The Production Process and Costs

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 (increases with capital intensity)

• 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

Optimal Level of Inputs

Marginal Principle: continue to hire as long as marginal benefit > marginal cost of the input, stop when MB = MC.

MC (Pinput, monetary units): cost of hiring the last unit of input.

MB (MPinput, physical units): contribution of the last unit of input hired to the total product.

MB (VMPinput, monetary units): value of the output produced by the last unit of input = MPinput * Poutput .

Downward sloping portion of the VMP curve is the demand for input.

VMPL

Q

L

PL

L*

Demand for labor

Solve the following equation for L*:

MBL = VMPL = MPL*PQ = PL = MCL

The Long Run Production Function:

Q = 10K1/2L1/2

Isoquants and the Production Surface

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

MPMP

KQ

LQ

LKL

LQK

KQ

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

KL

aa

LKabfor

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

PMP

PMP

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

MPMP

PP

MRS

Returns to Scale• Return (MP): How TP changes when one input increases• RTS: How TP changes when all inputs increase by the same multiple λ > 0• Q = f(K, L)

•

• Q = 50K½L½

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

ScaletoReturnsDecreasingConstant

IncreasingQL)K,f(If

Expansion path and Long-Run Total Cost

Long-Run Total Cost is the least cost combination of inputs for each production quantity (derives from the expansion path)

K*PK + L*PL =

LTC = 10Q-.6Q2+.02Q3

QLTCLAC

dQLTCdLMC )(

12

12

QQLTCLTC

QLTCLMCarc

Effect of a Fixed Input on Cost of Production

In the short run K is fixed at K0. Any input L other than L0 will result in other than least TC. If I1 is required, input L will be reduced to point E, associated with TCmuch higher than optimal at point A.

LTC as a Lower Envelope of STC

• Every point on LTC represents a least-cost combination.

• In the short run one ormore inputs are fixed so that only a single point on STC is a least-cost combination of inputs.

• STC curves intersect cost axis at the value of the TFC.

STC = TFC + TVC = 1000+80Q-6Q2+.2Q3

SAC = STC / Q

= TFC/Q + TVC/Q

= AFC + AVC

AFC = 1000/Q

AVC = 80-6Q+.2Q2

SMC = dSTC/dQ

= dTFC/dQ + dTVC/dQ

= dTVC/dQ

= 80-12Q+.6Q2

Productivity of Variable Input and Short-Run Cost

= Q = f(L)

Short-Run Total Cost, Total Variable Cost & Total Fixed Cost

= PL * L

= PK * K

= TFC + TVC

Average Product and Average Variable Cost

Marginal Product and Short-Run Marginal Cost

LAC as a Lower Envelope of SAC• In the long run all

total costs represent least-costs.

• All average costs must be least cost as well.

• Various short-run cost curves for various values of the fixed input.

• In the short run only one point represents least cost.

Economies of scale (minimum SAC of in the smaller facility greater than SAC in the larger facility) exist up to the minimum LAC (downward sloping portion of LAC curve).

Beyond minimum LAC diseconomies of scale.

Economiesof Scale

Diseconomiesof Scale

Long-Run Average Cost and Returns to Scale

Economiesof Scale

Diseconomiesof Scale

Increasing Returns to Scale: Economies of Scale:Q1 = f(K = 20, L = 10) = 100 PK = 20, PL = 50

LTC1 = 20*20 + 50*10 = 900LAC1 = 900 / 100 = 9

Q2 = f(K = 40, L = 20) = 300 > 2Q2 LTC2 = 20*40 + 50*20 = 1,800LAC2 = 1,800 / 300 = 6 < LAC1

Economies of Scopeand Cost Complementarity

• Cheaper to produce outputs jointly than separately:C(Q1, Q2) < C(Q1, 0) + C(0, Q2)

• MC of producing good 1 declines as more of good 2 is produced:MC1 / Q2 < 0

• Example: Joint processing of deposit accounts and loans in banksScope: Single financial advisor eliminates duplicate common factors of production (computers, loan production offices)

Complementarity: Account and credit information developed for deposits lowers credit check and monitoring cost for loans. Expansion of deposit base lowers cost of providing loans.

Quadratic Multi-Product Cost Function

• C(Q1, Q2) = f + aQ1Q2 + (Q1 )2 + (Q2 )2 MCi(Qi, Qj) = aQj + 2Qi

• Economies of scope (cheaper joint product) if :f > aQ1Q2

C(Q1, 0) + C(0, Q2 ) = f + (Q1)2 + f + (Q2)2

• Cost complementarity exists if: a < 0MCi/ Qj = a < 0

A Numerical Example:• C(Q1, Q2) = 90 - 2Q1Q2 + (Q1 )2 + (Q2 )2

MC1(Q1, Q2) = -2Q2 + 2Q1

• Economies of Scope?Yes, since 90 > -2Q1Q2

• Cost Complementarity?Yes, since a = -2 < 0

• Implications for Merger?