# Profit Maximising Midel

[See Chap 11]

1

Profit Maximization

• A profit-maximizing firm chooses both its inputs and its outputs with the goal of achieving maximum economic profits

2

Model

• Firm has inputs (z1,z2). Prices (r1,r2).

– Price taker on input market.

• Firm has output q=f(z1,z2). Price p.

– Price taker in output market.

• Firm’s problem:

– Choose output q and inputs (z1,z2) to maximise profits. Where:

π = pq - r1z1 – r2z2

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1

One-Step Solution

• Choose (z1,z2) to maximise π = pf(z1,z2) - r1z1 – r2z2 • This is unconstrained maximization problem. • FOCs are p ∂f ( z1 , z 2 ) = r1 and z1 p ∂f ( z1 , z 2 ) = r2 z2

• Together these yield optimal inputs zi*(p,r1,r2). • Output is q*(p,r1,r2) = f(z1*, z2*). This is usually called the supply function. • Profit is π(p,r1,r2) = pq* - r1z1* - r2z2* 4

Example: f(z1,z2)=z11/3z21/3

• Profit is π = pz11/3z21/3 - r1z1 – r2z2 • FOCs are

1 − 2 / 3 1/ 3 pz1 z 2 = r1 and 3

1 p3 27 r12 r2

1 1/ 3 − 2 / 3 pz1 z 2 = r2 3

1 p3 27 r1r22

• Solving these two eqns, optimal inputs are

* z1 ( p, r1 , r2 ) =

and

* z2 ( p, r1 , r2 ) =

• Optimal output • Profits

* * q * ( p, r1 , r2 ) = ( z1 )1/ 3 ( z 2 )1 / 3 =

1 p2 9 r1r2

1 p3 27 r1r2

5

* π * ( p, r1 , r2 ) = pq* − r1 z1* − r2 z2 =

Two-Step Solution

Step 1: Find cheapest way to obtain output q. c(r1,r2,q) = minz1,z2 r1z1+r2z2 s.t f(z1,z2) ≥ q Step 2: Find profit maximizing output. π(p,r1,r2) = maxq pq - c(r1,r2,q) This is unconstrained maximization problem. • Solving yields optimal output q*(r1,r2,p). • Profit is π(p,r1,r2) = pq* - c(r1,r2,q*). 6

2

Step 2: Output Choice

• We wish to maximize pq - c(r1,r2,q) • The FOC is p = dc(r1,r2,q)/dq • That is, p = MC(q) • Intuition: produce more if revenue from unit exceeds the cost from the unit. • SOC: MC’(q)≥0, so MC curve must be upward sloping at optimum. 7

Example: f(z1,z2)=z11/3z21/3

• From cost slides (p18), c(r1,r2,q) = 2(r1r2)1/2 q3/2 • We wish to maximize π = pq - 2(r1r2)1/2 q3/2 • FOC is p = 3(r1r2)1/2 q1/2 • Rearranging, optimal output is q* ( p, r1 , r2 ) =

• Profits are

1 p2 9 r1r2

1 p3 27 r1r2

8

π * ( p, r1 , r2 ) = pq * − c( r1 , r2 , q * ) =

Profit Function

• Profits are given by π = pq* - c(q*) • We can write this as π = pq* - AC(q*)q* = [p-AC(q*)]q* • We can also write this as π = pq* − ∫ MC ( x)dx − F = ∫ [ p − MC ( x)]dx − F 0 0 q q

where F is fixed cost

9

3

Profit Function

Left: Profit is distance between two lines. Right: Max profit equals A+B+C. If no fixed cost, this equals A+B+D+E. 10

Supply Functions

11

Supply with Fixed Cost

price

MC

p*

AC

Maximum profit occurs where p = MC

q*

output

12

4

Supply with Fixed Cost

price

MC

p*

AC

Since p > AC, we have π > 0.

q*

output

13

Supply with Fixed Cost

price

p** p* AC MC

If the price rises to p**, the firm will produce q** and π > 0 q* q**

output

14

Supply with Fixed Cost

price

MC

If the price falls to p***, we might think the firm chooses q***. AC

p* = MR

p***

But π < 0 so firm prefers q=0.

q***

q*

output

15

5

Supply Curve

• We can use the marginal cost curve to show how much the firm will produce at every possible market price. • The firm can always choose q=0 – Firm only operates if revenue covers costs. – Firm chooses q=0 if pqAVC. • Firm shuts down if p

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