Profit Maximising Midel
PROFIT MAXIMIZATION
[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
3
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*).
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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
[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
3
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