Transcript MICROECONOMIC THEORY - University of North Texas

Production
Technology
• The physical laws of nature and limits of
material availability and human
understanding that govern what is possible
in converting inputs into output.
Inputs, Factors of Production
• Land (incl. raw materials)
• Labor (including human capital)
• Capital (physical capital, like machinery and
buildings)
Production Function
• A firm’s production function for a particular
good (q) shows the maximum amount of the
good that can be produced using alternative
combinations of capital (K) and labor (L).
q = f(K, L)
• Producing less than the maximum is always
possible and all levels of output below the
maximum are feasible and define the
“production set.”
Production Function
q
q = f(K, L)
K
L
Production set
q = f(K, L)
q
K
All points “under”
the production
function
L
Production Function and Isoquants
q = f(K, L)
q
In the long
run, all
combinations
of inputs are
possible
K
L
Isoquants are
horizontal cross
sections of the
production function
projected on the base
plane.
Short Run, Long Run
• Long Run, quantities of ALL inputs used in production
can be varied.
• Short Run, the quantity of at least one input used in
production is fixed.
• ALL production takes place in a short run environment.
• You can think of the long run as the ability to move from
one short run environment to another.
• Actual time it takes to make this move depends on many
factors, technical, economic and regulatory.
The model
• Standard basic model to think of production
as a function of K and L.
• L variable in the short run while K is fixed.
Short run, hold K fixed.
q = f(K, L)
q
In the short
run, K is fixed
and only L
can vary
K
L
The cross
section of the
production
function at a
fixed K is the
short run
production
function
More, fixed K
q = f(K, L)
q
In the short
run, K is fixed
and only L
can vary
K
L
The cross
section of the
production
function at a
fixed K is the
short run
production
function
Three levels of K
q = f(K=K3, L)
q
In the short
run, we
assume, the
quantity of at
least one
input used -but not all -is fixed.
q = f(K=K2, L)
q = f(K=K1, L)
L
L constant
q = f(K,L=L3)
q
L and K are
just names
for inputs.
Either one
could be
fixed in the
short run.
Just intuitive
that K is fixed
and L
variable in
the SR.
q = f(K,L=L2)
q = f(K,L=L1)
K
SR and then LR
• First we’ll think about the short run, and then
turn to the long run.
Marginal Physical Product
• Marginal Product is the additional output that
can be produced by employing one more unit
of that input
– holding other inputs constant, so a short run
concept
q
marginal product of labor: MPL   fL
L
q
marginal product of capital: MPK   fK
K
Marginal Productivity Assumptions
• We assume managers are not going to allow
employees in the building if they bring total
output down.
q
MPL   fL  0
L
q
MPK   fK  0
K
• However, over the range where profit is
maximized, marginal products are positive.
Increasing and Diminishing Marginal
Product (assumes something is fixed)
• Empirically, economists find that most production
processes exhibit (as L increases from zero):
– Increasing Marginal Returns – each worker added causes
output to increase by more than the previous worker
(workers are not able to gain from specialization, K is
fixed)
– And then…
– Decreasing Marginal Returns –workers added to
production add less to output than the previous worker
(workers crowd each other as they try to share a fixed
amount of capital)
Marginal Productivity Assumptions
• Because of IMR and DMR, these are possible:
MPL 2 f
 2  fLL , ,  0
L
L
MPK 2 f
 2  fKK , ,  0
K
K
• Whether MP is always diminishing or whether it first
increases and then diminishes depends on the
context of the economic discussion.
• In economics classes, we think of increasing marginal
returns and then diminishing marginal returns (need
this for a U-shaped MC curve).
MP Assumptions
• As revenue or profit max means producing where
MC is rising (MPL is falling), theoretically, we tend
to ignore IMR and assume DMR
MPL  2 f
 2  fLL  0
L
L
MPK 2 f
 2  fKK  0
K
K
Malthus and Diminishing Marginal
Productivity
• He argued that population growth meant
declining marginal labor productivity
– His mistake was holding all else (except labor, i.e.
population) constant.
– Ignored technological growth.
– Productivity was actually growing exponentially, but
at such a slow rate that he did not see it.
Per Capita
Output
Watts’s Steam Engine
Essay on the Principle of
Population, 1st ed (1798)
Economic growth of IR
first noticed in the
1830s
Malthus Dies, 1834
1800
1840
1880
Year
Effect of Technology
• If we think of higher technology as being like
having MORE capital, then you can think of the
industrial revolution the result of fLK > 0 and a
rapid expansion of K.
Average Physical Product
• Labor productivity is often measured by
average productivity.
output
q f(K,L)
APL 
= 
labor input L
L
Specific Function
• Suppose the production function for tennis
balls can be represented by
3 3
KL
q  f K,L   12K L 
100
2 2
• To construct MPL and APL, we must assume a
value for K
– let K = 10
• The production function becomes
q  f K,L   1,200L2  10L3
SR Production Function (K = 10)
q  1,200L  10L
2
q
3000000
3
2500000
2000000
q(K=10)
1500000
1000000
500000
0
0
20
40
60
80
100
120
L
Marginal Product
• The marginal product function is
MPL  2,400L  30L
2
MPL  2,400L  30L2  0
30L2  2,400L
30L  2,400
L=80
• When MPL = 0, total product is maximized at
L = 80.
SR Production Function (K = 10)
q
q  1,200L2  10L3
3000000
2500000
2000000
q(K=10)
1500000
1000000
500000
0
0
20
40
60
Slope of function is MPL at that level of L
80
100
120
L
Inflection Point
• Output where MPL goes from increasing to
decreasing (inflection point)
fL  MPL  2,400L  30L2
dMPL
fLL 
 2,400  60L
dL
2,400  60L  0
LI =40
SR Production Function (K = 10)
At inflection point, MPL is at its highest
q
q  1,200L2  10L3
3000000
MPL  1,280,000 at L = 40
2500000
2000000
q(K=10)
1500000
1000000
500000
0
0
20
40
LI
60
80
100
120
L
Average Product
• To find average productivity, we hold K=10
and solve
f K,L 
APL 
 1,200L  10L2
L
dAPL
Maximized where
=0
dL
dAPL
 1,200  20L
dL
1,200  20L  0
20L  1,200
L A  60
SR Production Function (K = 10)
q
Slope of ray from
origin to curve at
any L is = APL
3000000
Slope of this ray =36,000
So APL =36,000 when L= 60
2500000
2000000
q(K=10)
1500000
q  1,200L2  10L3
1000000
APL  1200L  10L2
500000
0
0
20
40
60
LA
80
100
120
L
MPL and APL
• In fact, when L = 60, both APL and MPL are equal to
36,000
• Thus, when APL is at its maximum, APL and MPL are
equal
• So long as a worker hired has a MPL higher than the
overall APL, the APL will continue to rise.
• If the MPL = APL,
• But if a worker hired has a MPL below the overall APL,
the APL will fall.
MPL and APL
60000
40000
20000
0
0
-20000
-40000
-60000
-80000
20
40
60
LI
LA
80
100
120
MPL
APL
MPL and APL
q(K=10)
3000000
Where the
ray is also
tangent,
MPL = APL
2500000
2000000
1500000
q(K=10)
1000000
500000
0
0
60000
50000
40000
30000
20000
10000
0
-10000 0
-20000
20
40
60
80
100
MPL
APL
20
40
60
80
100
Long Run
• All mixes of K and L are possible.
• Daily decisions about production always
have some fixed inputs, so the long run is a
planning time horizon.
Isoquant Map
• Each isoquant represents a different level
of output, q0 = f(K0,L0), q1 = f(K1,L1)
K
dK
0
dL
q1 = 30
q0 = 20
L
Marginal Rate of Technical
Substitution (TRS, RTS, MRTS)
• The slope of an isoquant shows the rate at which
L can be substituted for K, or how much capital
must be hired to replace one Laborer.
dK
TRS  
dL qq0
K
KA
A
B
KB
q0 = 20
LA
LB
L
TRS and Marginal Productivities
• Take the total differential of the production
function:
f
f
dq 
 dL 
 dK
L
K
dq  MPL  dL  MPK  dK
• Along an isoquant dq = 0, so
MPL  dL  MPK  dK
MPL
dK
TRS defined as, TRS= 

dL qq0 MPK
Alternatively:
Implicit Function Rule
Take an implicit function: f(K,L)-q=0
If the conditions of the implict function therorem hold,
then there exists an equation K=g(L; q) and
f
MPL
fL
dK
dK
dK
L


or

or

f
MPK
dL
fK
dL
dL
K
MPL
dK
Again,TRS defined as, TRS= 

dL qq0 MPK
Diminishing TRS
• Again, for demand (this time of inputs) to be
well behaved, we need production technology
(akin to preferences) to be convex.
K
MPL
dK

dL
MPK
Which means, the slope rises,
gets closer to zero as L increases.
And means the TRS falls as L
increases.
dK
TRS  
dL
L
Diminishing TRS
• To show that isoquants are convex (that dK/dL
increases – gets closer to zero) along all
isoquants)
• That is, either:
• The level sets (isoquants) are strictly convex
• The production function is strictly quasi-concave
Convexity (level curves)
• dK/dL increases along all indifference curves
• We can use the explicit equation for an
isoquant, K=K(L, q0) and find
d2K
0
2
dL qq
0
to demonstrate convexity.
• That is, while negative, the slope is getting
larger as L increases (closer to zero).
• But we cannot always get a well defined
equation for an isoquant.
Alternatively (level curves)
• As above, starting with q0 =f(K,L),
fL (K,L)
dK

dL
fK (K,L)
fL (K,L)
TRS 
fK (K,L)
• So convexity if
 fL K,L  
d



2
fK K,L  
dK


 0 or
2
dL
dL
 fK (K,L) 
d

fL (K,L) 
dTRS


0
dL
dL
Convexity (level curves)
• And, that is
 fL (K,L) 
d 

2
2
2
f
(K,L)
2f
f
f

f
f

f
dK
 K
  LK L K K LL L fKK  0

dL2
dL
fK3
 fL (K,L) 
d

2
2
f
(K,L)

2f
f
f

f
f

f
dTRS

LK L K
K LL
L fKK
  K
0
3
dL
dL
fK
*Note that fK3 > 0
•
What of:
–
–
–
–
–
fL > 0, monotonacity
fK > 0, monotonacity
fLL < 0, diminishing marginal returns
fKK < 0, diminishing marginal returns
fLK = ?
Strict Quasi-Convexity
(production function)
• Also, convexity of technology will hold if the
production function is strictly quasi-concave
– A function is strictly quasi-concave if its bordered
Hessian
0
fL
fK
H  fL
fLL
fLK
fK
fKL
fKK
– is negative definite
0
0 fL
H
 0 and H  fL
fL fLL
fK
fL
fK
fLL
fKL
fLK  0
fKK
Negative Definite (production function)
• So the production function is strictly quasi-concave if
– 1. –fLfL < 0
– 2. 2fLfKfLK-fK2fLL -fL2fKK > 0
• Related to the level curve result:
– Remembering that a convex level set comes from this
d2K (2fL fK fLK  fK2 fLL  fL 2fKK )

0
2
3
dL
fK
– We can see that strict convexity of the level set and
strict quasi-concavity of the function are related, and
each is sufficient to demonstrate that both are true.
TRS and Marginal Productivities
• Intuitively, it seems reasonable that fLK should
be positive
– if workers have more capital, they will be more
productive
• But some production functions have fKL < 0
over some input ranges
– assuming diminishing TRS means that MPL and MPK
diminish quickly enough to compensate for any
possible negative cross-productivity effects
TRS and MPL and MPK
• Back to our sample production function:
3 3
K
L
2 2
q  f K,L   12K L 
100
• For this production function
3 2
3K
L
2
MPL  24K L 
100
2 3
3K
L
2
MPK  24KL 
100
IMR and DMR vs. NMR
• Pull out a few terms
3K L
MPL  24K L 
100
3 2
3K
L
2
MPL  0, 24K L 
100
fL  0 if (2,400  3KL)  0
fL  0 if KL  800
2 3
3K
L
2
MPK  24KL 
100
2 3
3K
L
2
MPK  0, 24KL 
100
fK  0 if (2,400  3KL)  0
fK  0 if KL  800
fL  0 if KL  800
fK  0 if KL  800
3 2
2
• If K = 10, then MPL = 0 at L=80
IMR vs. DMR
• Because
3
6K L
fLL  24K 100
3
6K
L
2
fLL  0, 24K 
100
2
3
6KL
fKK  24L 
100
3
6KL
2
fKK  0, 24L 
100
2
• fLL> 0 and fKK > 0 if K*L < 400
• fLL< 0 and fKK < 0 if K*L > 400
• If K = 10, then inflection point at L=40
Cross Effect
•Cross differentiation of either of the marginal
productivity functions yields
9K2L2
fLK  fKL  48KL 
100
9K2L2
fLK  0, 48KL 
100
• fLK > 0 if KL < 533
• fLK < 0 if KL > 533
•If K = 10
• fLK> 0 when L < 53.3
• fLK< 0 when L > 53.3
A Diminishing TRS?
• Strictly Quasi-Concave if
+ ?
? ? ?
+ ?
f f  2fK fL fKL  f f  0
2
K LL
2
L KK
• Lots of parts that have different signs depending on
K and L.
Returns to Scale
• How does output respond to increases in all
inputs together?
– suppose that all inputs are doubled, would
output double?
• Returns to scale have been of interest to
economists since Adam Smith’s pin factory
Returns to Scale
• Two forces that occur as inputs are scaled
upwards
– greater division of labor and specialization of
function
– loss in efficiency (bureaucratic inertia)
• management may become more difficult
• fall of the Roman Empire?
• General Motors?
Returns to Scale
• Starting at very small scale and then expanding, firms
tend to exhibit increasing returns to scale at small scale,
which changes to constant returns over a range, and
then when they get larger, face decreasing returns to
scale.
• Obviously, the scale at each transition can vary.
–
–
–
–
Vacuum Cleaner Repair Shops
Steel Mills
Doughnut Shops
Automobile manufacture
• Empirical analysis reveals that established firms tend to
operate at a CRS scale.
Returns to Scale
• If the production function is given by q = f(K,L)
and all inputs are multiplied by the same
positive constant (t >1), then
Effect on Output
Returns to Scale
f(tK1,tL1) = tf(K1,L1)
Constant
f(tK1,tL1) < tf(K1,L1)
Decreasing
f(tK1,tL1) > tf(K1,L1)
Increasing
Returns to Scale
• Constant Returns to Scale
q = K.5L.5
What if we increase all inputs by a factor of t?
(tK).5(tL).5 = ?
t(K).5(L).5 = tq
• For t > 1, increase all inputs by a factor of t
and output increases by a factor of t
• I.e. increase all inputs by x% and output
increases by x%
Returns to Scale
• Decreasing Returns to Scale
q = K.25L.25
What if we increase all inputs by a factor of t?
(tK).25(tL).25 = ?
t.5(K).25(L).25 = t.5q, which is < tq
• For t > 1, increase all inputs by a factor of t
and output increases by a factor < t
• I.e. increase all inputs by x% and output
increases by less than x%
Returns to Scale
• Increasing Returns to Scale
q = K1L1
What if we increase all inputs by a factor of t?
(tK)1(tL)1 = ?
tq < t2(K)1(L)1 = t2q, which is > tq
• For t>1, increase all inputs by a factor of t and
output increases by a factor > t
• I.e. increase all inputs by x% and output
increases by more than x%
Returns to Scale
• Using the usual homogeneity notation,
alternatively, it is notated, for t > 0.
• That is, production is homogeneous of
degree k.
Effect on Output
Returns to Scale
k = 1, Constant
k
f(tK1,tL1) = t f(K1,L1)
k < 1, Decreasing
k > 1, Increasing
Returns to Scale, Example
• Solve for k
•
•
•
•
•
•
•
•
•
q = K.4L.4
tkq = (tK).4(tL).4 = t.8(K).4(L).4
k ln(t) + ln(Q) = .8ln(t)+.4ln(K)+.4ln(L)
k ln(t) = .8ln(t)+.4ln(K)+.4ln(L) - ln(Q)
k ln(t) = .8ln(t)+.4ln(K)+.4ln(L)-.4ln(K)-.4ln(L)
k ln(t) = .8ln(t)
k ln(t) = .8ln(t)/ln(t)
k = .8, production is Homogeneous of degree .8
k < 1 so DRS
Returns to Scale by Elasticity
• What is the % change in output for a t%
increase in all inputs?
• Generally evaluated at t = 1
f(tK,tL)
t
eq,t 

, and evaluated at t=1
t
f(tK,tL)
CRS: q,t =1
DRS: q,t < 1
IRS: q,t > 1
Returns to Scale by Elasticity
• What is the % change in output for a t%
increase in all inputs? Evaluated at t = 1.
q  K.4 L.2  L
f (tK, tL)
t
eq,t 

t
f (tK, tL)
eq,t 

  tK 
.4
t
.6  K 
.4
eq,t 
 tL .2  tL 
 L .2  L 
t .4

t
 tK .4  tL .2  tL
t
t .6  K 
.4
 L .2  tL
.6t  K 
 L .2  tL
eq,t 
.4
.2
t  K   L   t1.4 L
.4
.2
.6  K   L   L
at t  1, eq,t 
 K .4  L .2  L
.4
• In this example, RTS varies by K and L.
Constant Returns to Scale is Special
• Empirically, firms operate at a CRS scale.
• If a function is HD1, then the first partials will
be HD0.
• If
f(tK,tL)  t  f(K,L)
• Then
f(K,L) f(tK,tL)
MPK 

K
K
f(K,L) f(tK,tL)
MPL 

L
L
Constant Returns to Scale is Special
• Obviously, if CRS, we can scale by any t > 0
• But let’s pick a specific scale factor, 1/L:
• If
• Then
1 1  K 
f  K, L   f  ,1 
L L  L 
K 
f  ,1 
L 

MPK 
,
K
K 
f  ,1 
L 

MPL 
L
• Which tells us that if production is CRS, then it is also
homothetic. Isoquants are radial expansions with the
RTS the same along all linear expansion paths.
Constant Returns to Scale
• The marginal productivity of any input
depends on the ratio of capital and labor
– not on the absolute levels of these inputs
• Therefore the TRS between K and L depends
only on the ratio of K to L, not the scale of
operation
• That is, increasing all inputs by x% does not
affect the TRS
• The production function will be homothetic
(TRS constant along ray from origin)
• Geometrically, this means all of the isoquants
are radial expansions of one another
Constant Returns to Scale
• Along a ray from the origin (constant K/L), the
TRS will be the same on all isoquants
K
The isoquants are equally
spaced as output expands
q=3
q=2
q=1
L
Economies of Scale
(not Returns to Scale)
• In the real world, firms rarely scale up or down
all inputs (e.g. management does not typically
scale up with production).
• Economies of scale: %ΔLRAC/%ΔQ
– Economies of scale if < 0
– Diseconomies of scale if > 0
Elasticity of Substitution
• The elasticity of substitution () measures the proportionate
change in K/L relative to the proportionate change in the TRS along
an isoquant
% K
d K
TRS
L
L



%TRS
dTRS K
L
   
 
• And as was demonstrated earlier, elasticity is the effect of a change
in one log on another.
dln K
dln K
L 
L

dlnTRS dln  fL 
 f 
 K
• The value of  will always be positive because K/L and TRS move in
the same direction
 
 
Elasticity of Substitution
• Both RTS and K/L will change as we move from point A
to point B
 is the ratio of these
K
proportional changes
 measures the
TRSA
A
(K/L)A
(K/L)B
curvature of the
isoquant
TRSB
B
q = q0
L
Elasticity of Substitution
• If  is low, the K/L will not change much relative
to TRS
– the isoquant will be relatively flat
• If  is high, the K/L will change by a substantial
amount as TRS changes
– the isoquant will be sharply curved
• More interesting when you remember that to
minimize cost, TRS = pL/pK so TRS changes with
input prices.
Elasticity of Substitution
K
q=g(K,L)
g >  f
q=f(K,L)
L
• It is possible for  to change along an isoquant
or as the scale of production changes
Elasticity of Substitution
• Solving for σ can be tricky, but, we can employ this calculus trick
(especially useful for homothetic production functions):
y
1

x
x
y
• This allows us to turn this problem

• Into the (sometimes) easier
 L
dln K
f
dln  L 
 fK 
1

f
dln  L 
 fK 
dln K
L
 
Elasticity of Substitution
CRS is Special Again
• For CRS production functions only we have
this option too
• Let q = f(K,L)
fK  fL

q  fK,L
Common Production Functions
• Linear (inputs are perfect substitutes)
• Fixed Proportions (inputs are perfect
compliments)
• Cobb-Douglas
• CES
• Generalized Leontief
The Linear Production Function
(inputs are perfect substitutes)
• Suppose that the production function is
q = f(K,L) = aK + bL
• This production function exhibits constant
returns to scale
f(tK,tL) = atK + btL = t(aK + bL) = tf(K,L)
• All isoquants are straight lines
Linear Production Function
q  aK  bL
fL b
TRS  
fK a
K
K
 ln  
 ln  
1
1
L
L






 ln  TRS 
  b 
  b  0
  ln      ln   
  a 
  a 
K
 ln  
L

The Linear Production Function
Capital and labor are perfect substitutes
TRS is constant as K/L changes
K
slope = -b/a
q1
q2
=
q3
L
Fixed Proportions
• Suppose that the production function is
q = min (aK,bL) a,b > 0
• Capital and labor must always be used in a
fixed ratio
– the firm will always operate along a ray where
K/L is constant
• Because K/L is constant,  = 0
Fixed Proportions
No substitution between labor and capital is possible
K/L is fixed at b/a
K
=0
q3
q3/a
q2
q1
q3/b
L
Cobb-Douglas Production Function
• Suppose that the production function is
q = f(K,L) = AKaLb A, a, b > 0
• This production function can exhibit any
returns to scale
f(tK,tL) = A(tK)a(tL)b = Ata+b KaLb = ta+bf(K,L)
– if a + b = 1  constant returns to scale
– if a + b > 1  increasing returns to scale
– if a + b < 1  decreasing returns to scale
Cobb-Douglas Production Function
q  AKaLb
fL bAKaLb1 bK
TRS  

a1 b
fK aAK L aL
K
K
 ln  
 ln  
1
1
L
L





 1
 bK 
 b
 b
K 
K  1


 ln     ln    ln   
  ln    ln   
 aL 
 L 
 L 
 a
 a
K
 ln  
L
Cobb-Douglas Production Function
•The Cobb-Douglas production function is linear
in logarithms
ln q = ln A + a ln K + b ln L
• a is the elasticity of output with respect to K
• b is the elasticity of output with respect to L
• Statistically, this is how we estimate production
functions via regression analysis.
CES Production Function
• Suppose that the production function is
q  K  L 




,   1,   0,   0
•  > 1  increasing returns to scale
•  = 1  constant returns to scale
•  < 1  decreasing returns to scale
CES Production Function
• TRS

q  K  L  
    1 1
fL  K  L  L



    1
fK  K  L  K1

fL  L 
TRS    
fK  K 
1

1
L 
 
K
 1

1
L
 
K
1 
K
 
L
• Note, not a function of scale, γ
1 
CES Production Function
• σ
K
 ln  
L 

TRS

K
 ln  
L
  K 1 
 ln   

 L 



1
  K 1 
 ln   

 L 



K
 ln  
L
1

1   

1
K
 1    ln  
L 
K
 
L

1
K
1


  
L
K
 
L
CES Production Function
• For CES
q  K  L 



,   1,   0,   0
1
1   
• At limit as  → 1, σ → ∞, linear production function
• At limit as  → -, σ → ∞, fixed proportions
production function
• When  = 0, Cobb-Douglas production function
A Generalized Leontief Production
Function
• Suppose that the production function is
q  K  L  2 KL
• TRS
0.5
0.5
0.5
K
K

L
1 
0.5
0.5
fL
K
L  L
TRS  
 0.5
0.5
0.5
0.5
L K
fK
L
L
1 
0.5
K
K
.5
K
TRS   
L
A Generalized Leontief Production
Function
• σ
 K
  ln 
L



0.5
 K 
  ln   
 L 


2
1
1
1


0.5
  K    .5 ln K .5
 
  ln   
L
 L 


K
 ln
 K
L
  ln 
 L
Technical Progress
• Methods of production change over time
• Following the development of superior
production techniques, the same level of
output can be produced with fewer inputs
– the isoquant shifts inward
Technical Progress
• Suppose that the production function is
q = A(t)f(K(t),L(t))
where A(t) represents all influences that go
into determining q other than K and L
– changes in A over time represent technical
progress
• A is shown as a function of time (t)
• dA/dt > 0
Technical Progress
• Differentiating the production function
q  A  t   f K  t  ,L  t  
with respect to time we get
dq dA(t)
 df dK df dL 

 f(K(t),L(t))  A(t)  


dt
dt
dK
dt
dL
dt


Which simplifies to
dq dA
 df dK df dL 
  f(K,L)  A  


dt dt
dK
dt
dL
dt


Technical Progress
dq dA
 df dK df dL 
  f(K,L)  A  


dt dt
dK
dt
dL
dt


• Since
q  A  f K,L  , then A 
q
q
, and f K,L  
f K,L 
A
• And so
dq dA q
q  dK
dL 
  
fK   fL  

dt dt A f(K,L)  dt
dt 
Technical Progress
dq dA q
q  dK
dL 
  
fK   fL  

dt dt A f(K,L)  dt
dt 
• Dividing by q gives us
dq
dt  dA  1   fK  dK  fL  dL 
q
dt A  f(K,L) dt f(K,L) dt 
Technical Progress
dq
dt  dA  1   fK  dK  fL  dL 
q
dt A  f(K,L) dt f(K,L) dt 
• Expand by strategically adding in K/K and L/L
dq / dt dA / dt
K dK / dt
L dL / dt

 fK 

 fL 

q
A
f(K,L)
K
f(K,L)
L
Technical Progress
dq / dt dA / dt
K dK / dt
L dL / dt

 fK 

 fL 

q
A
f(K,L)
K
f(K,L)
L
• For any variable x, [(dx/dt)/x] is the
proportional growth rate in x
– denote this by Gx
• Then, we can write the equation in terms of
growth rates
K
L
Gq  GA  fK 
 GK  fL 
 GL
f(K,L)
f(K,L)
Technical Progress
• Note the elasticities
K
L
Gq  GA  fK 
 GK  fL 
 GL
f(K,L)
f(K,L)
f(K,L) K
f(K,L) L
eq,K 

, eq,L 

K f(K,L)
L f(K,L)
• Yielding
Gq  GA  eq,KGK  eq,L GL
• Growth is a function of technical change and growth
in the use of inputs.
Solow, US Growth 1909-1949
• Solow estimated the following
•
•
•
•
•
Gq = 2.75%
GL = 1.00%
GK = 1.75%
eq,L = .65
eq,K = .35
• Plug these in
Gq  GA  eq,KGK  eq,L GL
GA  Gq  eq,KGK  eq,L GL
• And GA = 1.5%
• Conclusion, technology grew at a 1.5% rate from 19091949. 55% of GDP growth in the period.
Appendix
• Full derivations of TRS and convexity in
production.
RTS and Marginal Productivities:
Implicit Function Rule
Generalize the implicit function: f(K,L)-q=0 to F(K,L,q)  0
F
F
F
dK  dL  dq  0
q
L
K
If the conditions of the implict function therorem hold,
then there exists an equation K=g(L,q) and
g
g
dK  dL  dq
q
L
Substitute
g
g
Subsitute dK  dL  dq into
L
q
F
F
F
dK  dL  dq  0 to get
K
L
q
F  g
g  F
F
dL

dq

dL

dq  0


K  L
q  L
q
F g
F
F g
F
dL  dL 
dq  dq  0
K L
L
K q
q
 F g F 
 F g F 
  dL  
  dq  0

 K L L 
 K q q 
And get to…
 F g F 
 F g F 

dL

  dq  0



 K L L 
 K q q 
Since the dL and dq are independent, the bracketed
expressions must = 0 for the equation to hold. That is:
 F g F 
   0 and

 K L L 
We are interested in:
 F g F 
 K q  q   0


 F g F 
 0

 K L L 
F
g L

 0 and K=g(L,q) and F(K,L,q)  f(K,L)-q=0
L F
K
And get to…
F
g L
 F g F 
Solve this 
   0 to get

0
L F
 K L L 
K
g dK
And since K=g(L,q) with q held constant,

and
L dL
F f
F f
F(K,L,q)  f(K,L)-q=0, so

and

L L
K K
f(K,L)
dK
  L
f(K,L)
dL
K
MP
dK
 L
dL
MPK
Convexity, Increasing dK/dL
 fL L,K  
 dL dK 
dK  
 dL
d


f
f
f

f
f

f


 LL dL LK dL  K L  KL dL KK dL  
fK L,K  
fL
d2K
dK







,
Note:


dL2
dL
fK2
dL
fK
 f L,K   

f 
f 
d  L
  fLL  fLK L  fK  fL  fLK  fKK L 
fK 
fK 
 fK L,K    

dL
UK2
 f L,K  
fL2
d  L
 fLL fK  fLK fL  fKL fL  fKK
f
L,K


fK
K


dL
fK2
f
Multiply by: K
fK
 f L,K  
d  L

2
2
f
L,K


K

  fLL fK  fLK fL fK  fKL fL fK  fKK fL
dL
fK3
 f L,K  
d  L

2
2
 fK L,K    2fLK fL fK  fLL fK  fKK fL  0
dL
fK3
Diminishing TRS
• TRS diminishing if this < 0
 fL L,K  
d 
 2
2
f
L,K


d  TRS 
f
f

2f
f
f

f
K


K LL
K L KL
L fKK


0
3
dL
dL
fK
• Which is the same thing.
Alternatively, the Bordered Hessian
Strictly Quasi-Concave if
det
and
0
fL
fL
f LL
 f L  0
2
0
fL
fK
det f L
fK
f LL
f KL
f LK  2f LK f L f K  f L 2 f KK  f K 2 f LL  0
f KK
which looks a lot like the negative of this:
2f K f L f KL  f L2f KK  f K2 f LL
dRTS
 0 if
0
3
dL
fK