Producer Theory: Concept & Formula
Guide
Cheat Sheet based on Varian’s Microeconomics
1. Technology 3. Cost Minimization
The firm converts inputs (x1 , x2 ) into output y via the The goal: Produce y at the lowest cost.
**Production Function** y = f (x1 , x2 ).
min w1 x1 + w2 x2 s.t. f (x1 , x2 ) = y
Marginal Product (MP)
The Tangency Condition
Extra output from one more unit of input.
For Cobb-Douglas and smooth curves:
∂f ∂f
M P1 = M P2 =
∂x1 ∂x2 M P1 w1
=
M P2 w2
Diminishing MP: Usually ∂M P
∂x < 0. Adding more labor
to a fixed machine helps less and less. This defines the optimal mix of inputs.
Technical Rate of Substitution (TRS)
Conditional Factor Demands
The slope of the **Isoquant**.
The derived demand for inputs based on the output target
dx2 M P1 y: x1 (w1 , w2 , y).
T RS = =−
dx1 M P2
Measures the technical trade-off: how much capital can I 4. Cost Curves
reduce if I add one worker to keep output constant?
Once we find the optimal mix, we calculate the Cost Func-
tion C(y).
Returns to Scale (RTS)
What happens if we scale all inputs by t > 1? Definitions
• CRS: f (tx1 , tx2 ) = tf (x1 , x2 ). (Double inputs → • Total Cost: C(y) = V C(y) + F C.
Double output).
• Avg Cost (AC): C(y)/y.
• IRS: f (tx1 , tx2 ) > tf (x1 , x2 ). (Double inputs →
Triple output).
• Marginal Cost (MC): ∂C/∂y.
• DRS: f (tx1 , tx2 ) < tf (x1 , x2 ).
Geometry of Costs
2. Profit Maximization
M C = AC at the minimum of AC
Profit π = Revenue − Cost.
π = pf (x1 , x2 ) − w1 x1 − w2 x2 • If M C < AC, AC is falling.
• If M C > AC, AC is rising.
Short Run (SR)
One factor is fixed (usually x2 = k̄). Firm chooses x1 .
5. Firm Supply
p · M P1 = w1
A competitive firm takes price p as given.
Value of Marginal Product = Factor Price.
The Supply Rule
Long Run (LR)
All factors are variable. p = M C(y)
p · M P1 = w1 and p · M P2 = w2 Provided that the slope of MC is positive.
1
Guide
Cheat Sheet based on Varian’s Microeconomics
1. Technology 3. Cost Minimization
The firm converts inputs (x1 , x2 ) into output y via the The goal: Produce y at the lowest cost.
**Production Function** y = f (x1 , x2 ).
min w1 x1 + w2 x2 s.t. f (x1 , x2 ) = y
Marginal Product (MP)
The Tangency Condition
Extra output from one more unit of input.
For Cobb-Douglas and smooth curves:
∂f ∂f
M P1 = M P2 =
∂x1 ∂x2 M P1 w1
=
M P2 w2
Diminishing MP: Usually ∂M P
∂x < 0. Adding more labor
to a fixed machine helps less and less. This defines the optimal mix of inputs.
Technical Rate of Substitution (TRS)
Conditional Factor Demands
The slope of the **Isoquant**.
The derived demand for inputs based on the output target
dx2 M P1 y: x1 (w1 , w2 , y).
T RS = =−
dx1 M P2
Measures the technical trade-off: how much capital can I 4. Cost Curves
reduce if I add one worker to keep output constant?
Once we find the optimal mix, we calculate the Cost Func-
tion C(y).
Returns to Scale (RTS)
What happens if we scale all inputs by t > 1? Definitions
• CRS: f (tx1 , tx2 ) = tf (x1 , x2 ). (Double inputs → • Total Cost: C(y) = V C(y) + F C.
Double output).
• Avg Cost (AC): C(y)/y.
• IRS: f (tx1 , tx2 ) > tf (x1 , x2 ). (Double inputs →
Triple output).
• Marginal Cost (MC): ∂C/∂y.
• DRS: f (tx1 , tx2 ) < tf (x1 , x2 ).
Geometry of Costs
2. Profit Maximization
M C = AC at the minimum of AC
Profit π = Revenue − Cost.
π = pf (x1 , x2 ) − w1 x1 − w2 x2 • If M C < AC, AC is falling.
• If M C > AC, AC is rising.
Short Run (SR)
One factor is fixed (usually x2 = k̄). Firm chooses x1 .
5. Firm Supply
p · M P1 = w1
A competitive firm takes price p as given.
Value of Marginal Product = Factor Price.
The Supply Rule
Long Run (LR)
All factors are variable. p = M C(y)
p · M P1 = w1 and p · M P2 = w2 Provided that the slope of MC is positive.
1
