Assignment #12: IRS and competitive equilibrium
Ÿ IRS and competitive equilibrium: basic setup
Firms. Consider a one sector economy. There is continuum of length one of identical firms. Each firm has access to the following technology
Y HiL = A KHiLΑ LHiL Β
A > 0; Α + Β > 1; i Î @0, ..., 1D
(1)
where Y HiL denotes output of firm i, KHiL physical capital employed by firm i, and LHiL is labor employed by firm i. Notice that
there are increasing returns to scale (IRS) in Y -production at the level of the individual firm.
Households. On the household side there is mass one of identical households who own the capital stock K and are endowed
with L units of labor, which are supplied inelastically to the labor market.
Question. Does total household income equal total output? Stated differently, is the assumption of IRS compatible with a
competitive equilibrium? Provide a concise economic reasoning.
Ÿ Solution (basic setup)
Firms maximize profits by taking the interest rate r and the wage rate w as given. This implies
r = pY Α
Y
and
w = pY Β
K
Y
(2)
L
Capital and labor earn their value marginal products, respectively. This factor payment scheme would, however, represent a
logically inconsistency since total factor earnings would exceed the value of overall production
pY Α
Y
K
K + pY Β
r
L = HΑ + ΒL pY Y > pY Y
Y
L
w
(3)
Þ The assumption of IRS is not compatible with a competitive equilibrium.
Ÿ IRS and competitive equilibrium: Marshallian externalities
Firms. Consider a perfectly competitive, one sector economy. There is a continuum of length one of identical firms. The output
technology of the individual firm reads as follows
Y HiL = KHiLΑ LHiL Β K L
a
Α + Β = 1; a, b > 0; i Î @0, ..., 1D
b
(4)
where K = Ù0 KHiL â i and L = Ù0 LHiL â i denote the average (across firms) levels of capital and labor, respectively. (Notice that K
1
1
doesn’t change if KHiL changes by one unit since the individual is of mass zero. The discrete analogue of this statement is as
follows. Imagine an economy which comprises, say, 10 000 firms. If one firm would increase its KHiL by one unit, then the
average stock of capital,
1
10 000
weight of any individual firm,
000
Ú10
i=1 KHiL =
1
,
10 000
1
10 000
KH1L + ... +
1
10 000
KH10 000L, remains approximately constant because the
is small. More specifically, if an individual firm increases its capital stock by one unit, then
the average capital stock increases by
1
units.
10 000
The same reasoning applies to L and LHiL.) The implicit assumption is that
there are positive spill-over effects in the production sphere. Total factor productivity of the individual firm is higher the higher
is the overall input of capital and labor. Notice that there are constant returns to scale (CRS) at the level of the individual firm
but IRS at the aggregate level.
Households. On the household side there is mass one of identical households who own the capital stock K and are endowed
with L units of labor, which are supplied inelastically to the labor market.
Question. Does total household income equal total output? Stated differently, is the assumption of IRS compatible with a
competitive equilibrium? Provide a concise economic reasoning.
Ÿ Solution (Marshallian externalities)
Ÿ IRS and competitive equilibrium: Monopolistic competition
Firms. The economy comprises two sectors. In the final output sector (CRS, perfectly competitive) there is mass one of identical firms. The output technology reads
Y H jL = à xHiLΛ â i
1
0
1Λ
0 < Λ < 1; j Î @0, .., 1D
(9)
In the intermediate goods sector (IRS, monopolistic competition) there is mass one of identical firms. Each firm has access to
the following technology
xHiL = KHiLΑ LHiL Β
Α, Β > 0; Α + Β > 1
(10)
Households. On the household side there is mass one of identical households who own the capital stock K and are endowed
with L units of labor, which is supplied inelastically to the labor market. Households are the owners of the firms. Hence, total
2
AM_Assignments_12(1).nb
Households. On the household side there is mass one of identical households who own the capital stock K and are endowed
with L units of labor, which is supplied inelastically to the labor market. Households are the owners of the firms. Hence, total
earnings of the representative household is given by
Earnings =
ΠY
¨
+
ΠHiL
©
profitof typicalY-firm
+rK +wL
(11)
profitof typicalx-firm
Market structure. Factor markets are perfectly competitive. The final output sector is perfectly competitive. The intermediate
good sector is monopolistically competitive.
Question. Does total household income equal total output? Stated differently, is the assumption of IRS compatible with a
competitive equilibrium? Provide a concise economic reasoning.
Ÿ Solution (Monopolistic competition)
Final output sector. Final output firms maximize profits by taking pHiL as given (pY = 1)
max : à xHiLΛ â i
1
xHiL
1Λ
0
1
(12)
0
Y
Hà xHiLΛ â iL Λ -1 Λ xHiLΛ-1 - pHiL = 0
1
FOC :
- à pHiL xHiL â i>
1
Λ
for all i Î @0, ..., 1D
1
0
YΛ
IY Λ M Λ
1
-1
(13)
xHiLΛ-1 - pHiL = 0
(14)
Hinverse demandL
pHiL = Y 1-Λ xHiLΛ-1
(15)
Intermediate goods sector. Intermediate goods firm maximize profits by taking r and w as given
max :
K,L
Y 1-Λ xHiLΛ-1
xHiL - r KHiL - w LHiL>
(16)
pHiL accord. inversedemand function
max 9Y 1-Λ KHiLΛΑ LHiLΛΒ - r KHiL - w LHiL=
(17)
K,L
pHiL
FOC1 : Y 1-Λ Λ Α KHiLΛΑ-1 LHiLΛΒ - r = 0
Þ
r=ΛΑ
Y
1-Λ
Λ
xHiL
Y
1-Λ
=ΛΑ
KHiL
x HiLΛ-1 xHiL
pHiL xHiL
=ΛΑ
KHiL
(18)
KHiL
pHiL
FOC2 : Y 1-Λ KHiLΛΑ Λ Β LHiLΛΒ-1 - w = 0
Þ
w=Λ Β
Y
1-Λ
Λ
xHiL
Y
1-Λ
=Λ Β
LHiL
x HiLΛ-1 xHiL
LHiL
pHiL xHiL
=Λ Β
(19)
LHiL
Remark. Notice that K and L are underpaid relative to the perfect-competition equilibrium. This can be seen by writing, say,
r=ΛΑ
Y
KHiL
(due to, in equilibrium, Y = pHiL xHiL). The reason is that when deciding upon the optimal amount of, say, capital, the
monopolist takes into account that an increased employment of capital leads to more output and hence deteriorates the price
pHiL.
Equilibrium. Since the final output sector is perfectly competitive, profits ΠY must vanish in equilibrium
ΠY = Y - à pHiL xHiL â i = 0
1
Y = à pHiL xHiL â i
1
Þ
0
0
=
§
pHiL xHiL
(20)
in symmetricequ.
Profits of the typical intermediate good firm ΠHiL may then be expressed as follows
ΠHiL = pHiL xHiL - §
r KHiL - §
w LHiL = Y - Λ Α Y - Λ Β Y = H1 - ΛHΑ + ΒLL Y
(21)
H1 - ΛHΑ + ΒLL Y + Λ Α Y + Λ Β Y = Y
(22)
ΛΑ
Y
KHiL
ΛΒ
Y
LHiL
Total earnings of the representative household may accordingly be expressed as follows
ΠHiL
rK
wL
Þ The assumption of IRS is compatible with a competitive equilibrium.