4 THE Profit Function - Lecture notes 1 PDF

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THEORY OF THE FIRM IV: THE PROFIF FUNCTION.LECTURE FOUR.INTRODUCTIONHaving discussed the firm’s technology and the costs associated, in this lesson we now turn to the firm’s solution to its optimization problem. The solution is expressed as a profit function.LECTURE OBJECTIVESAt the end of this less...

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THEORY OF THE FIRM IV: THE PROFIF FUNCTION. LECTURE FOUR.

INTRODUCTION

Having discussed the firm’s technology and the costs associated, in this lesson we now turn to the firm’s solution to its optimization problem. The solution is expressed as a profit function.

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LECTURE OBJECTIVES

At the end of this lesson, the learner should: •

Understand the definition and properties of the profit function.

•

Understand how to derive the profit function

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THE PROFIT FUNCTION. It represents the solution to the firm’s optimization problem. It’s expressed as a function of output prices of input prices. i.e.

 ( p , w) = max pf (x ) − wx  Direct function

 ( p, w) = max p( y ) − C (w, y ) Indirect function Where p is the output price which the producer takes as given.

w is the vector of strictly the input prices. x is a vector of inputs y is the output

PROPERTIES OF THE PROFIT FUNCTION (i) The profits are non-negative i.e. a producer never accepts negative profits in the long run (ii)

Its non-decreasing in output price, if p and p' are two output prices and that p  p' the profits evaluated:  ( p , w )   ( p ' w )

(iii)

The profit function is non-increasing in input prices i.e. if w and w' are two vectors or sets of input prices and w  w' , then  ( p, w)   ( pw')

(iv)

The profit function is positively, linearly homogenous in both input and output prices. i.e.  (p, w) =  ( p , w ) where   0 . If output and input prices is double, then the profits will double. This property is a further consequence of the principle that only relative prices matter in economics.

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(v)

If profit function  ( p, w) is differentiable in p & w , then there exists a unique profit maximizing supply and derived demand functions given as follows: Supply y (p, w ) =

 (p, w ) .........(i ) p

Demand x( p, w) =

 ( p, w ) .........(ii) wi

(First Hotellingslemma ) (Second Hotellingslemma)

The Hotellings Lemma states that: •

If the profit function is well behaved, the first partial positive derivative of the profit function with respect to the output prices is the firms supply function.

•

For a well behaved profit function (twice differentiable) the first negative partial derivative of the profit function with respect to the input price gives the unconditional factor demand. (vi) The profit function is convex in all prices if and only if the production function is strictly concave. i.e. the Hessian matrix of the profit function is positive semi-definite.

p

w1

w2

  y y  y p   2  2  2 2  pw1 pw2   p w1 w2  p    2  2 2 x1 x1     =  x1  >0 H = w1    2 p w 1 w2  w1 p w1 w2   w1    2  x 2   2   2  2   x 2 x 2    w2  p dw w   w2 p w2 w1 w2   1 2

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       p   = p p 2 2

    2  2 = − wi  wi wi  

=

y which is the slope of a supply curve and is positive, and p

   = − x i which is positive since x i is the slope of a demand and is  wi wi  

negative. DERIVATION OF THE PROFIT FUNCTION Given a production function y = f (x1 x 2 ) x1 and x2 are two inputs Let p be price of output

w1 & w2 be prices of x1 x 2 respectively The firm aims at maximizing  given by

max  = py (x 1x 2 ) − w 1x 1 − w 2x 2

(Unconstrained profit maximization problem)

Finding the first order conditions and equating them to zero;

 = pMPx1 − w1 = 0...........(i )  x1  = pMPx2 − w2 = 0............(ii )  x2 Solving equations (i) & (ii ) simultaneously, we will find the optimal amount of inputs that maximize profit. x1* ( p , w1 , w2 ) x2* ( p , w1 , w2 )

Unconditional factor demands

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Substituting the unconditional factor demand in the profit equation, provides the profit function

 * (p , w ) = p y (x1* , x2* )− w1 x1* − w2 x *2 For example, consider a production function Y = x a where 0  a  1 Let p = price of output

w = Price of input max  = p.x a − wx  = ap. x a −1 − w = 0 x ap .x a−1 = w x a −1 =

w a .p 1

 w  a−1  x =   a. p  *

( )

y* = x*

a a

 w  a− 1  y =   a. p   = py * − wx * *

a

1

 w  a− 1  w a − 1  w  −   = p  a. p   a. p  a

1

1

1 − a w a− 1 1 a −1 1 a − 1  =   p 1 p a −1 − w.wa − 1 a p  a

 =p

−

1 a −1

a

1

a  1  a −1  w  a −1    − w a −1  a  a. p 

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a

a

w a−1

= p

1 a−1

a a−1

a

w a −1

−

1

1

p a −1 a a − 1

a

  w a −1  1 1   = 1 − 1   −a a− 1 a 1 p a a−1  a

 = kw

a a −1

p

 1  −   a −1 

Note that the profit function is positively linearly homogeneous and convex. i.e. a

1

−  = (tw )a −1 * (tp ) a −1 a

t a−1

−

1 a−1

=1

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SUMMARY

In this lesson, we have learnt that the profit function results from a combination of the firm’s production function and the cost function.

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NOTE.

The function that gives the maximum profits of a firm as a function of the prices is called the profit function of the firm.

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SELF-TEST QUESTIONS

Given the production function in the last lesson’s question, derive the firm’s profit function.

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REFERENCES

 Varian, H.R. Microeconomic Analysis. W.W. Norton and Co. London  Chiang, A Fundamental Methods of Mathematical Economics (McGraw Hill).  Baumo, W. Economic Theory and Operations Analysis (Prentice Hall)  Silberberg, E. The structure of Economics: A Mathematical Analysis (McGraw Hill)  E. Herdeson, J. and R. Quandt, Microeconomic Theory: A Mathematical Approach (MacGraw).  Koutsoyiannis, J. Modern Microeconomic (McMillan)  Nicholson, W. Microeconomic Theory; Basic Principles and Extensions (Dryden Press). (Additional references will be provided in classes).  Pindyck. S.R& Rubifield D.L (1996) Microeconomics 3rd Edition: New Delhi: Prentice Hallindia Private Ltd.

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