kuliah 5 fungsi produksi dan teknologi ekonomi mikro pasca sarjana – ilmu akuntansi fe-ui 2010 dr....
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KULIAH 5KULIAH 5
FUNGSI PRODUKSI DAN TEKNOLOGI
Ekonomi MikroPasca Sarjana – Ilmu Akuntansi
FE-UI2010
Dr. Amalia A. Widyasanti
Marginal ProductivityMarginal ProductivityProduction function: Marginal physical product:
◦ Marginal physical product of capital: ◦ Marginal physical product of labour:
Diminishing marginal productivity:
Average physical productivity:
( , )q f k l
k k
qMP f
k
l l
qMP f
l
2
2
2
2
0
0
kkk
lll
MP qf
k k
MP qf
l l
( , )l
output q f k lAP
labor input l l
ExampleExampleSuppose the production function: Find:Marginal productMarginal physical productAverage physical productivity
2 2 3 3( , ) 600q f k l k l k l
Isoquant and RTS(Rate of Isoquant and RTS(Rate of Technical Substitution)Technical Substitution)
Isoquant: combinations of k and l that are able to produce a given quantity of output
Rate of technical substitution: the rate at which labor can be substituted for capital while holding output constant = ratio of MPl to MPk
0
(l for k) l
q q k
MPdkRTS
dl MP
Proof!
Isoquant MapsIsoquant MapsTo illustrate the possible
substitution of one input for another, we use an isoquant map
An isoquant shows those combinations of k and l that can produce a given level of output (q0)
f(k,l) = q0
Isoquant MapIsoquant Map
l per period
k per period
• Each isoquant represents a different level of output– output rises as we move northeast
q = 30
q = 20
Marginal Rate of Technical Marginal Rate of Technical Substitution (Substitution (RTSRTS))
l per period
k per period
q = 20
- slope = marginal rate of technical substitution (RTS)
• The slope of an isoquant shows the rate at which l can be substituted for k
lA
kA
kB
lB
A
B
RTS > 0 and is diminishing forincreasing inputs of labor
Marginal Rate of Technical Marginal Rate of Technical Substitution (Substitution (RTSRTS))
The marginal rate of technical substitution (RTS) shows the rate at which labor can be substituted for capital◦holding output constant along an isoquant
0
) for ( qqd
dkkRTS
ll
RTSRTS and Marginal Productivities and Marginal ProductivitiesTake the total differential of the
production function:
dkMPdMPdkk
fd
fdq k
lll l
• Along an isoquant dq = 0, so
dkMPdMP k ll
kqq MP
MP
d
dkkRTS l
ll
0
) for (
RTSRTS and Marginal and Marginal ProductivitiesProductivities
Because MPl and MPk will both be nonnegative, RTS will be positive (or zero)
However, it is generally not possible to derive a diminishing RTS from the assumption of diminishing marginal productivity alone
RTSRTS and Marginal and Marginal ProductivitiesProductivities
To show that isoquants are convex, we would like to show that d(RTS)/dl < 0
Since RTS = fl/fk
lll
d
ffd
d
dRTS k )/(
2)(
)]/()/([
k
kkkkk
f
ddkfffddkfff
d
dRTS ll
llllll
RTSRTS and Marginal and Marginal ProductivitiesProductivities
Using the fact that dk/dl = -fl/fk along an isoquant and Young’s theorem (fkl = flk)
3
22
)(
)2(
k
kkkkk
f
fffffff
d
dRTS lllll
l
• Because we have assumed fk > 0, the denominator is positive
• Because fll and fkk are both assumed to be negative, the ratio will be negative if fkl is positive
RTSRTS and Marginal and Marginal ProductivitiesProductivities
Intuitively, it seems reasonable that fkl = flk should be positive◦if workers have more capital, they will be
more productiveBut some production functions have
fkl < 0 over some input ranges◦assuming diminishing RTS means that MPl
and MPk diminish quickly enough to compensate for any possible negative cross-productivity effects
A Diminishing A Diminishing RTSRTSSuppose the production function is
q = f(k,l) = 600k 2l 2 - k 3l 3
For this production functionMPl = fl = 1200k 2l - 3k 3l 2
MPk = fk = 1200kl 2 - 3k 2l 3
◦these marginal productivities will be positive for values of k and l for which kl < 400
A Diminishing A Diminishing RTSRTSBecause
fll = 1200k 2 - 6k 3l
fkk = 1200l 2 - 6kl 3
this production function exhibits diminishing marginal productivities for sufficiently large values of k and l
◦fll and fkk < 0 if kl > 200
A Diminishing A Diminishing RTSRTSCross differentiation of either of
the marginal productivity functions yields
fkl = flk = 2400kl - 9k 2l 2
which is positive only for kl < 266
Returns to ScaleReturns 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 the days of Adam Smith
Returns to ScaleReturns to ScaleConstant Returns to Scale: f(tk,tl) =tf(k,l)
Decreasing Returns to Scale: f(tk,tl) < tf(k,l)
Increasing Returns to Scale: f(tk,tl) > tf(k,l)
Constant Returns to ScaleConstant Returns to ScaleConstant returns-to-scale production
functions are homogeneous of degree one in inputs
f(tk,tl) = t1f(k,l) = tqThe marginal productivity functions
are homogeneous of degree zero◦if a function is homogeneous of degree
k, its derivatives are homogeneous of degree k-1
Constant Returns to ScaleConstant Returns to Scale
l per period
k per period
• Along a ray from the origin (constant k/l), the RTS will be the same on all isoquants
q = 3
q = 2
q = 1
The isoquants are equallyspaced as output expands
Elasticity of Substitution Elasticity of Substitution (EOS)(EOS) EOS = the proportionate change in k/l relative to the
proportionate change in the RTS along an isoquant.
EOS is always positive, because…..
If production function is homothetic EOS will be the same along all isoquants
% (k/l) ( / ) ln( / ) ln( / )
% RTS / ln ln l
k
d k l RTS k l k lfdRTS k l RTS
f
• The value of will always be positive because k/l and RTS move in the same direction
Elasticity of SubstitutionElasticity of Substitution
l per period
k per period
• Both RTS and k/l will change as we move from point A to point B
A
B q = q0
RTSA
RTSB
(k/l)A
(k/l)B
is the ratio of theseproportional changes
measures thecurvature of theisoquant
Elasticity of SubstitutionElasticity of Substitution
If is high, the RTS will not change much relative to k/l◦ the isoquant will be relatively flat
If is low, the RTS will change by a substantial amount as k/l changes◦ the isoquant will be sharply curved
It is possible for to change along an isoquant or as the scale of production changes
Four Simple Production Four Simple Production FunctionsFunctions1. Linear: ( )
2. Fixed proportions: =0 capital and labour in a fixed ratio
3. Cobb-Douglas: ( )
4. CES:
( , )q f k l ak bl
min( , ) , 0q ak bl a b
1 ( , ) a bq f k l Ak l
( , )q f k l k l
1, 0, 0
1 increasing returns to scale
1 exhibits diminishing returns
The Linear Production The Linear Production FunctionFunction
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◦RTS is constant◦ =
The Linear Production FunctionThe Linear Production Function
l per period
k per period
q1q2 q3
Capital and labor are perfect substitutes
RTS is constant as k/l changes
slope = -b/a =
Fixed ProportionsFixed Proportions
Suppose that the production function is
q = min (ak,bl) a,b > 0Capital and labor must always be
used in a fixed ratio◦the firm will always operate along a
ray where k/l is constantBecause k/l is constant, = 0
Fixed ProportionsFixed Proportions
l per period
k per period
q1
q2
q3
No substitution between labor and capital is possible
= 0
k/l is fixed at b/a
q3/b
q3/a
Cobb-Douglas Production Cobb-Douglas Production FunctionFunction
Suppose that the production function isq = 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 Cobb-Douglas Production FunctionFunction
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
CES Production FunctionCES Production FunctionSuppose that the production function is
q = f(k,l) = [k + l] / 1, 0, > 0◦ > 1 increasing returns to scale◦ < 1 decreasing returns to scale
For this production function = 1/(1-)
◦ = 1 linear production function◦ = - fixed proportions production
function◦ = 0 Cobb-Douglas production function
Technical ProgressTechnical ProgressMethods of production change
over timeFollowing the development of
superior production techniques, the same level of output can be produced with fewer inputs◦the isoquant shifts in
Technical ProgressTechnical ProgressSuppose that the production function is
q = A(t)f(k,l)
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 ProgressTechnical ProgressDifferentiating the production function
with respect to time we get
dt
kdfAkf
dt
dA
dt
dq ),(),(
ll
dt
df
dt
dk
k
f
kf
q
A
q
dt
dA
dt
dq l
ll),(
Technical ProgressTechnical ProgressDividing by q gives us
dt
d
kf
f
dt
dk
kf
kf
A
dtdA
q
dtdq l
l
l
l
),(
/
),(
///
l
l
l
l
ll
dtd
kf
f
k
dtdk
kf
k
k
f
A
dtdA
q
dtdq /
),(
/
),(
//
Technical ProgressTechnical ProgressFor 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
ll
l
llG
kf
fG
kf
k
k
fGG kAq
),(),(
Technical ProgressTechnical ProgressSince
llGeGeGG qkkqAq ,,
kqeq
k
k
q
kf
k
k
f,),(
l
l
l
ll
l
l ,),( qeq
q
kf
f
Technical Progress in the Technical Progress in the Cobb-Douglas FunctionCobb-Douglas FunctionSuppose that the production function is
q = A(t)f(k,l) = A(t)k l 1-
If we assume that technical progress occurs at a constant exponential () then
A(t) = Aet
q = Aetk l 1-
Technical Progress in the Technical Progress in the Cobb-Douglas FunctionCobb-Douglas FunctionTaking logarithms and
differentiating with respect to t gives the growth equation
qGq
tq
t
q
q
q
t
q
/lnln
Technical Progress in the Technical Progress in the Cobb-Douglas FunctionCobb-Douglas Function
l
l
l
GGtt
kt
ktAG
k
q
)1(ln
)1(ln
)ln)1(ln(ln
Important Points to Note:Important Points to Note:If all but one of the inputs are
held constant, a relationship between the single variable input and output can be derived◦the marginal physical productivity is
the change in output resulting from a one-unit increase in the use of the input assumed to decline as use of the input
increases
Important Points to Note:Important Points to Note:The entire production function can
be illustrated by an isoquant map◦the slope of an isoquant is the
marginal rate of technical substitution (RTS) it shows how one input can be substituted
for another while holding output constant it is the ratio of the marginal physical
productivities of the two inputs
Important Points to Note:Important Points to Note:Isoquants are usually assumed to
be convex◦they obey the assumption of a
diminishing RTS this assumption cannot be derived
exclusively from the assumption of diminishing marginal productivity
one must be concerned with the effect of changes in one input on the marginal productivity of other inputs
Important Points to Note:Important Points to Note:
The returns to scale exhibited by a production function record how output responds to proportionate increases in all inputs◦if output increases proportionately with
input use, there are constant returns to scale
Important Points to Note:Important Points to Note:The elasticity of substitution ()
provides a measure of how easy it is to substitute one input for another in production◦a high implies nearly straight
isoquants◦a low implies that isoquants are
nearly L-shaped
Important Points to Note:Important Points to Note:Technical progress shifts the
entire production function and isoquant map◦technical improvements may arise
from the use of more productive inputs or better methods of economic organization