


Thus, we can write our cost function as

dependent on output, Q, alone. This can be written

as

Hence total costs can be written as

)(QcC

FCQcC )(


PRODUCTION COSTS


The marginal cost function can be written as

Q

FC

Q

Qc

Q

QcQMC v

)()(

)(

VCQ

QcQMC v

)(

)(


PRODUCTION COSTS


But note that the marginal cost measures the rate of change, hence we can define the marginal cost function as

Refresh your memory on the relationship between MC, AVC and AC.

dQ

Qcd

dQ

dTCQMC

)]([)(


PRODUCTION COSTS


PRODUCTION COSTS


The marginal cost curve, average variable

cost curve and average total cost curves are

generally U-shaped.

The U-shape in the short run is attributed to

increasing and diminishing returns from a

fixed-size plant, because the size of the plant is

not variable in the short run.


PRODUCTION COSTS


The marginal cost and average cost curves are

related:

When MC exceeds AC, average cost must

be rising

When MC is less than AC, average cost

must be falling

This relationship explains why marginal cost

curves always intersect average cost curves at

the minimum of the average cost curve.


PRODUCTION DECISION

Optimal Input Combinations

How will a competitive firm combine inputs to produce a given quantity of output?

As a first approximation, we assume that firms are out-and-out profit maximizers; that is to maximize the difference between revenue (R) and (economic) costs (C) incurred.


PRODUCTION DECISION


Thus, our typical firm seeks to maximize profits;

The assumption of profit maximization implies

that a firm will seek to minimize the costs of

producing a given output or seek to maximize

the output derived from a given level of cost.

CRmax


PRODUCTION DECISION


Also remember the assumption of perfect

factor markets, such that firms are price takers

in the input markets.

Thus, if we suppose there are two inputs,

labour (L) and capital (K), then what

combinations of L and K should the firm

choose?


PRODUCTION DECISION


If the wage rate (w) is the cost of labour and

the rental rate (r) is the cost of capital, then the

total cost outlay, C, is given by:

Lr

w

r

CK

rKwLC


PRODUCTION DECISION


Thus, the various combinations of L and K

that can be purchased, given input prices and

total outlays can be represented by a straight

line.

K

L C/w

C/r Slope = (w/r)

0


PRODUCTION DECISION


A family of Isocost lines can be illustrated

below

K

L 0

C(3)

C(2) C(1)

C(3) > C(2) > C(1)


Maximization of output for given cost

L

K

0

100

200

300

R

L*

K*


PRODUCTION DECISION

Optimization Condition

The firm maximizes output at point R, by

choosing L* and K* of labour and capital

respectively .

At point R, the isoquant is tangent to the

isoquant. Thus,

r

w

MP

MPMRTS

K

LKL ,


PRODUCTION DECISION


It follows that the optimal combination of inputs

is where.

Or

r

w

MP

MP

K

L

r

MP

w

MP KL


PRODUCTION DECISION


More generally, a firm will choose an input

combination such that.

Where MPa, MPb, ... ..., MPn are the marginal

products of inputs, a, b, ..., n, and Pa, Pb, ... ..., Pn

are input prices.

n

n

b

b

a

a

P

MP

P

MP

P

MP .......


Minimization of Cost for a Given

Output Level

L

K

0

400

Z

L*

K*


PRODUCTION DECISION


To minimize the cost of producing the output level, Q=400, the firm chooses point Z. Here too, the firm must equate the MRTS to the ratio of input prices

r

w

MP

MPMRTS

K

LKL ,


PRODUCTION DECISION

Constrained Optimization

The firm’s decision, as with the case of consumers, can be represented as a constrained optimization problem.

We first consider the case of constrained output maximization.

Thus, given Q = f(K, L) and C = rK + wL, we can set up the Lagrangian function:

)(),( CwLrKLKfZ 35 Econ 311 Microeconomics 1 Lecture Material Prepared by Dr. Emmanuel Codjoe

PRODUCTION DECISION


Z symbolises the Lagrangian function. Here

is an undetermined Lagrange multiplier

Also note that another formulation of the

Lagrangian function is:

)(),( wLrKCLKfZ

0


PRODUCTION DECISION


We set the first-order conditions, which are to

set to the partial derivatives of K, L, and λ equal to

zero.

)1(0

rfK

Zk

)2(0

wfL

Zl


PRODUCTION DECISION


From (1) and (2): moving the price terms to the

right and dividing (2) by (1):

)3(0

CwLrKZ

)4(r

w

f

f

k

l


PRODUCTION DECISION


From (4) the first order conditions state that the

ratio of marginal products must be equated with

their price ratios.

Solving (1) and (2) for λ, yields:

)5(w

f

r

f lk


PRODUCTION DECISION


(5) states that the contribution to output

of the last money outlay expended on each

input must equal λ.

The multiplier, λ, is the derivate of

output with respect to cost with prices

constant and output(s) variable.


PRODUCTION DECISION

Constrained Cost Minimization

The firm may desire to minimize the cost of

producing a prescribed level of output.

As with our earlier analysis, we form the

Lagrangian function, and set the partial

derivatives to zero for K, L, and λ.

),(( 0LKfQwLrKZ


PRODUCTION DECISION


An alternative formulation of the Lagrangian

function is:

In what follows, we set the various partial

derivatives to zero and obtain the optimal

conditions for input combination in production

)),(( 0QLKfwLrKZ


PRODUCTION DECISION


)'1(0

kfrK

Z

)'2(0

lfwL

Z

)'3(0),(0

LKfQZ


PRODUCTION DECISION


Since r and w are both positive, moving the

price terms and dividing (2’) by (1’), we obtain:

The first order conditions for the minimization

of cost subject to an output constraint are similar

to those for the maximization of output subject to

a cost constraint.

MRTSf

f

r

w

k

l


PRODUCTION DECISION


As noted, this condition above is same as that in output maximization

The multiplier, λ, in cost minimization is the derivative of cost with respect to output level (i.e., the firm finds the lowest isocost line which is tangent to the relevant isoquant.


MRTSf

f

r

w

k

l

PRODUCTION DECISION

Second-Order Conditions

We shall leave out the details of second-order

conditions. However, more generally, for output

maximization subject to a given cost, and for cost

minimization subject to a given output level, the

slope of the marginal product curves for the two

inputs must be negative.

or: 0;02

2

2

2

L

Q

K

Q 0;0 llkk ff


PRODUCTION DECISION


The second-order conditions ensure that we

are satisfied that the isoquants are convex to

the origin.

If the isoquant is concave, then we have a

corner solution.


PRODUCTION DECISION


e

e1

e2 L

K

0

Q=230


PRODUCTION DECISION


In the diagram above, output, Q=230, can be

produced at points e, e1 and e2.

This indicates different costs of producing the same level of output.

The lowest cost point is given by e2.


PRODUCTION DECISION

Profit Maximization: A Formal Analysis

A firm is usually free to vary the levels of both cost and output, with the ultimate objective being to maximize profits rather than a solution to a constrained maximum or constrained minimum problems.

The firm we have been analysing is assumed to operate in a competitive market. Its total revenue is given by the number of units of Q sold multiplied by the fixed unit price it receives.


PRODUCTION DECISION


The firm’s profit is thus defined as:

Given Q = f(K, L) and C = rK + wL, the firm’s profit function is given by

П = Pf(K, L) – rK - wL

Profit is a function of K and L and is thus maximised with respect to these variables.

CQP .


PRODUCTION DECISION


Differentiating the profit function with

respect to capital and labour, gives:

Moving input price items to the right, we

have:

)6(0 rPfkk

)7(0 wPf ll


PRODUCTION DECISION


Thus, Pfk and Pfl are the values of the

marginal product of K and L respectively, and

they represent the rate at which the firm’s revenue would increase with further increases

in K or L.

rPf k wPf l


PRODUCTION DECISION


Profit maximization requires that each input be utilised up to the point at which the value of its marginal product equals its price.

Profits can be increases as long as Pfk > r and Pfl > w.

That is, as long as the addition to revenue from employing an additional unit of input exceeds its cost.


PRODUCTION DECISION


The second-order conditions for profit maximization require that:

These suggest that profits must be decreasing with respect to further increases in L and K.

02

2

llPfL

02

2

kkPfK


PRODUCTION DECISION


Because P > 0, this suggests that the marginal

product of both L and K must be decreasing.

The conditions for first- and second-order

profit maximization require that the isoquant be

strictly concave in the neighbourhood of the a

point at which the first-order conditions are

satisfied with non-negative levels of inputs (K

and L). 56 Econ 311 Microeconomics 1 Lecture Material Prepared by Dr. Emmanuel Codjoe

PRODUCTION DECISION

Optimal Expansion Path in the Short-Run

In the short-run, K is fixed, the firm is therefore

forced to expand along a straight line parallel to

the horizontal axis.

With prices of factors constant, the firm does not

maximise profits in the short-run, due to the

constraint of given capital.

The optimal path would be along OA, but the

firm can only expand along in the short run. __

KK


PRODUCTION DECISION

Optimal Expansion Path in the Short-Run

0

_

K_

K

A

K

L 58 Econ 311 Microeconomics 1 Lecture Material Prepared by Dr. Emmanuel Codjoe

PRODUCTION DECISION

Optimal Expansion Path in the Long-Run

In the long-run all factors are variable.

Output can therefore be expanded without

limitation.

As is always the case, the firm’s objective of profit maximization, this means it chooses the

least-cost combination of inputs, which is

represented by the points of tangency between

the isocosts curves and isoquants.


PRODUCTION DECISION


0

E

K

L

150 120 80


PRODUCTION DECISION


The expansion path indicates how, as output rate changes (but input prices remain fixed),the quantity of each input changes.

If the production function is homogeneous, the expansion path will be a straight line through the origin, whose slope depends on the ratio of factor prices.

With only two inputs in production, it is also easy to derive the long-run cost function from the expansion path.


PRODUCTION DECISION


It is also worth noting that the maximum

profit-input combination lies on the expansion

path.

Given Pfl = w and Pfk = r, we note that this is a

special case of the constrained output

maximization discussed earlier.


PRODUCTION DECISION


That is, along the long-run expansion path, the

condition is satisfied.

Also the implies that profit is also maximised,

that is,

r

w

f

f

k

l

r

w

Pf

Pf

k

l


PRODUCTION DECISION

Input Demand Functions

The firm’s input demands are derived from the underlying demand for the goods and services it

produces.

Thus, the firm’s input demand functions are obtained by solving the first-order conditions for

profit maximization for L and K, as functions of

input prices and output price.


PRODUCTION DECISION


Therefore, more generally, given a production

function of the Cobb-Douglas form, we can obtain

the firm’s input demand functions

1:0,,

KALQ

rKwLKPAL


PRODUCTION DECISION


Solving for L and K, we obtain:

01 wKALPl

01

rKALPk

),,(;),,( **prwfKprwfL

0;0;0 prw fff66 Econ 311 Microeconomics 1 Lecture Material Prepared by Dr. Emmanuel Codjoe

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