Assignment 5
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chap005.pptxChapter 5
The Production Process and Costs
Copyright © 2014 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.
Chapter Outline
The production function
Short- versus long-run decisions
Measures of productivity
Manager’s role in production process
Algebraic forms of the production function and productivity
Isoquants and isocosts
Cost minimization and optimal input substitution
The cost function
Short-run costs
Average and marginal costs
Relations among costs
Fixed and sunk costs
Algebraic forms of cost functions
Long-run costs and economies of scale
Multiple-output cost functions
Economies of scope and cost complementarity
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Chapter Overview
2
Introduction
Chapter 4 focused on how consumers adjust consumption decisions in reaction to price and income changes. The theory developed illustrates the underlying principles of individual and market demand curves.
This chapter examines how managers select the optimal mix of inputs that minimize production costs.
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Chapter Overview
3
The Production Function
Mathematical function that defines the maximum amount of output that can be produced with a given set of inputs.
, where
is the level of output.
is the quantity of capital input.
is the quantity of labor input.
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The Production Function
4
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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The Production Function
5
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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The Production Function
6
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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The Production Function
7
Relation between Productivity Measures in Action
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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
8
The 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.
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
9
Manager’s Role in the Production Process in Action
Suppose a firm sells its output in a competitive market where its output is sold at $5 per unit. If workers are also hired at a competitive wage of $200, what is the marginal productivity of the last worker?
Since, and , then,
units
The marginal productivity of the last unit of labor is 40 units.
Alternatively, management should hire labor such that the last unit of labor produces 40 units.
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The Production Function
10
Algebraic Forms of Production Functions
Commonly used algebraic production function forms:
Linear: , where and are constants.
Leontief: , where and are constants.
Cobb-Douglas: , where and are constants.
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The Production Function
11
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
12
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
13
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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The Production Function
14
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
15
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
16
Diminishing Marginal Rate of Technical Substitution in Action
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Labor Input (L)
0
D
C
=100 units
Capital Input (K)
B
A
3
Slope (at C):
Slope (at A):
The Production Function
17
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
18
Isocost Line
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Labor Input (L)
0
Capital Input (K)
The Production Function
19
Changes in the Isocost Line
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Labor Input (L)
0
Capital Input (K)
The Production Function
Less expensive input
bundles
More expensive input
bundles
20
Changes in the Isocost Line
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Labor Input (L)
0
Capital Input (K)
The Production Function
Due to increase in wage rate
21
Cost-Minimization Input Rule in Action
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Labor Input (L)
0
=100 units
Capital Input (K)
The Production Function
22
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 inputs:
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
23
Cost-Minimizing Input Rule in Action
Suppose that labor and capital are hired at a competitive wage of $10 and $25, respectively. If the marginal product of capital is 6 units and the marginal product of labor is 3 units, is the firm hiring the cost-minimizing units of capital and labor?
Since , the marginal product per dollar spent on labor exceeds the marginal product per dollar spent on capital.
The firm is not minimizing costs and should use fewer units of capital and more labor.
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The Production Function
24
Optimal Input Substitution in Action
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Labor Input (L)
0
B
Capital Input (K)
New cost-minimizing
point due to higher wage
A
Initial point of cost minimization
The Production Function
H
I
F
J
G
25
The Cost Function
Mathematical relationship that relates cost to the cost-minimizing output associated with an isoquant.
Short-run costs
Fixed costs:
Sunk costs
Short-run variable costs:
Short-run total costs:
Long-run costs
All costs are variable
No fixed costs
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The Cost Function
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Short-Run Costs in Action
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Output
0
Total costs
Variable costs
Fixed costs
The Cost Function
27
Average and Marginal Costs
Average costs
Average fixed:
Average variable costs:
Average total cost:
Marginal cost
The (incremental) cost of producing an additional unit of output.
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The Cost Function
28
The Relationship between Average and Marginal Costs in Action
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Output
0
A
ATC, AVC, AFC
and MC ($)
Minimum of ATC
Minimum of AVC
The Cost Function
29
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.
Principle of Irrelevance of Sunk Costs
A decision maker should ignore sunk costs to maximize profits or minimize loses.
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The Cost Function
30
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 Total Costs in Action
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Output
0
LRAC ($)
The Cost Function
32
Economies of Scale
Economies of scale
Portion of the long-run average cost curve where long-run average costs decline as output increases.
Diseconomies of scale
Portion of the long-run average cost curve where long-run average costs increase as output increases.
Constant returns to scale
Portion of the long-run average cost curve that remains constant as output increases.
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The Cost Function
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Economies and Diseconomies of Scale in Action
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Output
0
LRAC ($)
The Cost Function
Economies of scale
Diseconomies of scale
34
Constant Returns to Scale in Action
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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
Multiple-Output Cost Function in Action
Suppose a firm produces two goods and has cost function given by
If the firm plans to produce 4 units of and 6 units of
Does this cost function exhibit cost complementarities?
Yes, cost complementarities exist since
Does this cost function exhibit economies of scope?
Yes, economies of scope exist since
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Multiple-Output Cost Function
37
Conclusion
To maximize profits (minimize costs) managers must use inputs such that the value of marginal product of each input reflects the price the firm must pay to employ the input.
The optimal mix of inputs is achieved when the .
Cost functions are the foundation for helping to determine profit-maximizing behavior in future chapters.
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Labor Input (L)
0
=100 units
Capital Input (K)
The Production Function
Cost-Minimization In Action
39
Diminishing Marginal Rate of Technical Substitution in Action
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Labor Input (L)
0
A
B
=100 units
Capital Input (K)
C
D
Slope (at A):
Slope (at C):
The Production Function
40
Suppose that capital and labor are hired at a competitive wage of $10 and $20, respectively. If the marginal product of capital is 5 units and the marginal product of labor is 10 units, is the firm hiring the cost-minimizing units of capital and labor?
Since , the cost-minimizing mix of capital and labor is being utilized.
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The Production Function
Cost-Minimization Input Rule In Action
41
Multiple-Output Cost Function in Action
Suppose a firm produces two goods and has cost function given by
If the firm plans to produce 4 units of and 6 units of
Does this cost function exhibit cost complementarities?
Yes, cost complementarities exist since
Does this cost function exhibit economies of scope?
Yes, economies of scope exist since
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Multiple-Output Cost Function
42
