The Production Process

Production Analysis• Production Function Q = f(K,L)

• Describes available technology and feasible means of converting inputs into maximum level of output, assuming efficient utilization of inputs:

• ensure firm operates on production function (incentives for workers to put max effort)

• use cost minimizing input mix

• Short-Run vs. Long-Run Decisions

• Fixed vs. Variable Inputs

Total Product

• Cobb-Douglas Production Function

• Example: Q = f(K,L) = K.5 L.5

• K is fixed at 16 units.

• Short run production function:

Q = (16).5 L.5 = 4 L.5

• Production when 100 units of labor are used?

Q = 4 (100).5 = 4(10) = 40 units

Marginal Product of Labor

• Continuous case: MPL = dQ/dL

• Discrete case: arc MPL = Q/L

• Measures the output produced by the last worker.

• Slope of the production function

Average Product of Labor

• APL = Q/L

• Measures the output of an “average” worker.

• Slope of the line from origin onto the production function

Law of Diminishing Returns (MPs)TP increases at an increasing rate (MP > 0 and ) until inflection , continues to increase at a diminishing rate (MP > 0 but ) until max and then decreases (MP < 0).

0

5

10

15

20

25

0 2 4 6 8 10 12

Input L

Tota

l Pro

duct

-4

-3

-2

-1

0

1

2

3

4

0 2 4 6 8 10 12

Three significant points are: Max MPL (TP inflects) Max APL = MPL

MPL = 0 (Max TP)

A line from the origin is tangent to Total Product curve at the maximum average product.

IncreasingMP

DiminishingMP

NegativeMP

LTP = Q = 10K1/2L1/2

Isoquant

• The combinations of inputs (K, L) that yield the producer the same level of output.

• The shape of an isoquant reflects the ease with which a producer can substitute among inputs while maintaining the same level of output.

• Slope or Marginal Rate of Technical Substitution can be derived using total differential of Q=f(K,L) set equal to zero (no change in Q along an isoquant)

K

L

MP

MP

KQ

LQ

L

KL

L

QK

K

Q

0

Cobb-Douglas Production Function• Q = KaLb

• Inputs are not perfectly substitutable (slope changes along the isoquant)

• Diminishing MRTS: slope becomes flatter

• Most production processes have isoquants of this shape

• Output requires both inputs

Q1

Q2

Q3K

-K1

||

-K2

L1 < L2 L

Increasing Output

K

L

a

a

L

Kabfor

1

,1

Linear Production Function

• Q = aK + bL

• Capital and labor are perfect substitutes (slope of isoquant is constant)y = ax + bK = Q/a - (b/a)L

• Output can be produced using only one input

Q3Q2Q1

Increasing Output

L

K

Leontief Production Function

• Q = min{aK, bL}

• Capital and labor are perfect complements and cannot be substituted (no MRTS <=> no slope)

• Capital and labor are used in fixed-proportions

• Both inputs needed to produce output

Q3

Q2

Q1

K

Increasing Output

Isocost• The combinations of inputs that

cost the same amount of moneyC = K*PK + L*PL

• For given input prices, isocosts farther from the origin are associated with higher costs.

• Changes in input prices change the slope (Market Rate of Substitution) of the isocost lineK = C/PK - (PL/PK)L

K

LC1C0

L

KNew Isocost for a decrease in the wage (labor price).

New Isocost for an increase in thebudget (total cost).

Long Run Cost Minimization

K

K

L

L

P

MP

P

MP

Q

L

K -PL/PK < -MPL/MPK

MPK/PK< MPL/PL

-PL/PK > -MPL/MPK

MPK/PK> MPL/PL

Point of CostMinimization

-PL/PK = -MPL/MPK

MPK/PK= MPL/PL

Min cost where isocost is tangent to isoquant (slopes are the same)

Expressed differently: MP (benefit) per dollar spent (cost) must be equal for all inputs

KLK

L

K

LKL MRTS

MP

MP

P

PMRS

Returns to Scale• Return (MP): How TP changes when one input increases

• RTS: How TP changes when all inputs increase by the same multiple > 0

• Q = f(K, L)

•

• Q = 50K½L½

Q = 100,000 + 500L + 100KQ = 0.01K3 + 4K2L + L2K + 0.0001L3

ScaletoReturns

Decreasing

Constant

Increasing

QL)K,f(If