Which of the following cost functions exhibits cost complementarity

.. ECON 563Managerial Economics

Module 4: Production and Cost Theory

Copyright 2017 Montclair State University

.. ECON 563 Managerial Economics Module 4a: Brief Overview

Learning Objectives

(1) Explain ways of measuring the productivity of inputs and the role of the manager in the production process.

(2) Calculate input demand and the cost-minimizing com- bination of inputs and use isoquant analysis to illustrate optimal input substitution.

(3) Calculate a cost function from a production function.

Learning Objectives

(4) Explain how economic costs differ from accounting costs. (5) Explain the difference between and the economic re-

levance of fixed costs, sunk costs, variable costs, and marginal costs.

(6) Calculate average and marginal costs from algebraic or tabular cost data and illustrate the relationship between average and marginal costs.

Learning Objectives

(7) Distinguish between short-run and long-run production decisions and illustrate their impact on costs and eco- nomies of scale.

(8) Conclude whether a multiple-output production process exhibits economies of scope or cost complementarities and explain their significance for managerial decisions.

.. ECON 563 Managerial Economics

Module 4b: Production Technology

Production Function Mathematical function that defines the maximum amount of output that can be produced with a given set of inputs.

Q = F (K,L),

• Q is the level of the output, • K is the level of the capital input, • L is the level of the labor input.

Short-run and Long-run Decisions Short-run

• Period of timewhere some factors of production (inputs) are fixed, and constrain a manager's decisions.

Long-run • Period of time over which all factors of production (in- puts) are variable, and can be adjusted by a manager.

Productivity Measures Total product (TP)

• Maximum level of output that can be produced with a given amount of inputs.

Average product (AP) • A measure of the output produced per unit of input.

• Average product of labor : APL = QL • Average product of capital : APK = QK

Example Consider the following production function when 5 units of labor and 10 units of capital are combined to produce :

Q = F (10, 5) = 150.

• Average product of labor : APL = 1505 = 30 units per worker

• Average product of capital : APK = 15010 = 15 units per unit of capital.

Marginal Product (MP)

The change in total product (output) attributable to the last unit of an input.

• Marginal product of labor : MPL = ∆Q∆L • Marginal product of capital : MPK = ∆Q∆K .

.. ECON 563 Managerial Economics

Module 4c: Decision at Margin

Manager's Role in the Production Process

• Produce output on the production function. • Aligning incentives to induce maximum worker effort. • Use the right mix of inputs to maximize profits.

Decisions at the Margin

To maximize profits when labor or capital vary in the short run, the manager will hire

• labor until the value of the marginal product of labor equals the wage rate (w) : VMPL = w where VMPL = P ×MPL.

• capital until the value of the marginal product of capital equals the rental rate (r) : VMPK = r where VMPK = P ×MPK .

Decisions at the Margin • Value marginal product : The value of the output produ- ced by the last unit of an input.

• Law of diminishing returns : The marginal product of an additional unit of input will at some point be lower than the marginal product of the previous unit.

Profit-Maximizing Input Usage • To maximize profits, use input levels at which marginal benefit equals marginal cost

• When the cost of each additional unit of labor is w, the manager should continue to employ labor up to the point where VMPL = w in the range of diminishing marginal product.

.. ECON 563 Managerial Economics

Module 4d: Production Functions

Common Production Function Forms • Linear : Assumes a perfect linear relationship between all inputs and total output

Q = F (K,L) = aK + bL,

where a and b are positive constants. • Leontief : Assumes that inputs are used in fixed propor- tions

Q = F (K,L) = min{aK, bL}, where a and b are positive constants.

Cobb-Douglas Production Technology

• It assumes some degree of substitutability among in- puts

Q = F (K,L) = KaLb,

• where a and b are positive constants.

Linear Production Technology • Suppose that a firm’s estimated production function is

Q = F (K,L) = 3K + 5L.

• How much output is produced when 5 units of capital and 10 units of labor are employed ?

Q = F (5, 10) = 3× 5 + 5× 10 = 15 + 50 = 65.

MP and AP • Linear production

MPK = a,MPL = b, APK = aK + bL

K ,APL =

aK + bL

L .

• Cobb-Douglas

MPK = aK a−1Lb,MPL = bK

aLb−1,

APK = K a−1Lb, APL = K

aLb−1.

Cobb- Douglas Technology • Let K = 1 be a fixed input and Q = K0.25L0.75. • What is the marginal product of labor when 16 units of labor is hired ?

MPL = 0.75(1) 0.25(16)−0.25 = 0.75× 0.5 = 0.375.

.. ECON 563 Managerial Economics

Module 4e: Isoquant and Isocost

Isoquants and MRTS Isoquant

• It capture the tradeoff between combinations of inputs that yield the same output in the long run, when all in- puts are variable.

Marginal rate of technical substitutions (MRTS) • The rate at which a producer can substitute between two inputs and maintain the same level of output.

• Absolute value of the slope of the isoquant.

MRTSKL = MPL MPK

.

K

L

Q=10 Q=30

Q=60 Q=90

Increasing Output

0

Isoquants and MRTS

K

L

∆L

∆L

0

Diminishing Marginal rate of technical substitutions

Isocost • Combination of inputs that yield the same cost.

wL+ rK = C.

• or, re-arranging to the intercept-slope formulation :

K = C

r − w

r L.

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

• Changes in input prices change the slopes of isocost lines.

K

LC w

C r

0

K = Cr − w r L

Isocost

K

LC1 w

C1

r

C0

w

C0

r

0

C1

C0

Changes in the isocosts C1 > C0

K

LCw0

C r

C w1

0

Changes in the isocost line w1 > w0

.. ECON 563 Managerial Economics Module 4f: Cost Minimization

Cost Minimization Producing at the lowest possible cost.

• Cost-minimizing input rule. • Produce at a given level of output where the marginal product per dollar spent is equal for all input.

MPL w

= MPK r

.

• Equivalently, a firm should employ inputs such that the marginal rate of technical substitution equals the ratio of input prices.

MPL MPK

= w

r .

K

K∗

C1 r

C0 w

LC0 w

C1 wL

∗

Q=60

0

Example : Minimum cost is C0, Inputs (K∗, L∗)

Optimal Input Bundle • To minimize the cost of producing a given level of out- put, the firm should use less of an input and more of other inputs when that input’s price rises.

• It is easy to draw isoquant - iscocost diagram to verify this observation.

Cost Function Mathematical relationship that relates cost to the cost-minimizing input associated with an isoquant.

• Short-run costs • Fixed costs (FC) : do not change with changes in output ; in- clude the costs of fixed inputs used in production.

• Sunk costs. • Variable costs, V C(Q) : costs that change with changes in outputs ; include the costs of inputs that vary with output.

Total Cost TC(Q) = FC + V C(Q).

• Long-run costs : • All costs are variable, and there is no fixed costs.

Average and Marginal Cost • Average costs

• Average Fixed cost : AFC = FCQ • Average Variable cost : AV C = V C(Q)Q • Average Total cost : AT C = C(Q)Q

• Marginal cost (MC) : • The (incremental) cost of producing an additional unit of out- put.

• MC = ∆C∆Q .

C(q)

q AFC

AVC AC

MC

0

Relation among AFC, AV C, AC, MC

Minimum of AVC and AC • Average variable cost attains it's minimum at the output q where MC = AV C.

• Average cost attains it's minimum at the output q where MC = AC.

.. ECON 563 Managerial Economics Module 4g: Cost Function

Fixed Cost and Sunk Cost Fixed costs

• Cost that does not change with output. Sunk costs

• Cost that is forever lost after it has been paid. Irrelevance of Sunk Costs

• A decision maker should ignore sunk costs to maximize profits or minimize losses.

Example of Cost Function Cubic cost function

• Costs are a cubic function of output ; provides a reaso- nable approximation to virtually any cost function.

C(Q) = F + aQ+ bQ2 + cQ3,

• where a, b, c, and F are constants and F represents the fixed cost.

Marginal cost function : MC(Q) = a+ 2bQ+ 3cQ2. Minimum AC

• AC attains it's minumum at output Q such that

AC = F

Q + a+ bQ+ cQ2 = a+ 2bQ+ 3cQ2 = MC.

Long-run Costs • In the long run, all costs are variable since a manager is free to adjust levels of all inputs.

Long-run average cost curve • A curve that defines the minimum average cost of pro- ducing alternative levels of output allowing for optimal selection of both fixed and variable factors of produc- tion.

Return to Scale Economies of scale

• Declining portion of the long-run average cost curve as output increases.

Diseconomies of scale • Rising portion of the long-run average cost curve as out- put increases.

Constant returns to scale • Portion of the long-run average cost curve that remains constant as output increases.

Multi-product Cost Function Economies of scope

• Exist when the total cost of producing Q1 and Q2 toge- ther is less than the total cost of producing each of the type of output separately.

C(Q1, 0) + C(0, Q2) > C(Q1, Q2).

Cost complementarity • Exists when the marginal cost of producing one type of output decreases when the output of another good is increased.

∆MC1(Q1, Q2)

∆Q2 < 0.

Example of Multi-product Cost Function

C(Q1, Q2) = F + aQ1Q2 +Q 2 1 +Q

2 2.

• For this cost function MC1 = aQ2 + 2Q1 and MC2 = aQ1 + 2Q2.

• When a < 0, an increase in Q2 reduces the marginal cost of producing product 1, and cost function exhibits cost complementarity.

• If a > 0, there are no cost complementarities. • Economy of scope exists whenever F − aQ1Q2 > 0.