# ECON 2X03 Lecture Notes - Lecture 4: Production Function, Marginal Product, Farad

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Published on 18 Apr 2013

Department

Economics

Course

ECON 2X03

Professor

Lecture 4

Individual Firm’s Supply

The supply function for a firm gives the profit maximizing quantity of output as a function of market

price. Found where SMC = P and SMC is upward-sloping and P>min of AVC (if P<min AVC, y = 0)

Ex. A firm has y = z11/2z21/2 and faces w1=w2=1. In the short run z2=4. What is its SR supply function?

- y=z11/241/2

- z1*= y2/4

- VC = w1z1* = y2/4

- SMC = dVC/dy = y/2

- SMC always upward-sloping

- AVC = VC/y=y/4 which is <y/2 = SMC for Vy>0

- So just set SMC = P

- y/2=P

- y*=2P

Ex. A firm has VC = y3/3-5y2+45y. What is the SR supply?

- SMC = dVC/dy = y2-10y+45

- dSMC/dy = 2y-10

- y= 5

- At y=5, SMC = 25-50+45 = 20

- AVC = VC/y = y2/3-5y+45

- dAVC/dy=2y/3-5 = 0

- y=15/2

- AVC(15/2) = 75/4-150/4+180/4=105/4

- So for P<=105/4, y*=0. For P>=105/4, P=SMC

- P = y2-10y+45

- y2-10y+45-P=0

- y2-10y+25+20-P=0

- (y-5)2+20-P=0

- (y-5)2=P-20

- Y=5+sqrt(P-20)

- So supply function: y* = {0 if P<=105/4 || 5+sqrt(P-20) if P>=105/4}

Market Supply Function (SR): total desired (π-maxing) output in a market as a function of market price.

Given a market with n firms with individual supply functions yi*(p) i=1,2,…n, the market supply is Y*(P)=

SUM(yi(P)) on [1,n]

Ex. A market has 200 firms. They each have z11/2z21/2 and w1=w2 =1. 100 of them have 4 units of input 2

(which is fixed in the SR). The other 100 firms have 25 units of input 2. What is the market supply

function?

- y = z11/2z21/2

- z1*=y2/z2

- VC = w1z1* = y2/z2

- SMC = dVC/dy = 2y/z2 (>AVC = y/z2)

- So P = SMC gives individual firm supply.

- P=2y/z2

- yi*(p)=Pz2/2

- Y*(P) = SUM(1-200) (yi(p)) = 100(P(4)/2) + 100(P(25)/2)

- Y*(P) = 200P + 1250P = 1450P market supply function

## Document Summary

The supply function for a firm gives the profit maximizing quantity of output as a function of market price. Found where smc = p and smc is upward-sloping and p>min of avc (if p0. So just set smc = p y/2=p y*=2p. What is the sr supply? y= 5 y=15/2. At y=5, smc = 25-50+45 = 20. So supply function: y* = {0 if p<=105/4 || 5+sqrt(p-20) if p>=105/4} Market supply function (sr): total desired ( -maxing) output in a market as a function of market price. Given a market with n firms with individual supply functions yi*(p) i=1,2, n, the market supply is y*(p)= 100 of them have 4 units of input 2 (which is fixed in the sr).