Doyoung Park
ECON 6213: Microeconomic Theory—Production and Supply
Lecture Note 5
Abstract
Page
1 Production Functions
1.1 Marginal Productivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Diminishing Marginal Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Average Productivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2
3
3
2 Isoquant Maps and the Rate of Technical Substitution
2.1 The Marginal Rate of Technical Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Relating MRTS to Marginal Productivities . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Diminishing MRTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
5
6
6
3 Returns to Scale
3.1 Constant Returns to Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Homothetic Production Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 The n-input Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
8
9
10
4 The Elasticity of Substitution
10
5 Four Commonly Used Production Functions
5.1 Linear (Perfect Substitutes, σv = ∞) . . . . . . . . . . . . . . .
5.2 Fixed Proportions (Leontief/Perfect Complements, σv = 0) . . .
5.3 Cobb-Douglas (σv = 1) . . . . . . . . . . . . . . . . . . . . . . .
5.4 Constant Elasticity of Substitution (CES) Production Function
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6 Technical Progress
16
6.1 Measuring technical progress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
6.2 Growth accounting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Page 1 of 19
Doyoung Park
ECON 6213: Microeconomic Theory—Production and Supply
Lecture Note 5
Chapter 9 Production Functions
This chapter explores production functions, the mathematical relationships that describe how firms transform
inputs such as labor and capital into output such as goods and services.
1
Production Functions
The production function shows the maximum amount of possible output that can be attained by the firm
for a given level of inputs. Production can rely on many different types of inputs, so the general form of the
production function is
q = f (k, l, m, . . .)
where q is the level of output by the firm, k is physical capital (machines) used by the firm per period, l is
the amount of labor used, and m is materials, with other inputs (such as land or human capital/knowledge)
potentially entering the function.
A simplified production function with just two inputs, capital and labor, takes the general form given by
q = f (k, l)
Recognizing that, in reality, firms use many inputs in their production processes, this two-input production
function is sufficient for our later characterization of the firm’s problem of optimally choosing combinations
of inputs.
1.1
Marginal Productivity
The marginal physical product of an input (or just marginal product, denoted with M P ) is the amount of
additional output that is obtained from using one more unit of that input (e.g., the additional output a firm
achieves from hiring an additional worker), holding all other inputs constant. Mathematically, the marginal
products of labor and capital are given by
∂q
= fk
∂k
∂q
Marginal product of labor = M Pl =
= fl .
∂l
Marginal product of capital = M Pk =
Note that the value of the marginal product of a particular input depends on how much of that particular
input is being used (for instance, the additional output for a firm from hiring an additional worker might be
different for the 10th worker versus the 100th worker).
Page 2 of 19
Doyoung Park
1.2
ECON 6213: Microeconomic Theory—Production and Supply
Lecture Note 5
Diminishing Marginal Product
It is generally the case that marginal product begins to decline after a certain point, which is the notion of
diminishing marginal product—adding more of an input adds a decreasing amount of output for the firm,
for high enough levels of that input. Mathematically, this is equivalent to saying that
∂M Pk
∂2f
= fkk < 0, for high enough k
=
∂k
∂k 2
∂M Pl
∂2f
= 2 = fll < 0, for high enough l.
∂l
∂l
It is also true that the marginal product of one input depends on how much of the other input is being
employed—for instance, the output added from a firm hiring a 100th worker depends on the size of its
factory or how much capital it has, or alternatively, the additional output from adding a new machine
depends on how many workers are currently employed. These relations are given by
∂M Pl
= flk
∂k
1.3
and
∂M Pk
= fkl .
∂l
Average Productivity
A related concept is the notion of average physical product (or average product for short, denoted by AP ) of
an input. The average product describes how much output is achieved per unit of an input that is employed.
For labor, the average product is given by
APl =
output
q
f (k, l)
= =
.
labor input
l
l
In words, this relation describes the average amount of output per worker, which is often otherwise referred
to as “labor productivity.”
Example. Suppose the production function for flyswatters can be represented by
q = f (k, l) = 600k 2 l2 − k 3 l3 .
To calculate the marginal and average product of labor (M Pl and APl ), we pick k = 10 so that we can find
the exact values of M Pl and APl . The production function then becomes
q = f (k, l) = 6000l2 − 1000l3 .
The marginal product of labor when k = 10 is then given by
M Pl =
∂q
= 12000l − 3000l2 .
∂l
Page 3 of 19
Doyoung Park
ECON 6213: Microeconomic Theory—Production and Supply
Lecture Note 5
Note that the marginal product of labor is positive (M Pl > 0) when
12000l − 3000l2 > 0,
or when l < 40 (each of the first 40 workers add additional output for the firm. On the other hand M Pl
becomes negative, meaning additional workers actually decrease the total level of output, when l > 40. The
level of output thus reaches a maximum at l = 40, which is given by q = 32 million.
Calculating the average product of labor (APl ) simply involves dividing the production function by l:
APl =
6000l2 − 1000l3
q
=
= 6000l − 1000l2 ,
l
l
which gives average amount of output per worker for any given level of l.
APl reaches a maximum when
∂APl
= 60000 − 2000l = 0,
∂l
or when l = 30. One thing to observe is that for l = 30, both APl and M Pl are equal to 900,000. It will
always be the case that average and marginal productivities of an input are equal to each other when the
average productivity of that input reaches a maximum.
2
Isoquant Maps and the Rate of Technical Substitution
Since we’re focusing on production functions of two inputs, k and l, it is possible to describe how a firm
can substitute one input for another to achieve a particular level of output. For instance, if a firm wants to
produce q = 10 units of output, it might be able to achieve that output using a relatively high amount of
labor and little capital, a relatively high amount of capital and little labor, or some other combination of
the two inputs.
An isoquant describes all combinations of k and l that together produce a given level of output, q0 . In
general, isoquants are described by the relation
f (k, l) = q0 .
That is, the isoquant for q0 is characterized by all of the combinations of k and l that allow a firm to produce
q0 . This concept is analogous to the indifference curves that described the consumer problem, in which we
described all of the combinations of goods x and y that gave a consumer some constant level of utility.
Page 4 of 19
Doyoung Park
ECON 6213: Microeconomic Theory—Production and Supply
Lecture Note 5
Isoquant Map
Capital (k)
kA
A
kB
q = 30
q = 20
B
q = 10
lA
lB
Labor (l)
The isoquant map above shows the isoquants for levels of output of q = 10, q = 20, and q = 30. Since
points A and B fall on the same isoquant, they represent different combinations of k and l that lead to
output of q = 10.
2.1
The Marginal Rate of Technical Substitution
If a firm changes its mix of inputs (for instance, uses less capital and more labor) while holding output
constant, this represents a movement along the isoquant. When the firm changes its input usage from
(lA , kA ) to (lB , kB ), it is substituting l for k, holding output fixed, and the slope of the isoquant along this
part of the curve represents the rate at which the firm substitutes l for k. The negative of this slope is the
marginal rate of technical substitution (M RT Sl,k ), and is equal to
M RT Sl,k (l for k) =
−dk
dl
.
q=q0
Analogously to the marginal rate of substitution for consumers (the rate at which a consumer would be
willing to give up good y to acquire more x, holding utility constant), the M RT Sl,k gives the rate at which
a firm substitutes labor for capital, holding output fixed.
Page 5 of 19
Doyoung Park
2.2
ECON 6213: Microeconomic Theory—Production and Supply
Lecture Note 5
Relating MRTS to Marginal Productivities
To relate the M RT Sl,k back to the notion of marginal product from earlier, consider the total derivative of
the production function q, when moving along an isoquant where q is fixed at q = q0 . Note that when q is
fixed, then k will be a function of l, since a given amount of labor employed will determine how much capital
the firm must use to produce q0 , so k = k (l). The total differential of q0 = f (k (l) , l), with q0 constant, is
then
dq0 = 0 = fk
dk
dk
+ fl = M P k
+ M Pl .
dl
dl
Manipulating,
M Pl · dl
= −M Pk · dk,
so the M RT Sl,k can be expressed as
M RT Sl,k (l for k) =
−dk
dl
=
q=q0
M Pl
fl
= .
M Pk
fk
That is, the rate at which the firm substitutes l for k is equal to the ratio of the respective inputs’ marginal
products. Intuitively, this relation shows that when labor has a high marginal product relative to that of
capital, the firm will be willing to give up a large amount of capital to add another worker, holding output
fixed. Alternatively, if the marginal product of labor is relatively low relative to that of capital, the firm will
be willing to give up only a small amount of capital to add more labor.
2.3
Diminishing MRTS
M RT Sl,k will always be positive (or zero) as long as M Pl and M Pk are nonnegative. To show that the
isoquants are convex, i.e. that M RT Sl,k diminishes as l increases, we can show that d (M RT Sl,k ) /dl < 0.
Since M RT Sl,k = fl /fk , and noting from earlier that k = k (l) along an isoquant—and thus fl = fl (k (l) , l)
and fk = fk (k (l) , l),
d fl fk−1
d (fl /fk )
dM RT Sl,k
=
=
,
dl
dl
dl
dk
dk
−1
−2
= fk
fll + flk
− fl fk
fkl + fkk
.
dl
dl
Page 6 of 19
Doyoung Park
ECON 6213: Microeconomic Theory—Production and Supply
Lecture Note 5
Further, noting that dk/dl = −fl /fk along an isoquant,
dM RT Sl,k
= fk−1 fll − fl fk−1 flk − fl fk−2 fkl − fl fk−1 fkk ,
dl
fl fkl
f 2 fkk
fl flk
fll
− 2 − 2 + l 3 ,
=
fk
fk
fk
fk
=
fk2 f11 − fl fk flk − fl fk fkl + fl2 fkk
.
fk3
And finally, since by Young’s theorem, flk = fkl , further simplification yields
fk2 fll − 2fk fl fkl + fl2 fkk
dM RT Sl,k
.
=
dl
fk3
The denominator of this expression is positive since fk > 0 by assumption. And since fll and fkk are assumed
to be negative (i.e, diminishing marginal returns of inputs), then the numerator will be negative if fkl = flk
is positive, which it typically will be (since the marginal product of an additional worker typically increases
the more capital is available). Some functions, however, might have fkl < 0 over some ranges.
Example. Suppose we have the production function q = f (k, l) = 600k 2 l2 −k 3 l3 , as in the previous example.
The marginal products of each input are given by
M Pl = fl = 1200k 2 l − 3k 3 l2
M Pk = fk = 1200kl2 − 3k 2 l3 .
The marginal product of each input will be positive as long as kl < 400.
To show what happens to these marginal productivities as more and more of an input is used, we can
take the second derivatives:
dM Pl
= fll = 1200k 2 − 6k 3 l
dl
dM Pk
= fkk = 1200l2 − 6kl3 ,
dk
which implies that marginal productivities begin to diminish (fll and fkk become negative) if kl > 200.
Finally, cross differentiation shows how the marginal product of one input depends on the amount of the
other input:
dM Pl
dM Pk
=
= flk = fkl = 2400kl − 9k 2 l2 ,
dk
dl
which is positive for kl < 266.
Page 7 of 19
Doyoung Park
3
ECON 6213: Microeconomic Theory—Production and Supply
Lecture Note 5
Returns to Scale
We can also use the production function to think about what happens to output when all inputs are scaled up
proportionally. For instance, when both labor and capital are doubled in the two input production function,
what is the resulting effect on output—does it double, more than double, or less than double?
In general, suppose we scale up all inputs proportionally by some factor t > 1 (if we were doubling the
amount of each input, this would be t = 2, but we want to think about this generally). Returns to scale are
defined as follows:
Returns to scale
Constant
Decreasing
Increasing
Effect on output
f (tk, tl) = tf (k, l) = tq
f (tk, tl) < tf (k, l) = tq
f (tk, tl) > tf (k, l) = tq
Intuition (for t = 2)
Inputs double, output doubles
Inputs double, output less than doubles
Inputs double, output more than doubles
In reality, firms’ production functions might exhibit constant, decreasing, or increasing returns to scale
depending on the scale of production and the levels of input usage.
Example. Suppose we have the production function q = f (k, l) = 20k 1/2 l1/4 , and we want to determine
whether this function exhibits constant, decreasing, or increasing returns to scale. To do this, we can scale
up the arguments in the function, k and l, by the same factor t:
1/2
f (tk, tl) = 20 (tk)
1/4
(tl)
.
manipulating the function so isolate the t terms,
f (tk, tl) = t3/4 20k 1/2 l1/4 = t3/4 f (k, l) < tf (k, l) .
Since t3/4 < t, this production function exhibits decreasing returns to scale—increasing the amount of each
input by the factor t led to output increasing by a factor less than t.
3.1
Constant Returns to Scale
Constant returns-to-scale production functions are homogeneous of degree one in inputs:
f (tk, tl) = t1 f (k, l) = tq.
And, since if a function is homogeneous of degree k, its derivatives are homogeneous of degree k − 1, then
the marginal product functions are homogeneous of degree zero—scaling up all inputs proportionally has no
Page 8 of 19
Doyoung Park
ECON 6213: Microeconomic Theory—Production and Supply
Lecture Note 5
effect on the marginal product of any given input, meaning that
∂f (k, l)
∂f (tk, tl)
=
∂k
∂k
∂f (k, l)
∂f (tk, tl)
M Pl =
=
, for any t > 0
∂l
∂l
M Pk =
If we let t = 1/l, then these marginal product functions become
∂f (k/l, 1)
∂f (k/l)
=
∂k
∂k
∂f (k/l, 1)
∂f (k/l)
M Pl =
=
.
∂l
∂l
M Pk =
This means that for a constant returns-to-scale production function, the marginal product of any input
depends solely on the ratio of the two inputs, not on their absolute levels. And as the figure shows, for a
given ratio k/l, the RT S will be the same for all levels of production (i.e. the isoquants will have the same
slope for a given k/l no matter the level of output).
Isoquant Map for a Constant Returns-to-Scale Production Function
Capital (k)
q=3
q=2
q=1
Labor (l)
3.2
Homothetic Production Functions
The result that for certain production functions, the RT S depends only on the ratio of capital to labor, but
not on the absolute levels of each input or the level of output describes homothetic production functions, for
which the isoquants are radial expansions of each other.
Page 9 of 19
Doyoung Park
ECON 6213: Microeconomic Theory—Production and Supply
Lecture Note 5
A production function need not exhibit constant returns to scale to be homothetic. Suppose f (k, l) is a
γ
constant returns-to-scale production function, and F (k, l) = [f (k, l)] , where γ > 0. If γ > 1, then
γ
γ
γ
F (tk, tl) = [f (tk, tl)] = [tf (k, l)] = tγ [f (k, l)] = tγ F (k, l) , for any t > 0.
This function F (k, l) thus exhibits increasing returns to scale (and we could do a similar exercise with γ < 1
to generate a function with decreasing returns to scale). This function F (k, l) remains homothetic, which
indicates that returns to scale do not necessarily determine the shape taken by the isoquants.
3.3
The n-input Case
Returns to scale can be generalized to production functions with more than two inputs. If q = f (x1 , x2 , . . . , xn ),
then if we scale up all n inputs by the factor t,
f (tx1 , tx2 , . . . , txn ) = tk f (x1 , x2 , . . . , xn ) = tk q.
The returns to scale can be determined from the value of k:
• If k = 1, constant returns to scale
• If k < 1, decreasing returns to scale
• If k > 1, increasing returns to scale
4
The Elasticity of Substitution
One important characteristic of production, and part of what determines the shape of a production function’s
isoquants, is the degree to which one input can be substituted for another (how well labor can take the place
of capital, or vice versa). For the production function the elasticity of substitution, denoted by σ, measures
how the ratio of capital to labor changes relative to changes in the RT S while moving along an isoquant. σ
(which is always nonnegative, since k/l and the RT S move in the same direction is defined as
σ=
%∆ (k/l)
d (k/l) RT S
d ln (k/l)
d ln (k/l)
=
×
=
=
.
%∆RT S
dRT S
k/l
d ln RT S
d ln (fl /fk )
As shown in the figure, as the firm moves from point A to point B—giving up capital to add labor, holding
output constant—both the RT S and the ratio k/l change. σ is the ratio of these proportional changes and
measures how curved the isoquant is.
Page 10 of 19
Doyoung Park
ECON 6213: Microeconomic Theory—Production and Supply
Lecture Note 5
Graphical Depiction of the Elasticity of Substitution
Capital (k)
A
RT SA
RT SB
B
(k/l)A
q0
(k/l)B
Labor (l)
The elasticity σ can range between 0 and ∞, and can also change while moving along an isoquant
(meaning that k and l become more or less substitutable for one another as the input mix changes). In
general,
• If σ is high, then the RT S does not change much relative to k/l, meaning that the isoquant is relatively
flat. This means that k and l are relatively easily substitutable for each other.
• If σ is low, then the RT S changes substantially as k/l changes, and the isoquant will be noticeably
curved. This means that k and l do not easily substitute for one another.
5
Four Commonly Used Production Functions
While there are infinitely many production functions that obey the assumptions laid out so far, there are
several production functions that are commonly used because they have desirable properties, or because they
describe unique production processes.
5.1
Linear (Perfect Substitutes, σv = ∞)
The linear production function takes the form
q = f (k, l) = αk + βl.
Page 11 of 19
Doyoung Park
ECON 6213: Microeconomic Theory—Production and Supply
Lecture Note 5
This production function describes production processes for which the two inputs are perfectly substitutable
for each other according to some fixed ratio (e.g., 2 worker can always do the job of 1 machine, and vice
versa).
Linear production functions exhibit constant returns to scale, and its isoquants are straight lines with
slope −β/α, since
RT S =
M Pl
β
= .
M Pk
α
For linear production functions, σ = ∞, since as k/l changes moving along an isoquant, %∆RT S = 0.
Linear Production Function
Capital (k)
Slope = −β/α
q1
q2
q3
Labor (l)
5.2
Fixed Proportions (Leontief/Perfect Complements, σv = 0)
The fixed proportions production function describes production processes for which each input must be
employed in fixed proportion, and adding more of a single input beyond the fixed ratio adds no additional
output (e.g. 2 workers for every machine, and adding additional workers without adding more machines has
no effect on output). This production function takes the general form
q = min (αk, βl) with α, β > 0.
Capital and labor must always be used in a fixed ratio, meaning that the firm will always operate along a
ray where k/l is constant. The firm will always employ labor and capital such that αk = βl (since if they
Page 12 of 19
Doyoung Park
ECON 6213: Microeconomic Theory—Production and Supply
Lecture Note 5
were not equal the firm is unnecessarily using too much of one of the inputs—e.g. hiring a third worker when
it only has one machine), meaning that the firm will always choose capital and labor such that k/l = β/α.
Fixed Proportions Production Function
Capital (k)
q3
q2
q2 /α
q1
q2 /β
Labor (l)
And, because the capital-labor ratio does not change, σ = 0, meaning it is impossible for a firm to
substitute labor for capital moving along an isoquant.
5.3
Cobb-Douglas (σv = 1)
As on the consumer side for utility functions, the Cobb-Douglas production function takes the form
q = f (k, l) = Ak α lβ , with A, α, β > 0.
The Cobb-Douglas production function can exhibit any kind of returns to scale, since
α
β
f (tk, tl) = A (tk) (tl) = Atα+β k α lβ = tα+β f (k, l) .
• If α + β = 1, this implies constant returns to scale.
• If α + β > 1, this implies increasing returns to scale.
• If α + β < 1, this implies decreasing returns to scale.
Page 13 of 19
Doyoung Park
ECON 6213: Microeconomic Theory—Production and Supply
Lecture Note 5
The Cobb-Douglas production function can also be easily log-linearized:
ln q = ln A + α ln k + β ln l,
which means that α is the elasticity of output with respect to k and β is the elasticity of output with respect
to l.
Cobb-Douglas Production Function
Capital (k)
q=3
q=2
q=1
Labor (l)
The fact that σ = 1 for a Cobb-Douglas production function implies that any given change in the RT S,
no matter where a firm is operating on an isoquant, will cause an equally sized change in the ratio k/l.
5.4
Constant Elasticity of Substitution (CES) Production Function
Constant elasticity of substitution production functions take the form
h 1
iγ(1−σ)
1
q = f (k, l) = k 1−σ + l 1−σ
, with γ > 0.
γ indicates the returns to scale of the function, with γ = 1 corresponding to constant returns to scale, γ > 1
corresponding to increasing returns to scale, and γ < 1 corresponding to decreasing returns to scale.
For convenience, the function is often rewritten as
f (k, l) = [k ρ + lρ ]
γ/ρ
, with ρ ≤ 1 and ρ 6= 0,
Page 14 of 19
Doyoung Park
ECON 6213: Microeconomic Theory—Production and Supply
Lecture Note 5
where ρ = (σ − 1) /σ (which implies that σ = 1/ (1 − ρ)). The CES production function can be shown to
converge to either of the three preceding production functions; it thus describes a wide range of different
production processes.
• As σ → ∞, the CES function becomes the linear (perfect substitutes) production function.
• As σ → 0, the CES function becomes the fixed proportions (perfect complements) production function.
• As σ → 1, the CES function becomes the Cobb-Douglas production function.
1/2
Example. Consider the production function q = f (k, l) = k + l + 2 (kl)
.
To find this function’s returns to scale,
1/2
f (tk, tl) = tk + tl + 2 (tk · tl)
1/2
= tk + tl + 2 t2 kl
h
i
1/2
= t k + l + 2 (kl)
= tf (k, l) .
This function thus exhibits constant returns to scale. The function’s marginal products of capital and labor
are given by
−1/2
M Pk = fk = 1 + (k/l)
M Pl = fl = 1 + (k/l)
1/2
.
The RT S is, then is equal to
1/2
RT S =
fl
1 + (k/l)
=
.
−1/2
fk
1 + (k/l)
It can also be shown that this function takes the CES form with ρ = 0.5 and γ = 1, since
1/2
f (k, l) = k + l + 2 (kl)
2
= k 1/2 + l1/2 ,
And the elasticity of substitution for this production function is then
σ=
1
1
=
= 2.
1−ρ
1/2
Page 15 of 19
Doyoung Park
6
ECON 6213: Microeconomic Theory—Production and Supply
Lecture Note 5
Technical Progress
Methods of production improve over time, and it is important to be able to capture these improvements
with the production function concept.
Figure 1: Technical Progress
A simplified view of such progress is provided by the above figure. Initially, isoquant q0 records those
combinations of capital and labor that can be used to produce an output level of q0 . Following the development of superior production techniques, this isoquant shifts to q00 . Now the same level of output can be
produced with fewer inputs.
One way to measure this improvement is by noting that with a level of capital input of, say, k1 , it
previously took l2 units of labor to produce q0 , whereas now it takes only l1 . Output per worker has risen
from q0 /l2 to q0 /l1 . But one must be careful in this type of calculation. An increase in capital input
to k2 would also have permitted a reduction in labor input to l1 along the original q0 isoquant. In this
case, output per worker would also rise, although there would have been no true technical progress. Use of
the production function concept can help to differentiate between these two concepts and therefore allow
economists to obtain an accurate estimate of the rate of technical change.
6.1
Measuring technical progress
Suppose that we let
q = A (t) f (k, l)
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Doyoung Park
ECON 6213: Microeconomic Theory—Production and Supply
Lecture Note 5
be the production function for some good (or perhaps for society’s output as a whole). The term A (t)
in the function represents all the influences that go into determining q other than k (machine-hours) and l
(labor-hours). Changes in A over time represent technical progress. For this reason, A is shown as a function
of time.
6.2
Growth accounting
Differentiating equation
q = A (t) f (k, l)
with respect to time, t, gives
dq
dA
df (k, l)
=
· f (k, l) + A (t) ·
,
dt
dt
dt
q
q
∂f dk ∂f dl
dA (t)
·
+
·
+
·
.
=
dt
A (t) f (k, l) ∂k dt
∂l dt
Dividing by qgives
dq/dt
dA/dt ∂f /∂k dk
∂f /∂l dl
=
+
·
+
· ,
q
A
f (k, l) dt
f (k, l) dt
k
dk/dt ∂f
l
dl/dt
dA/dt ∂f
+
·
·
+
·
·
.
=
A
∂k f (k, l)
k
∂l f (k, l)
l
Now, for any variable x,
dx/dt
dx/x
=
,
x
dt
which means the proportional changes (growth) in x per unit of time. We denote this by Gx . Hence, the
equation can be rewritten in terms of growth rates as
Gq = GA +
∂f
k
∂f
l
·
· Gk +
·
· Gl .
∂k f (k, l)
∂l f (k, l)
Note that
∂f
k
·
= ef,k ,
∂k f (k, l)
∂f
l
·
= ef,l ,
∂l f (k, l)
which both represent the input elasticity of output. Therefore, the growth equation becomes
Gq = GA + ef,k Gk + ef,l Gl ,
= GA + eq,k Gk + eq,l Gl since f = q.
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Doyoung Park
ECON 6213: Microeconomic Theory—Production and Supply
Lecture Note 5
This shows that the rate of growth in output can be broken down into the sum of two components: (a)
growth attributed to changes in inputs (k and l) and other “residual” growth—that is, changes in A—-that
represents technical progress.
R.M. Solow studied the U.S. economy between years 1909 and 1949 and recorded the following values for
the terms in the equation:
Gq = 2.75%/year,
Gl = 1%/year,
Gk = 1.75%/year,
eq,l = 0.65,
eq,k = 0.35.
Consequently,
GA = Gq − eq,k Gk − eq,l Gl ,
= 2.75 − (0.35)(1.75) − (0.65)(1),
= 1.5%.
The conclusion Solow reached was that technology advanced at a rate of 1.5% per year from 1909 to 1949.
Example. Technical Progress in the Cobb-Douglas Production Function
The Cobb-Douglas production function provides an especially easy avenue for illustrating technical
progress. Assuming constant returns to scale, such a production function with technical progress might
be represented by
q = A (t) f (k, l) = A (t) k α l1−α .
If we also assume the technical progress occurs at a constant exponential (θ), then we can write A (t) =
Aeθt and the production function becomes
q = Aeθt k α l1−α .
A particularly easy way to study the properties of this type of function over time is to use “logarithmic
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Doyoung Park
ECON 6213: Microeconomic Theory—Production and Supply
Lecture Note 5
differentiation”:
∂lnq
∂lnq ∂q
=
·
,
∂t
∂q ∂t
∂q/∂t
=
,
q
= Gq ,
∂ [lnA + θt + αlnk + (1 − α) lnl]
,
∂t
∂lnk
∂lnl
=θ+α
+ (1 − α)
,
∂t
∂t
∂lnk ∂k
∂lnl ∂l
=θ+α
·
+ (1 − α)
· ,
∂k
∂t
∂l ∂t
∂k/∂t
∂l/∂t
=θ+α
+ (1 − α) ·
,
k
l
=
Gq = θ + αGk + (1 − α) Gl .
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