Topic 6 - Firms and Productions (ECON 250 McGill)

8/10/2019 Topic 6 - Firms and Productions (ECON 250 McGill)

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ECON 250D1. Topic 6: FIRMS AND PRODUCTION

October 2014

(Prof. N.V. Long) Topic 6: Firms and Production O ctob er 2014


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Notes on Conference for Topic 6

For conference on Topic 6 (next week), please try to answer FIVEquestions: (3.4, 4.13, 4.15, 4.16, and 5.3) at the end of the chapterpages 203-205.

Do not attempt other questions.

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Ownership and Management

Most rms are privately owned.

Economists assume that the objective of private-owned rms is tomaximize prot.

This is not as simple as it may sound: there is a distinction betweenshort-run prot and long-run prot.

A rm may want to sacrice short-run prot so as to boost long-runprot.

However, at our level, we do not deal with this deep question.

We simply assume that in each period, the rm chooses its outputlevel to maximize the dierence between the its total revenue for thperiod and its total cost for the period.

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nputs and Outputs

We consider only a simple model: the rm produces only one outpuand uses several inputs.

We consider three main types of inputs:

(i) capital services (broadly dened to include land, building, andequipment),

(ii) labour services, and

(iii) raw materials.Inputs are also called factors of production.



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Production Functions

Suppose a rm uses two inputs, labour and capital, to produce anoutput.

Let qdenote the output level, and L and Kthe quantity of labourand capital the rm employs.

The production function q=f(L, K) tells us how output varies asthe rm varies its inputs.

It also contains information about the degree of substitutability of tinputs



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Short-Run and Long-Run

In the short-run (for example, one year), the rm cannot, or is notwilling to change the quantity of some inputs (for example, the sizethe boiler, or the number of elevators in its factory).

In the long-run, all inputs are variable.



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Short-Run Production Function

We assume that there are two inputs, say capital and labour.

In the short run, capital is a xed input, and labour is the only

variable input.Thus K is xed at some level K, and the rm can decide on the levof employment, L.

We dene the marginal product of labour, given K, by

MPL = f(L, K)

L

and the average product of labour, given K, by

APL = f(L, K)

L



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Long-Run Production Function

In the long-run, we assume that level of all inputs can be adjusted.

For simplicity, we often consider a model with just two inputs, forexample, capital and labour.

We often assume that the production function takes theCobb-Douglas form

q=ALK where 0 < < 1 and 0


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Long-Run Production Function

A rms capital is often interpreted as the aggregate value ofequipment and buildings that it uses.

for example six hand-held power saws (costing $100 each) mayrepresent more capital than ten handsaws (costing $40 each)

and two bench power saws (costing $400 each) may represent morecapital than six hand-held power saws.

This raises a deep question: what is the appropriate measure ofcapital when equipment and building are heterogeneous?

At the elementary theory level, we abstract from these complicationand assume that capital is a homogenous input. Similarly, we oftenassume labour is a homogeneous input, for simplicity.

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soquants

We normally restrict attention to production functions with just two

inputs.

Given a production function, say q=f(L, K), for any xed outputlevel q> 0, we dene the isoquant for qas the set of points (L, K)such that f(L, K) =q

In general, the two inputs need not be capital and labour. We candenote them as x1 and x2, and write q=f(x1, x2).

Since isoquants in production theory are indierence curves in thetheory of the consumer, we will not dwelve on their properties.

(Prof. N.V. Long) Topic 6: Firms and Production O ctob er 2014 1


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soquants

Figure 1: isoquant maps for a Cobb-Douglas production function.Other cases:

(i) The two inputs are perfect substitutes (Figure 2): To produce

orange juice, a rm may use a combination of Californian oranges aSouth African oranges:

f(x1, x2) =x1+ x2

(ii) The two inputs are perfect complements (Figure 3A): To producone hour of gondola ride, a rm need one gondola (for one hour) an

one gondolier (for one hour),

f(x1, x2) =min fx1, x2g

Generalize to xed coecient function q=minn

x1a1

, x2a2

oe.g. 1

bicycle frame, 2 bicycle wheels.



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Marginal Rate of Technical Substitution

consumer theory:MRS12 =U1U

2In production theory, we analogously dene the marginal rate oftechnical substitution between input 1 and input 2 at the point(x1, x2) as the slope of the isoquant passing through that point

MRTS12 =f1f2

where f1 and f2 are the marginal products of inputs 1 and 2respectively.



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Elasticity of Substitution

FIGURE 4 compares two isoquants for producing one unit of twodierent nal goods, say a manufactured good, denoted by qm , and

an agricultural good, denoted by qa

f(x1, x2) =qm =1

g(x1, x2) =qa =1

We assume the two production functions are dierent, and their

isoquants have dierent curvature, though both are convex to theorigin.

Suppose these two isoquants have a common point E, at which thejMRTSm12 j and jMRTS

a12 j have the same numerical value.



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Now let both jMRTSm12 j and jMRTSa12 j increase by t%.

This is a move from point E to point Malong the isoquant

f(x1, x2) =qm =1, and from point E to pointA along the isoquang(x1, x2) =qa =1.

The ratio x2/x1 at point A is greater than that at point M.

We say that x1 and x2 are more substitutable in the production ofnal good a than in the production of nal good m.

Roughly speaking, greater substitutability means that curvature of t

isoquant is not too pronounced. A formal denition of the elasticityof substitution between the two inputs is

jj= percentage change inx2/x1percentage change in jMRTS12 j



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That is

jj=(x2/x1)

x2/x1jMRTS12 jjMRTS12 j

More precisely,

jj=

d(x2/x1 )x2/x1

djMRTS12 jjMRTS12 j

= jMRTS12 j

x2/x1

d(x2/x1)

djMRTS12 j



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Cobb-Douglas production function has jj=1 always.CES production function

q=

x1 + x

2

1/where 6=0, < < 1

its has jj= 11

Limiting cases: jj ! : perfect substitutes, jj !0: perfectcomplements.



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Returns to Scale

A production function q=f(x1, x2) is said to display constantreturns to scale if and only if the following property holds: Given aninput combination (x1, x2) that results in an output level q (i.e.,q=f (x1, x2)), scaling up that input combination by a factor >(or scaling down that input combination by a factor 0 < < 1) wiresult in a new output level equal toq.

Loosely speaking, the constant returns to scale property means that

a rm doubles (or triples) all its inputs, its output will be doubled (tripled).



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Homogeneity of degree 1

A more concise denition: A production function f(x1, x2), denedfor all (x

1, x

2)(0, 0), displays constant returns to scale if and on

iff(x1,x2) =f(x1, x2) for all > 0.

This corresponds to the mathematical denition of homogeneity ofdegree t .

Denition M1 (Homogeneity of degree h) A function f(x1, x2)dened for all (x1, x2)(0, 0), is said to be homogeneous of degreh if and only iff(x1,x2) =

hf(x1, x2) for all > 0, where h is constant, independent of.

The concept ofconstant returns to scaleis equivalent to the concept ofhomogeneity of degree 1.



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Example

Example: The Cobb-Douglas production function q=A (x1) (x2)

where A > 0, 0 < < 1 and 0


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Homogeneity of degree greater than 1

If a rms production function is homogeneous of degree 2, then,

when it doubles all the inputs, (i.e., =2) then the output isquadrupled, i.e. q=22q=4q, and when it triples all the inputs, i.=3), then the output is nine times bigger than before, i.e.q=32q=9q.

Homogeneity of degree h > 1 therefore implies increasing returns tscale.

(Similarly, if the degree of homogeneity of a production function ispositive but less than 1, we say that the the production functiondisplays decreasing returns to scale)



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ncreasing Returns to Scale

Increasing returns to scale is more general than that of homogeneity

of degree h>

1.Denition D2 (Increasing Returns to Scale)A production function q=f(x1, x2) is said to display increasing returns scale if and only if the following property holds: Given any inputcombination (x1, x2) that results in an output level q (i.e.,q=f (x1, x2)), if the rm scales up that input combination by a factor > 1 it will obtain in a new output level greater than q.Loosely speaking, a rms production function shows increasing returns tscale if the doubling all inputs results in more than in an output level thas greater than two times the previous output level.

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23/27

Decreasing Returns to Scale

Denition D3 (Decreasing Returns to Scale)

A production function q=f(x1, x2) is said to display decreasing returns scale if and only if the following property holds: Given any inputcombination (x1, x2) that results in an output level q (i.e.,q=f (x1, x2)), if the rm scales up that input combination by a factor > 1 it will obtain in a new output level smaller than q.



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Topic 6 - Firms and Productions (ECON 250 McGill)