bridging economics chapter 3 subject guide

7/25/2019 Bridging Economics Chapter 3 Subject Guide

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Chapter 3: Production and the behaviour of the firm

93

Chapter 3: Production and the behaviour

of the firm

Learning outcomesAt the end of this chapter, you should be able to:

define the concepts of productive efficiency, isoquants and iso-cost,

marginal and average product, constant returns to scale, transaction

costs, diminishing marginal return fixed and variable costs, the

relationship between marginal and average costs

use these definitions to give examples of increasing returns to scale, the

interrelation between marginal cost and returns to scale, homogeneous

production function, and profit maximisation with respect to output

and the firms supply curve

construct cost-functions and derive their relation to the productionfunction

use diagrams to analyse problems involving short-run and long-run

average cost schedule.

Reading

BFD Chapters 6 and 7.

LC Chapter 6.

Production functions

Reading

LC Chapter 6 p.117.

Production is a process whereby the combined activity of various economic

goods creates another economic good. To simplify our explanations, we

will divide all goods used i n t h e p r o d u c t io n o f o t h e r g o o d s into two

categories: L a b o u r (L) and Capi ta l (C). Capital denotes all n o n - l a b o u r

inputs, such as machinery and other physical assets..

In the real world, theL and C inputs are typically aggregates of various kinds

of labour, and various kinds of capital, but again we will simplify and assume

that there is only one kind of labour input and only one kind of capital input.

The process of production simply relates a set of quantities of inputs K

andL (f a ct o r s o f p r o d u c t io n ) to a quantity of output,X. This can be

expressed as a function,X =f(K,L). The function gives the level of output

X for any combined level of capital and labour inputs.

From an analytical point of view, the production function is very much

the same as the utility function (see Chapter 2). In both cases, we use

economic goods to produce another economic good. The difference is that

in the case of the production function we produce t a n g i b l e economic

goods, while in the case of utility (which is an abstract notion) we

produced a non-tangible economic good.

Since we assume that factors of production are scarce, and that theoutput produced is desirable, we are interested in finding the optimal

combination of inputs that will produce the optimal amount of output a

state that we call pr odu c t ive e f fi ci ency.

Previously we have

used the letter C to

refer to Capital; from

here on we will use the

letter K, which is the

normal convention in

economics.

Contrast this with

utility functions,

where our goal was to

choose consumptionin such a way as

to maximise utility.

This corresponds to

allocative efficiency.


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02 Introduction to economics

94

In order to find the state of pr odu c t ive e f fi ci ency, we first have to look

at the circumstances of production.

Suppose that a certain commodityX is produced by using labour and

capital. How does this production process work?

In Figur e 3 .1 we set the horizontal axis to denote the quantity of

labour (L) used (measured in work-hours) while the vertical axis denotes

quantities of capital (K) used (measured in machine hours).

Figure 3.1: Combinations of production inputs.

AtA, the combination ofL0units of labour (measured in hours) andK

0

units of capital (measured in hours) can produceX0units ofX, soX0=f(K

0, L

0). At any point in quadrant III, the combination of labour and

capital will yield less units ofX thanX0. All points in quadrant I suggest

combinations of inputs which yieldX X0. There are points in quadrants

IV and II where the loss of output as a result of a fall in the level of one

input can be offset by an increase in the use of the other input (compare

this to our analysis of utility in Chapter 2).

Choose a point in the four quadrants at random. What can you say about output in the

quadrant you have pinned?

Put differently, we can increase the level of output if we increase oneor both inputs, just as we could increase utility if we increased our

consumption of one or more goods. However, as with utility analysis, this

implies two things about our production function: First, that technology

is continuous, and second, that the two inputs can be d iv ided and

s u b s t i t u t e d at all levels. This means that when we move from a point

such asB in quadrant III to a point like C in quadrant I, output increases

continuously as we continuously increase both inputs. Hence, as atB X

X0and at C X X

0, there is a point likeDwhereX =X

0. This means that

all the other points of equal output (called i s o q u a n t s) are located in

quadrants IV and II. Like ind i f fe re nce cu rves (see Chapter 2), they will

be located on a downward sloping line.

The assumptions of divisibil i ty and subs t i tu tab i l i ty are not as

straightforward as in the case of utility analysis. Divisibility simply means

that we can measure inputs in terms of their time contribution. If there


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is divisibility, we can talk of any fraction of a unit that can affect output.

While some may have difficulties with this, we shall not consider it as a

serious problem. Substitutability is a much more difficult problem.

Consider linear production processes, and assume that we have three

different such production processes, or technologies: T1, T

2and T

3. Each

technology requires different combinations of inputs in the production

process, as indicated by the slope of the rays through the origin.Furthermore, assume that we cannot increase output by merely increasing

one of the inputs. We must use both and maintain the same proportions

between them. (This is called a Leontief-style production function.) So,

in the case ofA in process 1, you needL10units of labour andK1

0units of

capital to produce 1 unit ofX (Figure 3 .2).

Figure 3.2: Production isoquants.

The isoquant is L-shaped because increasing only labour or capital

(pointsB or C) will not increase output. Many people think this is how

production technologies really look. But in such a case there is clearly

no subs t i tu tab i l i ty between the means of production. If you want to

produce 2 units ofX using process 1, you will need to move to a point like

D, where both labour and capital have been increased toL11.units of labour

andK11units of capital. The fixed proportion between the two means of

production is captured by the ratio of capital to labour (K/L) which is

depicted by the slope of the ray from the origin. At bothA andD.

However, if we take substitutability to mean that we can choose

combinations of technology, then we may be able to move closer to the

notion of substitutability. If pointsA andE represent the combination

of capital and labour required for the production of one unit ofX under

technologies 1 and 2 respectively, pointF represents the inputs required

to produce one unit ofX by a combination of the two technologies. The

greater the share of technology 1 has in the production, the nearerF

will be toA. The greater the share of technology 2 in the production, the

nearerFwill get toE.


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If, in addition, we have technology 3 to consider,F ' depicts the level of

inputs required for the production of one unit ofXwith the combination

of technologies 2 and 3. As before, the more of technology 2 we use,

the nearerF ' will be toE. The more of technology 3 we use, the nearer

F F ' will be to G. If indeed we can mix technologies, we can see that the

levels of inputs required for the production of one unit of output are

arranged along the heavy curve in figure n, which may remind you of

the indifference curve. This means that if we have a very large number

of technologies available, the move from one technology to another can

constitute the notion of input substitutability.

Therefore, we would claim that while smooth functions may not directly

capture the nature of the production process, they nevertheless constitute

a good abstraction of it. We will obviously be looking for a function that

will be able to generate similar properties as the mixture of processes (i.e.

smooth isoquants which are convex).

Such a production function is given byX =f(K,L), wheref is a real-

number, continuous and twice differentiable function.

Properties of the production function.

Figure 3.3: Properties of the production function.

a. It is increasing in L and K (): As we increase the amount of one orboth inputs used, output increases.


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Figure 3.4: Marginal product.

b. The marginal product (MP) of Capital and Labour: Defines the increase

in output for a one-unit increase of a particular input, keeping the

other one constant:

atL0for a given level of the other input (K)

We typically assume that the marginal product is increasing for low levels

of an input, but decreasing for high levels. This is referred to as increasing

and diminishing returns to a factor, respectively. This assumption yields

a graph as in Figure 3 .4 , which depicts the level of output attainable

for every level of one input (here, Labour), keeping the level of the other

input (here, Capital) constant

c. I s o q u a n t s : Combinations ofK andLwhich yield the same level of

output are arranged on a curve going through regions IV and II in

Figur e 3 .3 above. This curve is called the i s o q u a n t , and is defined

for every level of output.

d. The s l o p e o f t h e i s o q u a n t is defined as dK/dLwhenX is

unchanged. If we changeL by dL, output will change by dL MPL[HS1],

given property (b) above. In the same way, if we changeK by dK,

output will change by dK MPK. Along the isoquant, the change in

output as a result of a change in K has to equal the change in output asa result of a change inL. Hence:

dL MPL= dK MP

K

which implies

We thus have to change the amount ofKwe use by this much to

substitute a unit ofL for output to remain unchanged.

e. Di m i n i s h in g m a r g i n a l p r o d u c t : because of the diminishingMP.

In terms of Figure 3 .5 , this means that the slope of the isoquant isgetting flatter asL increases (and steeper asK increases).


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Figure 3.5: The slope of the isoquant.

Returns to scale

LetX denote output.K andL denote the two factors of production, capital

and labour respectively, andf represents the production function:

X =f(L,K).

Re t u r n s t o s c a l e is a measure of how effective an increase in the scale

of operation would be.

I n c r e a s i n g r e t u r n s t o s c a le means that the proportionate increase

in output is greater than the proportionate expansion of operations.

Co n s t a n t r e tu r n s t o s ca l e means the increase in output is

proportionately the same as the increase in operations.

De c r e a s i n g r e t u r n s t o s c a le means that the proportionate increase

in output is less than the proportionate expansion of operation.

In the context of production functions, the s c a le o f o p e r a t i o n is

captured by the amount of inputs used. When we talk about changes in

scale we normally mean a change across all inputs. However, you will see

later on that the composition of inputs depends on their relative price, so

the scale of operation would normally relate to the price of the output.

Therefore, a change in scale does n o t alter the composition of inputs, it

only affects the level of their use. An increase in the scale of production

means that we have increased a l l inputs by a certain proportion. Ther e t u r n of this change is measured by the proportionate increase in

output.


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Figure 3.6: Increasing production and returns to scale.

In Figure 3 .6 , increases in scale are depicted by a movement along a ray

from the origin. The slope of such rays, in this diagram, represents a given

capital-to-labour ratio (K/L) (also known as the i n p u t s c o m p o s i t io n ).

As we move along such a ray, the levels of inputs is increased by the

same proportion (otherwise the inputs composition will change and we

will be doing more than merely expanding our operation) and output

(represented by the relevant isoquants) will rise too.

We began at pointA in the above diagram. We may now wish to increase

the scale of our operation by a certain proportion, say (> 1). We

therefore increase both Labour and Capital by such thatKB= K

AandL

B

= LA

. Clearly the input composition will remain constant, thus:

At pointB, output too has increased. Let us suppose thatXB= X

A, which

means that output increased by . Whether or not there are increasing,

constant or decreasing returns to scale depends now on whether is

greater, equal, or less than .

A simple way to look at this issue is by examining functions which have a

special property: homogeneity. This can be defined as follows:

Definition 6

f (X1, , X

n) will be called homogenous of degreet if for all X

1, , X

nin its domain

and for all , f (X1, , X

n) = tf (X

1, , X

n).


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A h o m o g e n o u s p r o d u c t io n fu n c t io n means that if we increase a l l

inputs by a certain proportion, the increase in output can be described

as a function of the increase in inputs. That is to say, can be written as

a function of , (), for instance in the function where, if we multiply

all inputs by , output will rise by = t. We can translate this into a

production function, since represents the proportional increase in

output:

f(L, K) = tf(L,K) = tx

If t = 1, it means that an increase in a l l inputs by a certain proportion

will increase output by the same proportion (= t= ). This is the

property of c o n s t a n t r e t u r n s t o sc a l e; it means that as we increase

the scale of our operation by increasing all inputs, output will rise by

exactly the same proportion as the increase in inputs.

If t > 1, it means that an increase in all inputs by a certain proportion

will increase output by a greater proportion (= t> ). This is the

property of i n c r e a s i n g r e t u r n s t o s c a le .

Finally, if t < 1, then output will increase by a lesser proportion than

the increase in inputs (= t< ). This is the property of d e c r e a s i n gr e t u r n s t o s c a le .

Generally, we assume that a l l types of returns to scale are present in the

process of production. We believe that increasing returns to scale are

likely at the first stages of production, at low levels of production, and

decreasing returns to scale will appear as output grows. This will often

be due to human factors, such as organisation (i.e. hierarchy) as well as

control as some of the reasons why any operation is bound to encounter

decreasing returns at some stage.

Note : returns to sca le refers to a change in a l l inputs.

If we had increasing returns to scale throughout the production process, what do you

think the optimal number of firms producing a good would be? Why?

Figure 3 .7 depicts the production function where the returns to scale

increases at the beginning and decreases towards the end. To make it

simple, suppose that V is a composite input, comprised of both labour and

capital in a certain. proportion. We thus get the relationship between the

level of (composite) input. and the level of output as shown.

Figure 3.7: A typical production function, derived from isoquants.


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Notice (in the right-hand diagram) that at first, fixed increments in output.

require ever-decreasing increases in inputs (which means increasing

returns to scale). Later, one can see that fixed increases in output require

ever-increasing increases in inputs (in other words decreasing returns to

scale).

The corresponding points in the left-hand diagram depict the firms growth

of output when all inputs are increased by the same proportion, along theray through the origin. When production functions have the property of

homogeneity, this ray will also become the firms e x p a n s io n p a t h . This

path depicts all the o p t i m a l combinations of inputs with which one can

produce a chosen level of output for given factor prices. As both inputs

change here, this is called t h e l o n g -r u n e x p a n s i o n p a t h . We will

discuss the difference between long run and short run in a little while.

Returns to factor (marginal product)

By definition, if there is more than one factor of production, the

contribution of a single factor must be analysed when the other inputs are

held constant. Marginal product tells us how a change in a single input

affects output. IfX =f (L, K), then the marginal product of labour is thechange inX (dX) that results from a change inL (dL) whenK is constant.

MPLis thus defined by (dX/dL).

Can a function exhibit increasing returns to scale and diminishing marginal product at the

same time? Explain your answer.

Average product

We define the average product as the output per unit of input:AP =X/L.

Short-run

M

Figure 3.8: Short-run production functions.

In the short run we assume that not all inputs are variable, unlike the

case considered above, when we looked at the long-run expansion path.

In this case, where there are only two inputs, it means, for instance, that

the quantity of capital (K) is given and we can only change output by

changing the other input, labour. Figur e 3 .8 depicts the production

function when capital is fixed at a levelK0.


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At low levels of production, it is reasonable to suppose that any increase

in the variable input will add to output in increasing increments (i.e.

increasing marginal product). At a certain point the amount of the

variable input, relative to the fixed input, will increase so much that its

contribution to output will have to fall.

Thus, at first, the production function exhibits increasingMPL, and then

diminishingMPL. We can now clearly see thatMPL= dX/dLwill give us theslope of the production function (on the right-hand side) for the short-run

case, where the amount of capital is fixed (Ques t ion : Why?). Up to point

M, the slope increases, then, it diminishes.

In the left-hand diagram, we can see the corresponding points of

production. These form what one may consider the s h o r t - r u n

e x p a n s io n p a t h . Along this expansion path the level of capital is fixed

and the only way to increase output is through increases in labour inputs.

Average and marginal product

M

Figure 3.9: Average and marginal product.

In the right-hand graph of Figure 3 .8 , the slope of the ray from the origin

to any of the three pointsA,B, C is of the formX/L. This is precisely the

average product of an input, theAP. We can see how it increases between

A andB and diminishes afterwards. At pointB, where the slope (theAP), is

at its highest, the slope of the ray equals the gradient of f, and thus equalsMP

L. Figur e 3 .9 depicts these relationships.

The behaviour of the firm

Firms have to make two decisions: how much to produce and which

technology to use (that is, what the i n p u t c o m p o s i t i o n will be). The

choice of technology is, in principle, very similar to the way individuals

choose their consumption basket. The production function, which

determines the amount of output produced for given inputs, is similar to

the utility function faced by the consumer.


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We generally assume that firms wish to maximise profit. (Whether this

is really the case and what the consequences might be if it was not, is an

interesting subject which we will not consider here.)

Consider the profit function:

=pXX c(X)

wherepXXis the revenue from sellingX units of a good, while c(X)

represents the cost of producing them. We shall assume at this stage thatprices (of product as well as inputs) are given (that is, determined by

exogenous factors).

This formulation of the profit function clearly shows how the choice of

quantity and technology affects the profit generated. The quantity chosen will

affect both the revenue and the cost, while the choice of technology will affect

only the cost. We will now separate the effects of the choice of technology and

of quantity in order to understand their impact on the profit function.

One way of separating the effects is to fix a total cost of production, c0, and

then to find the highest level of output which is feasible. This is depicted

in Figure 3 .10 .

Why would this maximise profit? (Consider the form of the revenue function.)

Figure 3.10: Profit maximisation: maximising output for given cost.

For a given level of cost c0, the firm can choose all combinations of labour

(L) and capital (K) that are within its b u d g e t c o n s t r a i n t . The rate at

which the firm can substitute capital for labour is given by the market

exchange rate, which is the r e l a t i ve p r i ce s o f c a p i t a l a n d l a b o u r ,

here defined as w0/r

0units of capital per labour.

The highest level of output which is now feasible is given by the highest

isoquant. The choice of input combinations is therefore determined at

the point where the isoquant (derived from the production function) is

tangential to the isocost (the firms b u d g e t c o n s t r a i n t). At that point,

the slope of the isocost, which is the market rate of exchange betweencapital and labour, is the same as the slope of the isoquant. The latter is

really the technological rate of substitution between capital and labour.


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This solution implies that at the optimal point, the firm will gain no extra

profit by exchanging labour and capital at the margin. If the firm gives

up some labour, but wants to keep the level of output constant, then the

amount of extra capital needed will cost exactly the amount saved by

reducing labour, leaving the total cost of production unchanged. PointB in

Figure 3 .10 is clearly not optimal. If the firm gave up one unit of labour

it would need units of capital to remain at the same level of output. But

in the market place, it can get (+ ) units of capital per unit of labour.

This means that the firm can improve its performance through market

operations, changing the technology it uses.

Figure 3.11: Long-run expansion paths.

For any given relative factor prices we can get the set of all points where

the firm is producing optimally. These points, captured in Figure 3 .11 ,

are what we call the firms e x p a n s io n p a t h . It is a long-run expansion

path as all means of production vary in the process of expansion.

In the case where the production function is homogeneous, the expansion

path will be a straight line.

So far, however, we have only talked about the choice of technology (input

composition) which constitutes optimal choice. We have not yet dealt with

the question of how muchX to produce. The answer to this depends onthe relationship between output (X) and the isocost lines. This relationship

is explored in the next section.

The cost functions

1. The general form of the cos t func t ion in our two inputs model is:

C(K,L) = wL + rK

where w and r are the market prices of labour and capital respectively.

We shall assume that these prices are fixed.

2. Long-run cost function: We assume all inputs to be variable in the long

run. When we discussed production functions, we assumed that lowlevels of production exhibited increasing returns to scale. An increase

in output will require decreasing increases in inputs, and hence a

decreasing increase in cost. In Figure 3 .12 this is the case between


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the origin and pointA. We assumed, when discussing production

functions, that at first, it is most likely that the process of production

will be characterised by increasing returns to scale, hence, an increase

of a unit of output will require a decreasing increase in inputs, thus, a

decreasing increase in cost. For the production function below, this is

the case between the origin and pointA.

Figure 3.12: Production functions and the total cost curve.

For higher level of outputs (beyond pointA), we assume the process

exhibits decreasing returns to scale. Every further increase in output

will require ever-increasing increases in inputs. This means that, for

fixed prices, the total cost of production will increase faster and faster.

See Figur e 3 .12 for the derivation of the total cost curve from the

production function.

3. Marg ina l cos t s : the change in total cost C that results from a change

in outputX.

MC is therefore dC/dX. We can clearly see that this is the gradient

of the cost function in the above diagram. Notice that Marginal

Cost and Returns to Scale are inversely related. When there are

increasing returns to scale, there will be diminishing marginal cost.The production of every extra unit of output will require decreasing

increases in inputs and thus, decreasing cost per extra unit.


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4. Average costs: The cost per unit of output.

Again, asAC = C/X, we can see that average cost is the slope of the ray

from the origin to a point on the cost curve in the above diagram.1

5. Short run: In the short run, one of the means of production is fixed (the

capital used to set up the production facility, for example) and its costs

are independent of the quantity produced, because we cannot change

the quantity of it that is used. The cost function, therefore, is dividedinto two elements:

Fixed Costs (FC), and Variable Costs (VC), which remain a function of

output: VC(X). Hence, we define the Short Run Total Cost as:

SRC =FC + VC(X)

To see how the SRC function behaves, we only have to recall the short-

run production function. The short-run production function has the same

shape as the long-run production function, though for different reasons.

Translating it into a cost function repeats the argument we had for the

long-run. Assume that the amount of capital used is fixed, and that we

can only vary the amount of labour used in the production process.

Whenever marginal product is rising, the cost of an extra unit (the

m a r g i n a l co s t ) will be decreasing. This is so because increased

productivity means that one would need less labour than one needed

before for an extra unit of output.

Therefore, the VC(X) part of the SRC behaves in exactly the same

way as theLRC, and has the same general shape. The only difference,

therefore, will be the position of the SRC. Figure 3 .13 depicts the

relationship between the long- and the short-run cost functions:

Figure 3.13: Long- and short-run production and cost functions.

Given input prices w0and r

0, our long-run level of outputXwould have

been produced using aK/L ratio (representing a particular choice of

technology) according to the long-run expansion path, which connects

all points where the slope of the isocost lines is equal to the slope of

the isoquants (points such asA orB). If, say,K is fixed atK0in the

short run, the short-run expansion path is given by the horizontal line

atK0. Clearly, only at pointA (and, associated with it, an output level

X0) would the firm be able to use the same combination of K andL in

both the long run and the short run (so for this output level, the level

at which capital is fixed in the short run is exactly the level that would

have been chosen without such a constraint, i.e. in the long run).

1 Note from the figure

that MCis equal to

AConly at the point

where average cost is

at a minimum. Thus, the

marginal cost curve willcut the average cost

curve at the point where

ACis at a minimum.


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At other levels of output, say a lower level such asX1, the combinations

ofK andL used would be different in the long and short run, due to

the fixed amount of capital available. Given the shape of isoquants,

this means that the cost of producingX1in the short run (at C) must be

higher than the cost of producing the same amount in the long run (at

B): C is on a higher isocost line thanB.

Show that this holds true for output levels above X0as well.

We can see the intuition behind this by noting that firms try to

maximise profits given a number of constraints, such as their available

technologies, input prices and the price they can charge for the output

they produce. If we now introduce an extra constraint (for example the

constraint that capital is fixed in the short run), it is clear that we are

not making the firms job any easier. That is, we cannot be l o w e r i n g

its cost of production. At best (for an output levelX0), the cost stays the

same; for other output levels, it will increase.

6. The relationship between long-run and short-run average cost (LRAC

and SRAC):

Figure 3.14: The derivation of SRAC and LRAC.

The top part of Figur e 3 .14 again shows the Long-Run and Short-RunTotal Cost functions. As before, we can find the average costs for each

level of output by calculating the slope C/X of a ray from the origin. Given

the shape and position of the SRC andLRC discussed before, we can see


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that for any given level of output, the slope of the ray that reaches the SRC

must be at least as great as that of a ray reaching the LRC. Therefore, the

SRACwill be greater than or equal to theLRAC. Equality is achieved only

when the long run and the short run use the same input combinations.

Application

Let us analyse the effects of a fall in the wage rate on the long-run cost.

Figure 3.15: A fall in the wage rate.

Suppose we are producingX0with a total cost of C

0(pointA in the above

diagram). The slope of the isocost line is w/r. Now, as the wage rate

falls, the intercept of the isocost line (associated with the same total costas before, C

0) shifts to the right along theL-axis. PointB depicts the new

optimal level of production at the current level of cost if all means of

production are variable. We see that for the same level of cost we can now

produce more ofX.

In order to see what happens to the total costs at any level of output,

we must consider what the cost would be of producing optimally the

previous quantity ofX (X0). The broken line at point C, on the same

isoquant as before, gives us this information. Clearly, this broken line,

which has the same slope as our new isocost line, reflects a lower cost than

that going through pointA.

Hence, for any level of output, total cost will fall and so will the a v e r a g e

cos t . As the marginal product of any of the inputs has not changed, it will

also cost less to increase output by a single unit of output at any level of

production.

Producer behaviour with respect to output

So far we have mainly looked at the question of how a producer will choose

the combination of labour and capital which will bring about maximum

profit for any amount ofXhe might intend to produce. In other words, we

have minimised the cost of producing some output X. Naturally, there is also

the question of how much to produce in the first place. To analyse this, wesimply have to rewrite the profit function and take a closer look at the role

ofX(the level of output) in determining the maximum profit.


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The producers problem will now be written as:

max (X) =R(X) C(X)

Our analysis of how to minimise the cost of production for any level of X

told us all we need to know about the cost function C(X). However, we do

not know much about the r e v e n u e functionR(X).

The revenue each firm earns is simply the price of its output multiplied by

the quantity sold. Its general form is:

R(X) =pX(X)X

wherepX(X) is the inverse demand function which the producer confronts.

In a perfectly competitive market where his choice of output does not

influence the price R becomes simply:

R =pXX

What is max per unit of X produced on production of 500 units when total R = 6,250

and total C = 3,975?

In general, a change inR can be broken down to dR = dPX +dXP, wheredPX is the loss of revenue on previous sales (if the price of X decreased)

and dXP is the gain on new sales at the current price. By now, this should

sound very familiar. If not, read again the section on the price elasticity of

demand.

We now want to know how revenue changes when we produce one more

unit ofX. We call this change in revenue the m a r g in a l r e v e n u e and we

denote it byMR = dR/dX.

In the case of perfect competition, where the firms behaviour does not

influence the price, the sale of one more unit will increase revenue by the

price we get for that unit. In other words, the competitive firms revenue

will change according to the change in sales. The price will not be affected

by these changes. Hence:

dRPC=pXdX

If we divide this by dX, we get the marginal revenue:

Hence, the marginal revenue in the case of perfect competition is simply

the price of the output in the market.

For more general revenue functions, we can find the marginal revenue by

similar means. All we have to do is to take the derivative of the revenue

function,R(X) =pX(X)X, with respect toX. We find that:

where dP/dX is the derivative of the demand function. It shows how the

price will change if output is changed. As dP/dX


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Now, if we multiply and divide dP/dX X byP/P, we can write:

where is the price elasticity of demand.

In a perfectly competitive market, the price elasticity of demand which the

individual producer confronts goes to infinity. Therefore:

Now that we have studied the r e v e n u e function, we can add it to the cost

function which we studied before. Recall that (X) =R(X) C(X). For a

firm in a perfectly competitive market, we can thus derive the following

profit function:

Figure 3.16: Cost, revenue and the profit function for a perfectly competitivemarket.

TheR function of a competitive industry has a constant slope,PX, which

is the firmsMR (= dR/dX). The greatest distance between it and the cost

function is always where the gradient of C(X) (i.e.MC(X)) equals the

gradient ofR (MR). However, there are two points where the distance is

the furthest. In the one case we will be minimising profit; in the other, we

will be maximising it. Therefore, althoughMR(X) =MC(X) is a necessarycondition for profit maximisation, it is not a sufficient one.


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Instead of exploring the mathematics of second order conditions (which

would let us choose the output level which actually maximises profits),

let us take another look at the profit function. Clearly, what distinguishes

the profit maximisation point from the other optimal solution (albeit to a

different problem) is that profits are positive.

We can rewrite the profit function in the following way:

Clearly, for > 0, we need thatPX> AC(X).

We can summarise our analysis so far in two principles which will guide

the behaviour of the firm:

1. The decision over h o w m u c h to produce.

The answer is to produce such as quantity ofX that:

MC(X) =MRc=PX

2. The decision over w h e t h e r to produce at all.

The answer here is to produce as long as:

PXAC(X)

Hence, provided that condition (2) is satisfied (i.e. provided the firm is

willing to stay in the market), the supply of the firm will be guided by the

part of theMC(X) function which is above theAC(X) function. We shall

call this part the sup p ly cu rve of the firm.

Note : You should pay attention here to the difference between the short

and the long run and the roles of both average cost and average variable

cost.

To complete the analysis make sure that you can use Figure 3.17 alone to explain the

two principles which constitute the firms behaviour.

Figure 3.17: The firms supply curve.


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A numerical example

Production functions

Consider the production circumstances of a goodXwhich requires only

one input (labour) for production. The technology available has the

following results:

L TP = X AP = X/L MP = dX/dL

0 0 0

1 9 9 9

2 24 12 15

3 42 14 18

4 60 15 18

5 75 15 15

6 87 14.5 12

7 96 13.7 9

8 101 12.6 5

9 101 11.2 0

10 95 9.5 6

Figure 3 .18 depicts how t o t a l p r o d u c t (TP), a v e r a g e and m a r g i n a l

product (AP andMP) change with the change in input (L). You can also

see how they relate to each other.

Figure 3.18: TP, AP and MP: a numerical example.

Notice that as long as the marginal product is greater than the averageproduct, the latter is rising. This is because the marginal product describes

the contribution of the l a s t u n i t o f in p u t . As long as this contribution


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is greater than the average, the average will have to rise. In brief, what

you see is that theAP is at its highest when it is the same as the marginal

product. If the marginal product is diminishing, every extra unit of output

will require more and more inputs. Thus, the product per input will have

to fall. The reason why it does not fall immediately when marginal product

begins to fall is that the increases in output at the beginning were so great

that it takes a much sharper decline in productivity to change the direction

of the average product.

Cost functions

Suppose now that the production ofX requires a licence which costs

1,130.

A labour unit costs900 (for the duration of the production process). We

can therefore distinguish between fixed costs (FC) which are unaffected

by the level of output produced, and variable costs (VC), which reflect the

level of production. Together, these give the total cost (TC) of producing a

given level of outputX:

TC =FC + VC

The average cost (AC) is simply TC/X. Naturally, average cost is the sum

of theAFC (=FC/X) andAVC (= VC/X). The marginal cost (MC) is the

change in cost per extra unit of output. Evidently, this change will depend

on the productivity of labour. The more productive labour is, the less

labour units will be required for the production of one unit of output.

For instance, one unit of labour may produce 9 units ofX. Hence, oneX

would require 1/9 units of labour. Note from the previous table that 9 is

the marginal product of the first labour unit. Hence, the amount of labour

required for the production of one unit is always 1/MP. As we pay W =

900 per labour unit, the cost of one unit of output will be [900 1/9] =

100. In general, therefore, we can write:

Given this information, we can calculate the cost functions for the firm:

L TP = X FC VC TC AFC AVC AC MC

0 0 0 0 0 0 0 0 0

1 9 1130 900 2030 125.6 100.0 225.6 100

2 24 1130 1800 2930 47.1 75.0 122.1 60

3 42 1130 2700 3830 26.9 64.3 91.2 50

4 60 1130 3600 4730 18.8 60.0 78.8 50

5 75 1130 4500 5630 15.1 60.0 75.1 60

6 87 1130 5400 6530 13.0 62.1 75.1 75

7 96 1130 6300 7430 11.8 65.6 77.4 100

8 101 1130 7200 8330 11.2 71.3 82.5 180

The general generation of the U shapedAC function is depicted in the

lower graph of Figur e 3 .19 . Clearly,AC =AFC +AVC. SinceFC is

unchanged as output increases,FC/Xwill be declining asX increases. Due

to diminishing Marginal Product, every extra unit of output will require

ever-increasing cost. Hence, VC/X is rising with output. For very low levels

ofX,FC/X is a large number while VC/X is small. The shape of theACwill

thus be dominated byAFC. For largeX,FC/X is almost zero. The shape of

ACwill thus be dominated byAVC.


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Figure 3.19: Cost functions: a numerical example.

Profit maximisation

Suppose now that the firm can sell a unit ofX for 100. We shall also assume

that whatever the firm does, it will not affect the market price as the firm is

too small. Hence, the revenue of the firm isPXX and the marginal revenue(the revenue of the last unit sold) will be the priceP

X. The following table

describes the situation of the firm under various levels of production.

TP P R TVC TC MR MC M AVC AC

0 100 0 0 1130 130 0 0 0

9 100 900 900 2030 1130 100 100 0 100.0 225.6

24 100 2400 1800 2930 530 100 60 40 75.0 122.1

42 100 4200 2700 3830 370 100 50 50 64.3 91.2

60 100 6000 3600 4730 1270 100 50 50 60.0 78.8

75 100 7500 4500 5630 1870 100 60 40 60.0 75.1

87 100 8700 5400 6530 2170 100 75 25 62.1 75.1

9 6 1 0 0 9 6 0 0 6 3 0 0 7 4 3 0 2 1 7 0 1 0 0 1 0 0 0 6 5 .6 7 7 .4

101 100 10100 7200 8330 1770 100 180 80 71.3 82.5

Notice that profit is maximised (M= 0 and > 0) whenMR =MC and

whenP =MR > AC.

The firm as an organisation: a note

In everything we have said so far, we treated the firm as an abstract object,

a profit maximising agent. But as we all know, firms are nothing like this.

They are large-scale organisations involving a great number of people with

different skills, wants and background. Can we really say that the firm is

simply a profit maximising agent?


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115

Since our purpose in using economics is to describe the world around us,

we have to be aware that the firm is a much more complicated structure

than the abstract notion of a profit maximiser might suggest. This does not

necessarily mean that our representation of those firms as simple profit

maximisers is not true: it might be a fairly good description of how firms

actually behave. Nonetheless, it is useful for us to spend some time looking

at how the organisation of a firm might influence (and be influenced by)

its economic environment.

There are two separate issues which we have to consider when we

examine the organisation of the firm. First, given the current structure of

corporations, where ownership is in the hands of shareholders who are not

the managers, it is not obvious that the managers would necessarily have

the interest of the shareholders close to their hearts.

There is little doubt that the shareholders would want the managers to

maximise profit. Most shareholders are not involved in the firm, and

they have no other consideration apart from profit maximisation. The

managers, on the other hand, are working in the corporation and must

consider the interests of other groups with whom they are in daily contact.

They are salaried, and so their earnings may be slightly less sensitive to

changes in the performance of the corporation than the income of the

shareholders would be (shareholders receive their income in the form of

dividends or of capital gains from selling shares).

Consequently, the shareholders, who have the power to appoint or sack

the managers, will face what is called the principal-agent problem. The

shareholders are the principals who want their agents (the managers) to

maximise profit. The managers have a great informational advantage over

the shareholders, who are less familiar with the issues associated with

running the corporation and are therefore susceptible to all kind of excuses

and stories which the managers can put forward to justify their actions

(and the subsequent reduced profit). The question for the shareholders,

therefore, is how to write a contract that would give the managers the

incentive to maximise profits. One of the most common incentives is some

form of p e r f o r m a n c e - r e l a t e d p a y, but whether this actually provides a

sufficient incentive for the managers is a different story.

The second and far more important issue is how the firm (or corporation)

evolved and how it might change. Put differently, why do we have

corporations in the first place?

These are very difficult and important questions to which all economists

must pay attention. Unfortunately, economic theory has not produced any

theories that would do justice to the importance of the question. If weunderstood why corporations exist, we would be able to understand how

they operate, what will make them succeed and what will change them.

In this context, I would like to draw your attention to two approaches

to the problem. First we have the r e a l e v o lu t i o n a r y approach. This

approach examines how and why corporations have been formed over

the years. For instance, when comparing the evolution of Russian and

Indian village communities, some authors have found that while the

sense of kinship among Indian village communities has decreased, that

among Russian communities remains very strong. A possible explanation

for this phenomenon is that while Indian village communities are located

in relatively populated areas, the Russian villages were located in a vast,

much emptier area. This allowed dissenting groups of people to move

away from existing village communities and set up new ones.


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The consequences of this difference for the evolution of the institution of

private property were immense. In India, the outcome was the emergence

of private ownership of land, while in Russia, the arable land had been

periodically redistributed and, as one scholar puts it, the village artificer, even

should he carry his tool to a distance, works for the profit of his co-villager

(Maine, H. (1987) The Early History of Institution, London, John Murray, p.81

though admittedly this was a long time ago). Along similar lines, though in

a much more complex setting, one can examine how corporations have been

formed. For instance, one must inquire why it is that the English bankruptcy

laws are so different from the American ones, and what consequences this

might have on the organisation of productive activities.

The second approach is basically trying to explain the corporation from an

analytical point of view. Recall that in Chapter 1 we discussed specialisation

and trade. We saw there that once people specialise, there will be a gap

between how much it costs them to produce a good and how much others are

willing to pay for it. Suppose that you specialise in potato production and you

come to the market once a week with your sack of potatoes. Suppose too that

you would like to find in the market, in return for your potatoes, a nourishing

breakfast. There is a very large gap between what you are willing to pay forthat breakfast and what it really cost to produce. However, if you wanted to

research the breakfast market thoroughly, you would need to invest a great

deal of time. You would need to visit all other producers in the community

and ask them about the properties of the goods which they produce. You will

need to study nutrition to create the bundle of goods which will constitute

your breakfast. During all the time you are researching into the question, you

are not selling potatoes and, if you have to go to the library in the village, you

will even be producing fewer potatoes. So matching your desires with what is

brought to the market will cost you a great deal.

If, instead, someone took on the role of entrepreneur, they would do

the work for you, finding out what you want and roaming the village todiscover whether the combination of goods can be produced, leaving you

to carry on producing potatoes. Alternatively, they may persuade someone

to produce the goods you want, so that next time you go to the market,

they will be there. Given the difference between what you were willing

to pay and what it costs others to produce, the entrepreneur can make a

profit. This is a case where the middle man is most important. Without

the entrepreneur, it would cost you much more to exchange potatoes for

breakfast. The entrepreneur could operate because they were cutting your

t r a n s a c t io n c o s t . As long as this is possible, there will be room for an

organisation like the firm which will do this work for you.

There is an additional problem, however. If the job of the entrepreneur issimply to sign contracts with various agents to provide a good at a lower

rate than that which the consumer is willing to pay, the source of efficiency

for the organisation is simply the contracts. So again we have the issue of

designing contracts that will create the incentive for the people connected

with the corporation to do their best. However, as contracts do not cover

all eventualities (they are i n c o m p l e t e ), there will be instances where

the contract does not specify who is entitled to what, and the question of

ownership emerges.

We have thus come full circle: from explaining private ownership as a

social institution which developed from the crumbling tribal sense of

community, to finding that this institution holds the key to the life of thecorporation. Unfortunately, these areas of research have not yet come

together, but as you can see, there is still a lot to be done to develop the

language with which we discuss economic and social organisation.


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Self-assessment

Check your knowledge

Check back through the text if you are not sure about any of these.

Define the concepts of productive efficiency, isoquants and iso-cost,

marginal and average product, constant returns to scale, transaction

costs, diminishing marginal return fixed and variable costs, the

relationship between marginal and average costs.

Use these definitions to give examples of increasing returns to scale, the

interrelation between marginal cost and returns to scale, homogeneous

production function, and profit maximisation with respect to output

and the firms supply curve.

Construct cost-functions and derive their relation to the production

function.

Use diagrams to analyse problems involving short-run and long-run

average cost schedule.

Give an example of:

increasing returns to scale

the inverse relationship between marginal cost and returns to scale

a homogeneous production function.

Test your understanding

In this section, you will find a set of problems of the kind you will meet in

the exam. The answers follow on page 118.

Question 1

Short-run average cost always lies above long-run average costexcept at one point. However, while it is clear that with a fixed

amount of one input the firm cannot expand along its long-run

expansion path, it can always use less of it and follow its long-

run expansion path. Therefore, short-run average cost should be

exactly the same as long-run average cost up to a certain point.

a. Derive the long-run average cost schedule.

b. What is the difference between the short and the long runs?

c. Derive the short-run average cost schedule.

d. Comment on the above statement.

Question 2

a. Under which conditions will the long-run average cost and marginal

cost be the same?

b. Will the short-run average cost be the same as the long-run average

cost?

c. Will the short-run marginal cost be the same as the short-run average

cost?

Question 3

a. Explain, using diagrams, why the short-run cost function is tangent to

the long-run cost function at one point only.b. Assuming upward sloping expansion paths, what will happen to the

level of output at which such tangency occurs, the higher is the level of

fixed capital? Does this mean that the level of capital which generates


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118

the coincidence of the short-run and the long-run minimum average

costs is an optimal level of fixed capital?

c. Will the long- and the short-run average cost of a production process

which exhibits only increasing returns to scale (at the relevant section

of output) have similar U-shapes? Explain.

AnswersQuestion 1

a. Der iving long-ru n a ve rage cos t . The key issues here are:

associating the shape of the long-run cost function with the relevant

properties of the production function; recognising that average costs

can be depicted by the ray from the origin; deriving the average cost

from the change in the slope of that ray from the origin. All of these are

in the domain of testing ones familiarity with various models.

b. Here, as in part (a), we need a simple exposition of material which is

covered in great detail in the present chapter. First, students should

demonstrate that they recognise the role of fixed costs in the distinctionbetween the long and the short run. An explanation of the short-run

cost curve and its position relative to the long-run curve is essential.

c. The derivation process should be explained carefully, where we

compare the ray from the origin (the average cost) which is associated

with the long-run cost curve with that ray which is associated with the

short-run curve.

d. Here, the more analytical part of the question begins. The statement

suggests that as we can always produce less with those means of

production which we have, there is no reason why producing less

than the level of output for which both short- and long-run cost

coincide, should cost more than it would if we could vary all means ofproduction.

The choice of framework here is crucial and, as you see below, it is the

firms optimal choice in the production factors plane:

Figure 3.20


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119

Tr a n s l a t in g t h e q u e s t i o n i n t o t h e l a n g u a g e o f t h e m o d e l: It is

proposed in the statement that there is no reason why the firm should

not produce at point C (with the long-run optimal mix of inputs) as it can

always reduce the amount of capital it uses. Obviously, while it is true that

the firm can use less capital, it will still have to pay for the idle means of

production. If atB the short-run total costs are:

C

B

1= r0K0+ w0L

B

1at C they will be:

CC1= r

0K

0+ w

0LC

1

AsLC1> LB

1, the actual costs at C are higher than atB and the configuration

around point C is not really feasible. It is not necessary to construct the

argument in this formal way but as you can see it is clearer and shorter.

Question 2

This is a question about (i) a constant return to scale production function;

(ii) the relationship between the properties of production functions and

those of the cost functions; and (iii) the difference between the short and

the long runs.

a. The main points here are these:

an understanding of the properties of the CRS production function

and its significance for the derivation of the corresponding long-run

cost function with its constant slope and 0-origin

the derivation of the long-run average and marginal cost functions

as depicted in Figure 3 .21 .

Figure 3.21

You are expected to demonstrate that you understand the geometrical

representation of both the average and marginal cost in the left-hand

diagram.

b. and c. An understanding is required here that returns to scale is a long-

run property of production and that it does not affect the short-run

principle of diminishing returns to factor.


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120

Figure 3.22

The answers to (b) and (c) are self-evident from Figure 3 .22 .

Question 3

a. The issue here is a recurring one: why should the short-run cost lie

everywhere above the long-run cost except at one point only? I am

sure that many of you will produce here the familiar picture shown in

Figure 3 .23 .

Figure 3.23

But this merely constitutes a restatement of what needs to be

explained. We need an e x p l a n a t i o n for this graph, pointing out

that there is only one combination of inputs which is optimal in the

long run for each level of output. In the short run, one of the means

of production is fixed. Hence, there will be only one level of output

which will be produced using the same input mix in both the long and

the short run. At any other level of output, as the short run a d d s a

constraint, the cost of production will have to be higher.

The above diagram is fully acceptable if accompanied by a graphicexplanation in theL,K space, where students are expected to produce

the long-run and the short-run expansion paths (Figure 3 .24 ).


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Figure 3.24

b. We normally assume that the long-run expansion path is upward

sloping. The short-run expansion path is horizontal. It is easy to see

that there is only one level of output for which the choice of inputs

will be on both the long-run and the short-run expansion path

(point C above). In addition, we expect students to show why at any

other point, the short-run costs are higher than the long-run costs.

Comparing the long-run cost of producing, say,X0(A) with that of the

short-run cost for producing the same level of output (B) will yield:

CA0= r

0K

0+ w

0LA

0< CB

0= r

0K

0+ w

0LB

0

Hence, the short-run cost lies everywhere above long-run cost with the

exception of point Cwhich is common to both expansion paths.

There are two elements to this section:

To show that the tangency between the long-run and the short-run

average cost curves occurs at higher levels of output the higher is

the level of fixed capital.

Figure 3.25


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122

As in Figure 3 .25 , this should be fairly obvious (a move fromA toB).

The second element is a much more important component in

this sub-question. It is the question whether the level ofKwhich

produces a tangency between the long-run and the short-run

average cost curves at their minimum can be considered as an

optimal level of fixed capital.

Here, you are expected to demonstrate that you understand therelationship between section (a) and the average cost curve. Ideally,

you are expected to produce diagrams as shown in Figure 3 .26 ,

where you explain how the short-run average cost relates to the

long-run average cost.

Figure 3.26

In addition, we would like you to demonstrate that you understand

that at the minimum of long-run average cost, the short-run average

cost is at its minimum too.

The main part of the answer is the explanation of the significance of

producing at the minimum average cost. We expect you to say that

only if the objective of the firm is to fully utilise its resources will the

minimum average cost be an optimal solution. As such, it can be said

that the level of capital which facilitates production at this level of cost

in the short run can be considered as optimal. However , as this is notreally the objective function of the firm, producing a level of output

where the short-run average cost is tangent to the long-run average


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cost at the minimum of both functions, doesnt have anything to do

with the concept of optimality.

c. This section is the more demanding part of the question, where we

expect students to produce the cost functions for the long run as well

as the short run in the case of increasing returns to scale. Figure 3 .27

shows what should emerge.

Figure 3.27

While the long-run average cost will be falling, the short-run average

cost should have its normal U-shape as increasing returns to scale is a

long-run property of the cost function.

bridging economics chapter 3 subject guide

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