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Production
ECO61 Microeconomic AnalysisUdayan Roy
Fall 2008
What is a firm?
• A firm is an organization that converts inputs such as labor, materials, energy, and capital into outputs, the goods and services that it sells.– Sole proprietorships are firms owned and run by a
single individual.– Partnerships are businesses jointly owned and
controlled by two or more people.– Corporations are owned by shareholders in
proportion to the numbers of shares of stock they hold.
What Owners Want?
• Main assumption: firm’s owners try to maximize profit
• Profit () is the difference between revenues, R, and costs, C:
= R – C
What are the categories of inputs?• Capital (K) - long-lived inputs.
– land, buildings (factories, stores), and equipment (machines, trucks)
• Labor (L) - human services – managers, skilled workers (architects, economists,
engineers, plumbers), and less-skilled workers (custodians, construction laborers, assembly-line workers)
• Materials (M) - raw goods (oil, water, wheat) and processed products (aluminum, plastic, paper, steel)
How firms combine the inputs?
• Production function is the relationship between the quantities of inputs used and the maximum quantity of output that can be produced, given current knowledge about technology and organization
Production Function
• Formally,q = f(L, K)
– where q units of output are produced using L units of labor services and K units of capital (the number of conveyor belts).
ProductionFunction
q = f(L, K)
Output q
Inputs(L, K)
The production function may simply be a table of numbers
The production function may be an algebraic formula
}12,24{min18
1224
15 4.06.0
KLQ
KLQ
KLQ
Just plug in numbers for L and K to get Q.
Marginal Product of Labor• Marginal product of labor (MPL ) - the change in
total output, Q, resulting from using an extra unit of labor, L, holding other factors constant:
L
QMPL
Average Product of Labor• Average product of labor (APL ) - the ratio of
output, Q, to the number of workers, L, used to produce that output:
L
QAPL
Production with Two Variable Inputs
• When a firm has more than one variable input it can produce a given amount of output with many different combinations of inputs– E.g., by substituting K for L
7-11
Isoquants
• An isoquant identifies all input combinations (bundles) that efficiently produce a given level of output– Note the close similarity to indifference curves– Can think of isoquants as contour lines for the
“hill” created by the production function
7-12
Family of Isoquants
K, U
nits
of c
apita
l per
day
e
b
a
d
fc
63210 L, Workers per day
6
3
2
1
q = 14
q = 24
q = 35
The production function above yields the isoquants on the left.
Properties of Isoquants
• Isoquants are thin• Do not slope upward• Two isoquants do not cross• Higher-output isoquants lie farther from the
origin
7-15
Figure 7.10: Properties of Isoquants
7-16
Figure 7.10: Properties of Isoquants
7-17
Substitution Between Inputs• Rate that one input can be substituted for another is an
important factor for managers in choosing best mix of inputs• Shape of isoquant captures information about input
substitution– Points on an isoquant have same output but different input mix– Rate of substitution for labor with capital is equal to negative the
slope
7-18
Marginal Rate of Technical Substitution
• Marginal Rate of Technical Substitution for labor with capital (MRTSLK): the amount of capital needed to replace labor while keeping output unchanged, per unit of replaced labor– Let K be the amount of capital that can replace L units
of labor in a way such that total output ― Q = F(L,K) ― is unchanged.
– Then, MRTSLK = - K / L, and– K / L is the slope of the isoquant at the pre-change
inputs bundle. – Therefore, MRTSLK = - slope of the isoquant
Marginal Rate of Technical Substitution
• marginal rate of technical substitution (MRTS) - the number of extra units of one input needed to replace one unit of another input that enables a firm to keep the amount of output it produces constant
L
KMRTS
laborin increase
capitalin increase
Slope of Isoquant!
How the Marginal Rate of Technical Substitution Varies Along an Isoquant
K,
Uni
ts o
f ca
pita
l per
day
L, Workers per day
45
7
10
16a
b
cd
e
q = 10
K = –6
L = 1
0 1
1
1
1
2 3
–3
–2
–1
4 5 6 7 8 9 10
MRTS in a Printing and Pu blishing U.S. Firm
Substitutability of Inputs and Marginal Products.
• Along an isoquant output doesn’t change (q = 0), or:
(MPL x ΔL) + (MPK x ΔK) = 0.
– or
Increase in q per extra unit of labor
Extra units of labor
Increase in q per extra unit of capital
Extra units of capital
isoquant of slope
0
MRTSMP
MP
L
K
LMP
MPK
LMPKMP
K
L
K
L
LK
Figure 7.12: MRTS
7-23
MRTS and Marginal Product
• Recall the relationship between MRS and marginal utility
• Parallel relationship exists between MRTS and marginal product
• The more productive labor is relative to capital, the more capital we must add to make up for any reduction in labor; the larger the MRTS
K
LLK MP
MPMRTS
7-24
Figure 7.13: Declining MRTS
• We often assume that MRTSLK decreases as we increase L and decrease K
• Why is this a reasonable assumption?
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Extreme Production Technologies
• Two inputs are perfect substitutes if their functions are identical– Firm is able to exchange one for another at a fixed rate– Each isoquant is a straight line, constant MRTS
• Two inputs are perfect complements when– They must be used in fixed proportions– Isoquants are L-shaped
7-26
Substitutability of Inputs
Substitutability of Inputs
Returns to Scale
Types of Returns to ScaleProportional change in
ALL inputs yields…What happens when all
inputs are doubled?
ConstantSame proportional change in
outputOutput doubles
IncreasingGreater than proportional
change in outputOutput more than
doubles
DecreasingLess than proportional
change in outputOutput less than doubles
7-29
Figure 7.17: Returns to Scale
7-30
Returns to Scale
• Constant returns to scale (CRS) - property of a production function whereby when all inputs are increased by a certain percentage, output increases by that same percentage.
f(2L, 2K) = 2f(L, K).
Returns to Scale (cont).
• Increasing returns to scale (IRS) - property of a production function whereby output rises more than in proportion to an equal increase in all inputs
f(2L, 2K) > 2f(L, K).
Returns to Scale (cont).
• Decreasing returns to scale (DRS) - property of a production function whereby output increases less than in proportion to an equal percentage increase in all inputs
f(2L, 2K) < 2f(L, K).
Productivity Differences and Technological Change
• A firm is more productive or has higher productivity when it can produce more output use the same amount of inputs– Its production function shifts upward at each
combination of inputs– May be either general change in productivity of
specifically linked to use of one input• Productivity improvement that leaves MRTS
unchanged is factor-neutral
7-34
The Cobb-Douglas Production Function
• It one is the most popular estimated functions.
q = ALK
Cobb-Douglas Production Function
• A shows firm’s general productivity level• and affect relative productivities of labor and
capital
• Substitution between inputs:
1
1
KALMP
KALMP
K
L
KALKLFQ ,
L
KMRTSLK
7-36
Figure: 7.16: Cobb-Douglas Production Function
7-37