Economis
The Production Process and Costs
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Chapter 5
Learning Objectives
Explain alternative ways of measuring the productivity of inputs and the role of the manager in the production process.
Calculate input demand and the cost-minimizing combination of inputs and use isoquant analysis to illustrate optimal input substitution.
Calculate a cost function from a production function and explain how economic costs differ from accounting costs.
Explain the difference between and the economic relevance of fixed costs, sunk costs, variable costs, and marginal costs.
Calculate average and marginal costs from algebraic or tabular cost data and illustrate the relationship between average and marginal costs.
Distinguish between short-run and long-run production decisions and illustrate their impact on costs and economies of scale.
Conclude whether a multiple-output production process exhibits economies of scope or cost complementarities and explain their significance for managerial decisions.
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2
The Production Function
Mathematical function that defines the maximum amount of output that can be produced with a given set of inputs.
is the level of output.
is the quantity of capital input.
is the quantity of labor input.
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The Production Function
3
Short-Run versus Long-Run Decisions: Fixed and Variable Inputs
Short-run
Period of time where some factors of production (inputs) are fixed, and constrain a manager’s decisions.
Long-run
Period of time over which all factors of production (inputs) are variable, and can be adjusted by a manager.
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5-4
The Production Function
4
Measures of Productivity
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:
Average product of capital:
Marginal product (MP)
The change in total product (output) attributable to the last unit of an input.
Marginal product of labor:
Marginal product of capital:
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5-5
The Production Function
5
Measures of Productivity in Action
Consider the following production function when 5 units of labor and 10 units of capital are combined produce: .
Compute the average product of labor.
units per worker
Compute the average product of capital.
units capital unit
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5-6
The Production Function
6
Increasing, Decreasing, and Negative Marginal Returns
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Labor input
(holding capital constant)
0
Total product
Average product
Marginal product
Total product (TP)
Average product (APL)
Marginal product (MPL)
Increasing
marginal
returns to labor
Decreasing
marginal
returns to labor
Negative
marginal
returns to labor
The Production Function
7
The Role of the Manager 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.
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: , where
Capital until the value of the marginal product of capital equals the rental rate: , where
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The Production Function
8
The Role of the Manager in the Production Process
Value marginal product: The value of the output produced by the last unit of an input.
Law of diminishing returns: The marginal product of an additional unit of output will at some point be lower than the marginal product of the previous unit.
Profit-Maximization 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.
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5-9
The Production Function
9
Algebraic Forms of Production Functions
Commonly used algebraic production function forms:
Linear: Assumes a perfect linear relationship between all inputs and total output
, where and are constants.
Leontief: Assumes that inputs are used in fixed proportions
, where and are constants.
Cobb-Douglas: Assumes some degree of substitutability among inputs
, where and are constants.
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The Production Function
10
Algebraic Forms of Production Functions in Action
Suppose that a firm’s estimated production function is:
How much output is produced when 3 units of capital and 7 units of labor are employed?
units
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The Production Function
11
Algebraic Measures of Productivity
Given the commonly used algebraic production function forms, we can compute the measures of productivity as follows:
Linear:
Marginal products: and
Average products: and
Cobb-Douglas:
Marginal products: and
Average products: and
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The Production Function
12
Algebraic Measures of Productivity in Action
Suppose that a firm produces output according to the production function
Which is the fixed input?
Capital is the fixed input.
What is the marginal product of labor when 16 units of labor is hired?
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5-13
The Production Function
13
Isoquants and Marginal Rate of Technical Substitution
Isoquants capture the tradeoff between combinations of inputs that yield the same output in the long run, when all inputs 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.
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The Production Function
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Isoquants and Marginal Rate of Technical Substitution in Action
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Labor Input
0
A
B
=100 units of output
Substituting labor for capital
200 units of output
300 units of output
Increasing output
Capital Input
The Production Function
15
Diminishing Marginal Rate of Technical Substitution
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Labor Input
0
D
C
=100 units
Capital Input
B
A
3
Slope:
Slope:
The Production Function
16
Isocost and Changes in Isocost Lines
Isocost
Combination of inputs that yield cost the same cost.
or, re-arranging to the intercept-slope formulation:
Changes in isocosts
For given input prices, isocosts farther from the origin are associated with higher costs.
Changes in input prices change the slopes of isocost lines.
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The Production Function
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Isocosts
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Labor Input
0
Capital Input
The Production Function
18
Changes in the Isocosts
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Labor Input
0
Capital Input
The Production Function
Less expensive input
bundles
More expensive input
bundles
19
Changes in the Isocost Line
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5-20
Labor Input
0
Capital Input
The Production Function
Due to increase in wage rate
20
Cost Minimization and the Cost-Minimizing Input Rule
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:
Equivalently, a firm should employ inputs such that the marginal rate of technical substitution equals the ratio of input prices:
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The Production Function
21
Cost-Minimization Input Rule in Action
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5-22
Labor Input
0
=100 units
Capital Input
The Production Function
22
Optimal Input Substitution
To minimize the cost of producing a given level of output, the firm should use less of an input and more of other inputs when that input’s price rises.
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23
The Production Function
Optimal Input Substitution in Action
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5-24
Labor Input
0
B
Capital Input
New cost-minimizing
point due to higher wage
A
Initial point of cost minimization
The Production Function
H
I
F
J
G
24
The Cost Function
Mathematical relationship that relates cost to the cost-minimizing output associated with an isoquant.
Short-run costs
Fixed costs (): do not change with changes in output; include the costs of fixed inputs used in production
Sunk costs
Variable costs []: costs that change with changes in outputs; include the costs of inputs that vary with output
Total costs:
Long-run costs
All costs are variable
No fixed costs
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The Cost Function
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Short-Run Costs
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5-26
Output
0
Total costs
Variable costs
Fixed costs
The Cost Function
26
Average and Marginal Costs
Average costs
Average fixed cost:
Average variable costs:
Average total cost:
Marginal cost (MC)
The (incremental) cost of producing an additional unit of output.
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The Cost Function
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The Relationship between Average and Marginal Costs
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Output
0
A
ATC, AVC, AFC
and MC ($)
Minimum of ATC
Minimum of AVC
The Cost Function
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Fixed and Sunk Costs
Fixed costs
Cost that does not change with output.
Sunk cost
Cost that is forever lost after it has been paid.
Irrelevance of Sunk Costs
A decision maker should ignore sunk costs to maximize profits or minimize loses.
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5-29
The Cost Function
29
Algebraic Forms of Cost Functions
The cubic cost function: costs are a cubic function of output; provides a reasonable approximation to virtually any cost function.
C(Q) = F + aQ + bQ2 + cQ3
where a, b, c, and f are constants and f represents fixed costs
Marginal cost function is:
MC(Q) = a + 2bQ + 3cQ2
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30
The Cost Function
Long-Run Costs
In the long run, all costs are variable since a manager is free to adjust levels of all inputs.
Long-run average cost curve
A curve that defines the minimum average cost of producing alternative levels of output allowing for optimal selection of both fixed and variable factors of production.
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The Cost Function
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Long-Run Average Cost
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Output
0
LRAC ($)
The Cost Function
32
Economies of Scale
Economies of scale
Declining portion of the long-run average cost curve as output increase.
Diseconomies of scale
Rising portion of the long-run average cost curve as output increases.
Constant returns to scale
Portion of the long-run average cost curve that remains constant as output increases.
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The Cost Function
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Economies and Diseconomies of Scale
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Output
0
LRAC ($)
The Cost Function
Economies of scale
Diseconomies of scale
34
Constant Returns to Scale
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5-35
Output
0
LRAC ($)
The Cost Function
35
Multiple-Output Cost Function
Economies of scope
Exist when the total cost of producing and together is less than the total cost of producing each of the type of output separately.
Cost complementarity
Exist when the marginal cost of producing one type of output decreases when the output of another good is increased.
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Multiple-Output Cost Function
36
Algebraic Form for a Multiproduct Cost Function
For this cost function:
MC1 = aQ2 + 2Q1
When a < 0, an increase in Q2 reduces the marginal cost of producing product 1.
If a < 0, this cost function exhibits cost complementarity
If a > 0, there are no cost complementarities
Exhibits economies of scope whenever f - > 0
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Multiple-Output Cost Function
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