ECON 500 Producer Theory

8/13/2019 ECON 500 Producer Theory

1/59

ECON 500

ECON 500Microeconomic Theory

Producer Theory


2/59

ECON 500

Producer Theory

A theory of how firms coordinate the transformation of inputs into outputs

with a goal of maximizing profits in the meantime.

Part I Production Functions

Part II Cost FunctionsPart III Profit Maximization


3/59

ECON 500

Part I. Production Functions

The principal activity of any firm is to turn inputs into outputs.In the theory of producer behavior the relationship between inputs and

outputs is formalized by a production function of the form

where qrepresents the firms output of a particular good during a period, k

represents the machine (that is, capital) usage during the period, l

represents hours of labor input, mrepresents raw materials used, and represents the possibility of other variables affecting the production

process.


4/59

ECON 500


Most of our analysis will involve two inputs kand l, capital and laborrespectively

Where qis the maximum amount of a good that can be produced by using

alternative combinations of kand l.

Since it is possible to produce the same amount of a good using differentcombinations of capital and labor, it is important to understand their

respective contributions to the production process at various levels of

utilization.


5/59

ECON 500


Marginal Physical Product:

The marginal physical product of an input is the additional output that can

be produced by employing one more unit of that input while holding all

other inputs constant.

Where the use partial derivatives reflect the fact that all other input usage

is held constant while the input of interest is being varied.


6/59

ECON 500


Diminishing Marginal Productivity:

We assume that the marginal physical product of an input depends on how

much of that input is used. We also postulate that inputs cannot be added

indefinitely to a production process without eventually exhibiting some

deterioration in its productivity. Mathematically:


7/59

ECON 500


Diminishing Marginal Productivity

Changes in the marginal productivity of labor depend not only on how

labor input is growing, but also on changes in capital.

Therefore, we must also be concerned with MPl / kIn most cases, this cross partial derivative is positive indicating that the

marginal productivity of an input increasing in the other input.

Diminishing marginal productivity of a single input can be offset by the

increases in other inputs. The gloomy prediction of Malthus that ledeconomics to be called a dismal science is misplaced.


8/59

ECON 500


Average Physical Productivity

APlis easily measured and is of much empirical significance. The

relationship between average and marginal physical productivity is also of

significance.

WhenAPlis at its maximum, it equalsMPl


9/59

ECON 500


Isoquants

An isoquant shows those combinations of kand lthat can produce a given

level of output. Mathematically, an isoquant records the set of kand lthat

satisfies:


10/59

ECON 500



11/59

ECON 500


Marginal Rate of Technical Substitution

The marginal rate of technical substitution (RTS) shows the rate at which

labor can be substituted for capital while holding output constant along an

isoquant.


12/59

ECON 500


RTS and Marginal Productivities

The total differential of the production function is

Along and isoquant, we know that dq=0, therefore

Hence


13/59

ECON 500


Diminishing RTS

Isoquants naturally have a negative slope but they are also as convex

curves. This means along any one of the isoquants, the RTS is diminishing

However, it is notpossible to derive a diminishing RTS from theassumption of diminishing marginal productivity alone since Marginal

Physical Product depends on the level of both inputs and cross

productivity effects are present.

To show that isoquants are convex, we would like to show thatdRTS/dl< 0. Since RTS = fl/fk, we have


14/59

ECON 500


Diminishing RTS

Because fland fkare functions of both k and l, we must be careful in

taking the derivative of this expression

Using the fact that dk/dl = - fl/fkalong an isoquant and

Youngs theorem (fkl

= flk

), we have


15/59

ECON 500


Diminishing RTS

Because we have assumed fk> 0, the denominator of this function is

positive. Hence the whole fraction will be negative if the numerator is

negative. Because flland fkkare both assumed to be negative, the

numerator definitely will be negative if fklis positive.

When we assume a diminishing RTS we are assuming that marginal

productivities diminish rapidly enough to compensate for anypossible

negative cross-productivity effects.


16/59

ECON 500


Returns to Scale

If the production function is given by q = f (k,l)and if all inputs are

multiplied by the same positive constant t (where t > 1), then we classify

the returns to scale of the production function by


17/59

ECON 500


Constant Returns to Scale

CRS production functions are homogenous of degree 1:

Derivatives of a CRS function are homogenous of degree 0:

And CRS functions are homothetic


18/59

ECON 500


Elasticity of Substitution

If the RTS does not change for changes in k/l, we say that substitution is

easy because the ratio of the marginal productivities of the two inputs does

not change as the input mix changes.

Alternatively, if the RTS changes rapidly for small changes in k/l, we

would say that substitution is difficult because minor variations in the

input mix will have a substantial effect on the inputs relative

productivities.

A scale-free measure of this responsiveness is


19/59

ECON 500



20/59

ECON 500


21/59

ECON 500

Part II. Cost Functions

Accounting Costs vs. Economics Costs

Accountants emphasize out-of-pocket expenses, historical costs,

depreciation, and other bookkeeping entries.

Economists on the other hand define cost by the size of the payment

necessary to keep the resource in its present employment, or alternatively,

by the remuneration that input would earn in its next best use.

Labor costs = hourly wage = wCapital costs = rental rate of capital = v

Cost of entrepreneurial services = forgone earnings


22/59

ECON 500


Cost Minimization

The firm seeks to minimize total costs given q = f (k, l) = q0.

First order conditions for this constrained minimum are


23/59

ECON 500


In order to minimize costs, the firm should choose inputs such that the rateat which k can be traded for l in production to the rate at which they can

be traded in the marketplace.

Alternatively, the firm should chose inputs such that the marginal

productivity per dollar spent should be the same for all inputs.


24/59

ECON 500



25/59

ECON 500



26/59

ECON 500


Total Cost Function

Average Cost Function

Marginal Cost Function


27/59

ECON 500


CRS Production Function and its Costs


28/59

ECON 500


Cubic Cost Function


29/59

ECON 500


Properties of Cost Functions

Non-decreasing in v, w, and q

Homogenous of degree 1 in input prices

Concave in input prices


30/59

ECON 500



31/59

ECON 500


Contingent Demand for Inputs and Shephards Lemma

Shephards lemma claimsthat C/w =l

Suppose that the price of labor (w) were to increase slightly. Costs would

rise by approximately the amount of labor lthat the firm was currentlyhiring.

Along the pseudo cost function all inputs are heldconstant, so an

increase in the wage increases costs in direct proportion to the amount of

labor used.

Because the true cost function is tangent to the pseudo-function at the

current wage, its slope (that is, its partial derivative) also will show the

current amount of labor input demanded.


32/59

ECON 500


Contingent Demand for Inputs and ShephardsLemma


33/59

ECON 500


Short run vs. Long run

In the short run, the firm is able to alter only its labor input while capital

input is fixed at some level k1.

With this formulation, payments to capital attain a fixed cost nature in the

short run and do not vary with the amount produced. Short run cost

function becomes:


34/59

ECON 500


Short run vs. Long run

In the short run, to change output, firms are forced to use input

combinations that are non-optimal.

Unavailability of input substitution prevents the firm from finding theinput mix where RTS equals the ratio of input prices.

Hence, in the short run, costs are not necessarily minimized.


35/59

ECON 500



36/59

ECON 500


Long-run total costs are always less than short-run total costs, except atthat output level for which the assumed fixed capital input is at the

appropriate level to ensure long-run cost minimization.

Therefore, long-run total cost curves constitute an envelope oftheir

respective short-run curves. A family of short run cost curves can beobtained by varying the capital level in

Long run cost curve can be recovered by combining the above SC with


37/59

ECON 500



38/59

ECON 500



39/59

ECON 500

Part III. Profit Maximization

A profit-maximizing firm chooses both its inputs and its outputs with thesole goal of achieving maximum economic profits. That is, the firm seeks

to make the difference between its total revenues and its total economic

costs as large as possible.

This holistic approach that treats the firm as a single decision-makingunit and sweeps away all the complicated behavioral issues about the

relationships (contractual or implicit) among input providers.

The profit maximizing assumption has a long history in economic

literature. It is plausible because firm owners may indeed seek to maketheir asset as valuable as possible and because competitive markets may

punish firms that do not maximize profits. The assumption also yields

interesting theoretical results that can explain actual firms decisions.


40/59

ECON 500


Marginalism and Profit Maximization

Firms perform the conceptual experiment of adjusting those variables that

can be controlled until it is impossible to increase profits further. This

involves, say, looking at the incremental, or marginal, profit obtainable

from producing one more unit of output. As long as this incremental profitis positive, the extra output will be produced. When the incremental profit

of an activity becomes zero, the firm has pushed output far enough, and it

would not be profitable to go further.


41/59

ECON 500


Firms choose the level of output that maximizes

The first order condition for a maximum is

In order to maximize economic profits, the firm should choose that output

for which marginal revenue is equal to marginal cost


42/59

ECON 500


MR=MC condition is only a necessary condition for maximum profits.

The second order condition that requires marginal profit to be

decreasing at the optimal level of output must also hold for sufficiency.


43/59

ECON 500



44/59

ECON 500


Marginal Revenue

If the firm can sell all it wishes without having any effect on market price,

the market price will indeed be the extra revenue obtained from selling

one more unit. Total revenue will be linear in output, marginal and

average revenue will be equal to price.

However, a firm may not always be able to sell all it wants at the

prevailing market price. If it faces a downward-sloping demand curve for

its product, the revenue obtained from selling one more unit will be less

than the price of that unit because, in order to get consumers to take theextra unit, the price of all other units must be lowered. Marginal revenue

be below price for any quantity q>1.


45/59

ECON 500


Marginal Revenue

If price does not change as quantity increases dp/dq = 0, marginal revenue

will be equal to price. In this case we say that the firm is a price taker

because its output decisions do not influence the price it receives.

On the other hand, if price falls as quantity increases dp/dq < 0, marginal

revenue will be less than price.


46/59

ECON 500



47/59

ECON 500


Marginal Revenue and Elasticity


48/59

ECON 500


Price Marginal Cost Markup

Using the MR elasticity relationship and equating MR to MC

The more elastic demand becomes, the lower the markup over MC


49/59

ECON 500


Supply Decision of a Price Taking Firm

If a firm is sufficiently small such that its output choice has no market on

the market price, the marginal revenue becomes equal to the market price.

The profit maximizing level of output q*can be found by

P =MR = MC

As long as at this level of output average variable cost is below the market

price, the firm would continue to operate in the short run.

The short run supply curve for a price taking firm is the positively sloped

segment of the firms short-run marginal cost above the point of minimum

average variable cost.


50/59

ECON 500



51/59

ECON 500


CobbDouglas Example


52/59

ECON 500


I - The supply curve is positively sloped - increases in P cause the firm to

produce more because it is willing to incur a higher marginal cost

II - The supply curve is shifted to the left by increases in the wage rate,

III - The supply curve is shifted outward by increases in capital input

IV - The rental rate of capital, v, is irrelevant to short-run supply decisions

CO 00


53/59

ECON 500


Profit Functions

We can represent the firms (maximized) profits asdepending only on the

prices that the firm faces by a profit function of the form

With the properties:

IHomogenous of degree 0

IINon-increasing in input prices

IIINon-decreasing in output price

IVConvex in output prices

ECON 500


54/59

ECON 500


Cobb Douglas Example

ECON 500


55/59

ECON 500


Input Demand

ECON 500


56/59

ECON 500


Comparative Statics for Input DemandSingle Input

ECON 500


57/59

ECON 500


Comparative Statics for Input DemandTwo Inputs

ECON 500


58/59

ECON 500


ECON 500


59/59

ECON 500

ECON 500 Producer Theory

Documents

Transcript of ECON 500 Producer Theory