**Unformatted text preview: **Intermediate Microeconomics
Production Instructor: Bin Xie
Spring 2015 Prot Maximization
A rm is an organization that converts inputs (labor,
materials, and capital) into outputs.
We only study for-prot rms and assume these rms' owners
are driven to maximize prot.
Prot is the dierence between revenue (R ), what it earns
from selling its product, and cost (C), what it pays for labor,
materials, and other inputs.
π = R − C . where R = pq . To maximize prots, a rm must produce as eciently as
possible, where ecient production means it cannot produce
its current level of output with fewer inputs. Production
The various ways that a rm can transform inputs into the
maximum amount of output are summarized in the production
function.
Assuming labor (L) and capital (K ) are the only inputs, the
production function is q = f (L, K ). A rm can more easily adjust its inputs in the long run than in
the short run.
The short run is a period of time so brief that at least one
factor of production cannot be varied (the xed input).
The long run is a long enough period of time that all inputs
can be varied (All inputs are variable inputs). Short-Run Production
In the short run (SR), we assume that capital is a xed input
and labor is a variable input.
¯
SR Production Function: q = fSR (L, K ). One variable input
and one xed input.
q is output, but also called total product; the short run
production function is also called the total product of labor.
The marginal product of labor is the additional output
produced by an additional unit of labor, holding all other
∂f (L, K )
∂q
=
.
factors constant: MPL =
∂L
∂L
The average product of labor is the ratio of output to the
q
amount of labor employed: APL = .
L Short-Run Production (Cont.) Output (quantity produced) q increases when the MPL > 0.
APL and MPL both rst rise and then fall as L increases.
Initial increases due to specialization of activities; more
workers are a good thing.
Eventual declines result when workers begin to get in each
other's way as they struggle with having a xed capital stock.
MPL curve rst pulls APL curve up and then pulls it down,
thus, MPL intersects APL at its maximum. Law of Diminishing Marginal Returns The law holds that, if a rm keeps increasing an input, holding
all other inputs and technology constant, the corresponding
increases in output will eventually becomes smaller.
∂MPL
∂(∂q/∂L)
∂2q
∂ 2 f (L, K )
=
=
=
< 0.
2
∂L
∂L
∂L
∂L2 Note that when MPL begins to fall, TP is still increasing.
Law of Diminishing Marginal Returns (LDMR) is really an
empirical regularity more than a law. Long-Run Production: Two Variable Inputs In the long run (LR ), we assume that both labor and capital
are variable inputs.
The freedom to vary both inputs provides rms with many
choices of how to produce (labor-intensive vs. capital-intensive
methods).
Consider a Cobb-Douglas production function where A, a, and
b are constants: q = ALa K b . Long-Run Production Isoquant A production isoquant graphically summarizes the ecient
combinations of inputs (labor and capital) that will produce a
specic level of output. (Analogous to indierence curve)
There could be innite numbers of production isoquants. Long-Run Production Isoquant (Cont.) Properties of isoquant (Still analogous to Indierence curve):
The farther an isoquant is from the origin, the greater the level
of output.
Isoquants do not cross.
Isoquants slope downward.
Isoquants must be thin. The shape of isoquants (curvature) indicates how readily a
rm can substitute between inputs in the production process. Long-Run Production Isoquant (Cont.) Types of isoquants:
Perfect substitutes. e.g. q = X + Y .
Fixed-proportions. e.g. q = min(X , Y ).
Convex. e.g. Cobb-Douglasq = X 0.5 Y 0.5 . Substituting Inputs
The slope of an isoquant shows the ability of a rm to replace
one input with another (holding output constant).
Marginal Rate of Technical Substitution (MRTS) is the
slope of an isoquant at a single point.
∆K
dK
change in capital
=
=
.
MRTS =
change in labor ∆L
dL
MRTS tells us how many units of K the rm can replace with
an extra unit of L (q constant).
dq
¯
∂f
∂L ∂K
dK
=0=
+
= MPL + MPK
. MPL : marginal
dL
∂L ∂K ∂L
dL
product of labor; MPK : marginal product of capital.
dK
MPL
MRTS =
=−
.
dL
MPK Substituting Inputs (Cont.) MRTS diminishes along a convex isoquant. The more L the rm has, the harder it is to replace K with L. The Elasticity of Substitution
Elasticity of Substitution measures the ease with which a rm can
substitute capital for labor.
d(K /L)
d(K /L) MRTS
K /L
.
σ=
=
dMRTS
dMRTS K /L
MRTS CES (Constant elasticity of substitution) production function:
d
q = (aLρ + bK ρ ) ρ .
1 e.g. q = (Lρ + K ρ ) ρ .
L
MRTS = −( )ρ−1 , Constant elasticity: σ = 1
.
1−ρ
L
(Exercise: How to get the elasticity? Hint: Treat as a
K whole. e.g. replace it using a = L
)
K K Returns to Scale How much does output change if a rm increases all its inputs
proportionately?
Production function exhibits constant returns to scale when
a percentage increase in inputs is followed by the same
percentage increase in output. Doubling inputs, doubles
output f (2L, 2K ) = 2f (L, K ).
More generally, a production function is homogeneous of
degree γ if f (xL, xK ) = x γ f (L, K ) where x is a positive
constant. Returns to Scale (Cont.)
Production function exhibits increasing returns to scale
when a percentage increase in inputs is followed by a larger
percentage increase in output.
f (2L, 2K ) > 2f (L, K )
Occurs with greater specialization of L and K; one large plant
more productive than two small plants. Production function exhibits decreasing returns to scale
when a percentage increase in inputs is followed by a smaller
percentage increase in output.
f (2L, 2K ) < 2f (L, K )
Occurs because of diculty organizing and coordinating
activities as rm size increases. Returns to Scale (Cont.) Determine the following production functions: constant return
to scale? increasing return to scale? decreasing return to
scale?
q
q
q
q = 3L0.3 K 0.7
= 2LK
= 15L + √
K
√
= L + K (homogeneous of degree? ) Productivity and Technical Change Even if all rms are producing eciently (an assumption we
make in this chapter), rms may not be equally productive.
e.g. Cobb-Douglas function: for dierent rms, q = ALa K 1−a ,
A could have dierent values. An advance in rm knowledge that allows more output to be
produced with the same level of inputs is called technical
progress.
For a single rm with Cobb-Douglas production function,
q = ALa K 1−a , A could increase after a breakthrough in
technology. ...

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