macro06 mc

6. Monopolistic competition ISHISE, Hirokazu

6. Monopolistic competition

Goals

to master a useful model of price setting, monopolistic competition to understand the loss associated with monopolistic competition

1 Basic monopolistic competition model

Motivation

Under perfect competition, agents take prices as given Want to include a price setting behavior A (relatively) simple device is monopolistic competition Used in many models in growth theories, international trade theories, etc. Basic premise is to consider variety of goods e.g., not just bottled green tea but distinguishing each dierent brand

1.1 Household demand

1.1.1 Set-up of two goods case

The utility function of the HH is given by

u(c); (1)

where c is composed of two types of consumption goods, and they are not perfectly substi-tutable each other,

c =

c1

1 + c1

2

1

; (2)

where > 1. The budget constraint is

p1c1 + p2c2 = M; (3)

where M is the wealth.

Maximization of u(c) is achieved by maximizing c The function is called constant elasticity of substitution (CES)

1 Graduate Introductory Macroeconomics Summer 2014


1.1.2 Demand of two goods case

Set-up a Lagrangian,

L =c1

1 + c1

2

1

+ (M p1c1 p2c2) : (4)

FOCs for j = 1; 2

c1

1 + c1

2

11

c11

j = pj; (5)

and then c1

1 + c1

2

11

c1

j = pjcj: (6)

Summing up the condition for j = 1; 2 to obtain

c1

1 + c1

2

11

c1

1 + c1

2

| {z }

c

= (p1c1 + p2c2)| {z }M

(7)

that is,

= c=M: (8)

Then,

c1 c1

j = pjc=M; (9)

or

cj = c(1)Mpj : (10)

" Demand if # price of the good " Demand if " total wealth

1.1.3 Alternative representation of the demand function

From FOCs, c1

1 + c1

2

11

c1

j = pj; (11)

or



c1

1 + c1

2

11

c1

j = 1p1j ; (12)

Summing up the equation for j = 1; 2 to obtainc1

1 + c1

2

11+1

= 1p11 + p

12

; (13)

or

1

=p11 + p

12

11 : (14)

That is, 1= is an average price of the two goods. Let p denote 1=. Since = c=M ,

pc = M: (15)

p is the price of the composite consumption good, c.Using this p,

cj =

p

pj

c: (16)

1.1.4 Many goods case

Consider n goods extension as

c =

nX

j=1

c1

j

! 1

; (17)

and the budget constraint is

nXi=1

pjcj = M: (18)

The result is straightforward:

cj = c(1)Mpj : (19)

The price of the composite is

p =

nX

j=1

p1j

! 11

: (20)



1.1.5 Continuum of goods

Remember that summation and integral are similar (except for technical dierence). Consider

c =

Z n0

c(j)1 dj

1

; (21)

and the budget constraint is Z n0

p(j)c(j)dj = M: (22)

The result is straightforward:

c(j) = c(1)Mp(j): (23)

and

p =

Z n0

p(j)1dj 1

1: (24)

Moreover, when we consider a special case that n = 1,

c =

Z 10

c(j)1 dj

1

: (25)

The demand function is the same as (23).

Dierent from the discrete case, there are \innitely many" values between 0 and 1 (0:1,0:11, 0:111, ...)

1.2 Final good rm

The meaning of the average price p is simpler if we consider a nal good producing sector.

1.2.1 Final goods producer

Consider instead that there is a representative rm which produces the nal goods y usingy(j) as intermediate inputs. The nal goods rms sell the nal goods with price p.

py Z 10

p(j)y(j)dj (26)

where

y =

Z 10

y(j)1 dj

1

: (27)



1.2.2 Demand for intermediate goods

maxfy(j)g

p

Z 10

y(j)1 dj

1

Z 10

p(j)y(j)dj (28)

FOC

p

Z 10

y(j)1 dj

11

y(j)11 = p(j): (29)

From this equation,

py1 y(j)

1 = p(j); (30)

and hence the demand for good y(j) is

y(j) =

p

p(j)

y: (31)

The demand function is the same as before " Demand if # price of the input " Demand if " price of the output " Demand if " output production

1.2.3 Zero prots

From (29)

p

Z 10

y(j)1 dj

11

y(j)1 = p(j)y(j): (32)

Integrating this equation from j = 0 to j = 1 to obtain

p

Z 10

y(j)1 dj

11 Z 1

0

y(j)1 dj =

Z 10

p(j)y(j)dj; (33)

and hence

0 = p

Z 10

y(j)1 dj

1

Z 10

p(j)y(j)dj; (34)

the prots of the nal goods rms are zero.



1.2.4 Determination of output price

Another expression from (29) is

p

Z 10

y(j)1 dj

11

y(j)1 = p(j); (35)

or

p1Z 1

0

y(j)1 dj

1y(j)

1 = p(j)1; (36)

and integrating this equation to obtain

p1Z 1

0

y(j)1 dj

1 Z 10

y(j)1 dj =

Z 10

p(j)1dj; (37)

or

p =

Z 10

p(j)(1)dj 1(1)

: (38)

The price of the output is a (geometric) weighted average of the input prices.

1.3 Intermediate good rm

An intermediate good producer produces a type of intermediate goods facing the demandfunction of the nal good producer.

1.3.1 Simplied version of intermediate good rm's problem

Consider that intermediate good rm produces intermediate good using labor only,

y(j) = z(j)l(j); (39)

where z(j) is labor productivity (as well as TFP) of the rm. The prot maximizationproblem is to choose p(j), y(j), and l(j) to maximize

p(j)y(j) wl(j); (40)subject to the demand (31), taking w, p and y as given. That is

maxfp(j);y(j);l(j)g

p(j)y(j) wl(j)s.t. y(j) = z(j)l(j)

y(j) =

p

p(j)

y:

A specication of Lagrangian is



L = p(j)y(j) wl(j) + (j)

p

p(j)

y y(j)

+ '(j) (z(j)l(j) y(j)) ; (41)

FOCs

0 = y(j) (j)pp(j)1y (42)0 = p(j) (j) '(j) (43)0 = w + '(j)z(j) (44)

Note rst that

'(j) = w=z(j); (45)

this is the marginal and average costs of production. By eliminating (j) and using constraintsto obtain

p(j) =

1'(j) =

1w

z(j): (46)

This is the prot-maximizing price. Using (31) and (46), y(j) is easily derived.

Under perfect competition, p(j) = w=z(j), the marginal and average costs =( 1) > 1 because > 1 Price under monopolistic competition is higher than marginal cost The ratio is called the mark-up Price under monopolistic competition is mark-up times the marginal cost

Given above price setting behavior, prots of the rm is

p(j)y(j) wl(j)=

1w

z(j)z(j)l(j) wl(j)

=1

1wl(j): (47)

The rm earns positive prots.



1.3.2 Intermediate good rm with capital

Production function of intermediate goods rms

y(j) = z(j)k(j)l(j)1; (48)

The prot maximization problem is

p(j)y(j) rk(j) wl(j); (49)

subject to the demand (31), taking r, w, p and y as given.

L =p(j)y(j) rk(j) wl(j)

+(j)

p

p(j)

y y(j)

+ '(j)

z(j)k(j)l(j)1 y(j) ; (50)

FOCs

0 = p(j) (j) '(j) (51)0 = y(j) (j)y(j)=p(j) (52)0 = r + '(j)y(j)=k(j) (53)0 = w + '(j)y(j)=l(j) (54)

By eliminating (j) and using constraints to obtain

p(j) =

1'(j): (55)

This is exactly the same as before. What is '(j) now? From constraints and FOCs,

'(j) =rk(j) + wl(j)

y(j); (56)

that is, '(j) again represents the average cost of producing y(j). Since the production functionis CRS, the average cost is the same as the marginal cost. Specically, from FOCs,

'(j) =rw1

(1 )1z(j) : (57)

2 Monopolistic competition in the growth model

Let's now introduce the monopolistically competitive intermediate goods producer into thegrowth model. We normalize the output price to be one, pt = 1.



2.1 Decentralized economy

2.1.1 Household

Utility function

maxfct;kt+1(j);xt(j)g1t=0

E0

1Xt=0

tu(ct; lt): (58)

Period budget constraint

ct +

Z 10

xt(j)dj = wtlt + rt

Z 10

kt(j)dj +

Z 10

t(j)dj: (59)

Households own capital, and rent it to rms. Capital stock follows

kt+1(j) = (1 )kt(j) + xt(j); (60)Merging capital evolution and the period budget constraint into one

ct +

Z 10

kt+1(j)dj = wtlt + (rt + 1 )Z 10

kt(j)dj +

Z 10

t(j)dj (61)

Lagrangian

L =E01Xt=0

t

"u(ct; lt)

+ t

wtlt + (rt + 1 )

Z 10

kt(j)dj +

Z 10

t(j)dj ct Z 10

kt+1(j)dj

#(62)

First order conditions

uct = t; (63)

ult = twt; (64)Ett+1(rt+1 + 1 ) = t: (65)

TVC

limT!1

TTkT+1(j) = 0: (66)

By eliminating t in the FOCs, we have the labor supply condition and Euler equation

wt =ultuct

; (67)

uct = Et (rt+1 + 1 )uct+1: (68)



2.1.2 Final good rms

The nal good producer maximize the prots

ptyt Z 10

pt(j)yt(j)dj (69)

where

yt =

Z 10

yt(j)1 dj

1

: (70)

This is already analyzed in the previous section. The results are

yt(j) =

pt

pt(j)

yt; (71)

p+1t =Z 10

pt(j)+1dj (72)

Note that pt = 1 by normalization (called numeraire).

2.1.3 Intermediate good rms

Production function of intermediate goods rms

yt(j) = zt(j)kt(j)lt(j)

1; (73)

The prot maximization problem is

pt(j)yt(j) rtkt(j) wtlt(j); (74)

subject to the demand (31), taking r, w, p and y as given. This is again already described inthe previous section. The results are

rt = yt(j)

kt(j)

1

pt(j); (75)

wt = (1 )yt(j)lt(j)

1

pt(j) (76)

2.1.4 Market clearing conditions

ct +

Z 10

xt(j)dj = yt; (77)Z 10

lt(j)dj = lt: (78)



2.1.5 Denition of the competitive equilibrium

Denition: A Competitive equilibrium is a sequence of quantitiesfct; yt; fkt+1(j); lt(j); yt(j); xt(j); t(j)gj2(0;1)g1t=0 and prices fwt; rt; fpt(j)gj2(0;1)g1t=0for given fk0(j)gj2(0;1), ffzt(j)gj2(0;1)g1t=0 which satisfy household optimality conditions,nal goods production rm's optimality conditions, intermediate goods productionrm's optimization conditions, and market clearing conditions.

This is not a perfectly competitive equilibrium The intermediate goods markets are monopolistically competitive Other markets are perfectly competitive

2.1.6 Characterization of the competitive equilibrium

From HH and intermediate goods rm conditions

ultuct

= 1

pt(j)(1 )yt(j)lt(j)

(79)

together with the nal goods rm condition,

ultuct

= 1

yt

yt(j)

1

(1 )yt(j)lt(j)

: (80)

Similarly, from the Euler equation and conditions of rms

uct = Et

1

yt+1

yt+1(j)

1

yt+1(j)

kt+1(j)+ 1

!uct+1 (81)

2.2 Social planner's problem

2.2.1 Problem

maxfct;kt+1(j);xt(j)g1t=0

E0

1Xt=0

tu(ct; lt): (82)



Constraints are following:

kt+1(j) = (1 )kt(j) + xt(j); (83)

ct +

Z 10

xt(j)dj = yt; (84)Z 10

lt(j)dj = lt; (85)

yt(j) = zt(j)kt(j)lt(j)

1; (86)

yt =

Z 10

yt(j)1 dj

1

: (87)

2.2.2 Characterization of the Pareto ecient allocation

L =E01Xt=0

t

"u

ct;

Z 10

lt(j)dj

+ t

Z 10

yt(j)1 dj

1

+ (1 )Z 10

kt+1(j)dj ct Z 10

kt+1(j)dj

!

+

Z 10

t(j)

zt(j)kt(j)

lt(j)1 yt(j)

dj

#(88)

one constraint with t continuum of dierent constraints with t(j)

FOCs

uct = t; (89)

ult = t(j)(1 )zt(j)kt(j)lt(j); (90)

t

Z 10

yt(j)1 dj

11

yt(j)11 = t(j); (91)

Ett+1(1 ) + t+1(j)zt+1(j)kt+1(j)1lt+1(j)1

= t (92)

From them

ul;tuct

=

yt

yt(j)

1

(1 )yt(j)lt(j)

; (93)

uct = Et

yt+1

yt+1(j)

1

yt+1(j)

kt+1(j)+ 1

!uct+1 (94)

Compare this with conditions of CE



CE allocation is not PE Source of ineciency: Check what happens if !1

2.3 Steady states

Assume that

u(ct; lt) = ln ct + (1 ) ln(1 lt) zt(j) = zt zt = exp(z^t) z^t = z^t1 + "t, "t iid N(0; 1).

Under this assumption of the TFP, the model has no trend growth. Moreover, since zt(j) = zt,the problem of the intermediate rm is symmetric. Hence,

pt(j) = pt = 1: (95)

2.3.1 Steady state of the competitive equilibrium

Consider that zt = 1. Using the Euler equation,

r =1

1 + : (96)

Note that p = 1, the rest is the same as usual.

2.4 Impulse response functions

2.4.1 Calibration

What is the reasonable value of ?

Remember that =( 1) is mark-up ratio Based on micro evidences, mark-up ratio is around 10{20% = 6{11 Note also that if !1, then the economy is identical to perfectly competitive model



2.4.2 Log-linearization

The log-linearized version of the economy is following:

z^t+1 = z^t + "t (97)

k^t+1 = (1 )k^t + xk x^t (98) Etc^t+1 + rEtr^t+1 = c^t (99)

w^t = c^t +l

1 l l^t (100)y^t = z^t + k^t + (1 )l^t (101)yy^t = cc^t + xx^t (102)

r^t = y^t k^t (103)w^t = y^t l^t (104)

Notice that does not directly appear. It is included in l (see Exercise) and hence in othersteady state values.

2.4.3 Impulse response functions

In the gure, red-broken line shows the IRFs when = 6

Two lines are very close each other There is an eciency in the steady state (see Exercise) However, it does not make a big dierence in terms of the dynamics around the steadystate



0 10 20 300

0.5

1

A

0 10 20 300

0.5

1

K

0 10 20 300

1

2

Y

0 10 20 301

0

1

L

0 10 20 300

0.5

1

C

0 10 20 3010

0

10

X

0 10 20 302

0

2

R

0 10 20 300

0.5

1

W



3 Exercise

3.1 Many goods case

Consider the problem in section 1.1.4.

c =

nX

j=1

c1

j

! 1

; (105)

and the budget constraint is

nXi=1

pjcj = M: (106)

Let p denote the price of the aggregate goods

pc = M: (107)

3.1.1 Question 1

Set-up a Lagrangian and derive FOC.

L =

nXj=1

c1

j

! 1

+

M

nXj=1

pjcj

!: (108)

FOC nX

j=1

c1

j

! 11

c11

j = pj; (109)

3.1.2 Question 2

Conrm that the Lagrange multiplier of the problem is 1=p.

From FOC, nX

j=1

c1

j

! 11

c1

j = pjcj: (110)

Summing up the condition for all j to obtain

c = M (111)

Since pc = M , = 1=p.



3.1.3 Question 3

Show p =Pn

j=1 p1j

11

.

From FOC, nX

j=1

c1

j

! 11

c1

j = pj; (112)

or nX

j=1

c1

j

! 11

c1

j = 1p1j ; (113)

Summing up the equation for all j to obtain nX

j=1

c1

j

! 11+1

= 1

nXj=1

p1j

!; (114)

or

1

=

nX

j=1

p1j

! 11

: (115)

Since = 1=p, The price of the composite is

p =

nX

j=1

p1j

! 11

: (116)

3.1.4 Question 4

Show cj = (p=pj)c.

From FOC, nX

j=1

c1

j

! 11

c1

j = pj; (117)

or

c1 c1

j = pj=p; (118)

and the result is obtained by a straightforward rearrangement.



3.2 Two-step approach of price determination

Consider the problem of section 1.3.2. Instead of applying the Lagrangian to the entireproblem, let's consider a two-step approach. The rst step is to consider the following cost-minimization problem.

minfk(j);l(j)g

rk(j) + wl(j) (119)

s.t. z(j)k(j)l(j)1 = y(j): (120)

3.2.1 Question 1

Attach a Lagrange multiplier '(j) to the constraint. Set-up a Lagrangian and derive theFOCs.

L = rk(j) + wl(j) + '(j) y(j) z(j)k(j)l(j)1 (121)FOCs are

r = '(j)z(j)k(j)1l(j)1 (122)

w = '(j)(1 )z(j)k(j)l(j) (123)

3.2.2 Question 2

Combine the FOCs to express the Lagrange multiplier as a function of parameters and ex-ogenous variables (r, w and z(j)).

'(j) =rw1

(1 )1z(j) : (124)

3.2.3 Question 3

By comparing the results in section 1.3.2, you have shown that the Lagrange multiplierexpresses the marginal cost of production. Using the logic of the Lagrange method, explainwhy the multiplier captures the marginal cost of production.

The multiplier in general represents the marginal change of the objective when the con-straint is marginally relaxed. In this specication, the multiplier represents the marginalcost associated with an increase in the production. Hence, this should be the marginalcost of production.

3.2.4 Question 4

Consider the following second-step problem:

maxfp(j);y(j)g

p(j)y(j) '(j)y(j) (125)

s.t. y(j) =

p

p(j)

y: (126)



Set-up the Lagrangian, derive the FOCs, and express the price as a function of '(j).

L = p(j)y(j) '(j)y(j) + (j)

p

p(j)

y y(j)

(127)

FOCs

0 = p(j) '(j) (j); (128)0 = y(j) (j)pp(j)1y: (129)

From the second equation,

y(j)p(j) = (j)y(j); (130)

and hence

0 = p(j) '(j) p(j)

; (131)

or

p(j) =

1'(j): (132)

3.3 Increasing returns to scale

Suppose that for an operation, a rm needs to employ f > 0 units of labor adding to laborproportional to output. A rm chooses p(j), y(j) and l(j), for taking r, w, p, y and f asgiven.

maxfp(j);y(j);l(j)g

p(j)y(j) wl(j) wfs.t. y(j) = zl(j)

y(j) =

p

p(j)

y:

Note that the productivity z is the same across goods.

3.3.1 Question 1

Set-up a Lagrangian and derive FOCs.



L = p(j)y(j) wl(j) wf + (j)

p

p(j)

y y(j)

+ '(j) (zl(j) y(j)) ; (133)

FOCs

0 = y(j) (j)pp(j)1y (134)0 = p(j) (j) '(j) (135)0 = w + '(j)z (136)

3.3.2 Question 2

Derive the expression of p(j).

p(j) =

1w

z(137)

3.3.3 Question 3

Derive the expression of y(j) and l(j).

y(j) = ( 1)pywz (138)l(j) = ( 1)pywz1 (139)

3.3.4 Question 4

What is the marginal cost of production, that is, the additional cost for producing an addi-tional unit of output?

w=z (140)

3.3.5 Question 5

What is the average cost of production, that is, total cost divided by the total output?

wl(j) + wf

y(j)=

w

z+ f( 1)py1wz (141)

3.3.6 Question 6

Is the price proportional to the marginal cost? How about the average cost?

The price is proportional to the marginal cost, not the average cost.



3.3.7 Question 7

Derive the expression of the prots of the rm. Are the prots zero?

p(j)y(j) wl(j) wf = 1

w

zzl(j) wl(j) wf

=1

1wl(j) wf

=1

1w( 1)pywz1 wf

=( 1)1pyw+1z1 wf (142)

The prots are not zero in general.

3.3.8 Question 8

Do the price, output, labor and prots dier across rms? Explain the reason.

Since the right hand sides of the expressions do not dier across rms, price, output,labor and prots are the same across rms. Since each rm faces the symmetricallytreated, there are no dierences among rms.

3.4 An equilibrium of the economy with increasing returns to scale

Consider a production function of the previous question. Suppose that there are L represen-tative households who consume these output y(j) from 0 to n (not necessarily from 0 to 1) asconsumption goods. Each household supplies one unit of labor. Hence, the total labor supplyof the economy is L. The utility of the household is given as

u = c (143)

where

c =

Z n0

c(j)dj

1

: (144)

Since there are L households, the goods market clearing condition is

Lc(j) = y(j) for all j: (145)

3.4.1 Question 1

Write down the household budget constraint.

Z n0

p(j)c(j)dj = w: (146)



3.4.2 Question 2

Write down the labor market clearing condition.

Z n0

(l(j) + f) dj = L: (147)

3.4.3 Question 3

If a rm earns positive prots, then we expect that a new entrant appears to exploit thisopportunity. Hence, in the equilibrium, we impose an additional condition, zero-protscondition. Write down the zero-prots condition of this economy.

p(j)y(j) wl(j) wf = 0 for all j (148)

3.4.4 Question 4

Dene the equilibrium. Remember that you need to add the zero-prots condition adding toother conditions included in the usual denitions.

3.4.5 Question 5

Derive the demand function of the good c(j) (without using p). Combine the demand functionwith the market clearing condition, and then compare this with the demand function givenin the previous question. What is y?

c(j) = c+1wp(j): (149)

Since Lc(j) = y(j),

Lc+1wp(j) =

p

p(j)

y; (150)

or

y = Lpc+1w: (151)

3.4.6 Question 6

Derive the expression of c as a function of exogenous variables and parameters (hint: use thezero-prots condition and obtained expression of y).



The zero-prots condition implies that

( 1)1pyw+1z1 = wf; (152)

or

( 1)1pLpc+1ww+1z1 = wf; (153)

or

c = zL1

1f1

1

1 ( 1): (154)

3.4.7 Question 7

Derive the expression of y(j) and l(j).

y(j) =( 1)pywz=( 1)pLpc+1wwz=( 1)zf: (155)

l(j) = y(j)=z = ( 1)f: (156)

3.4.8 Question 8

Derive the expression of n (hint:R n0fdj = nf).

L =

Z n0

(l(j) + f) dj = nf:

Hence,

n = L=f: (157)

3.4.9 Question 8

Suppose that L increases. Determine the eect of an increase in L on n and u.



@n

@L

L

n= 1: (158)

One percent increase in L increases n by one percent.Remember that u = c, so the elasticity is

@u

@L

L

u=

1

1 > 0: (159)

One percent increase in L increases u by 1=( 1) percent.

3.5 Steady state labor of the competitive equilibrium

Consider a model described in section 2.1.

3.5.1 Question 1

Starting from the section 2.3.1, calculate steady state value of labor, l.

Using the intermediate good rm's FOC,

r =

kl

1 1

: (160)

From the resource constraint

c+ x = y; (161)

or

1 (1l)(1 )

kl

1

+

kl

l =

kl

l; (162)

or

1 (1l)(1 ) 1

+

r

1

l = l; (163)

or

l =

1 +

1

1

1

1

1 1 +

!!1: (164)

3.5.2 Question 2

Is l increasing or decreasing in ? Give an intuition.

l is increasing in . As described in the text, if ! 1, the model is the same asthe perfectly competitive model. Due to the ineciency in production coming from themonopolistic competition, labor supply is lower, and production is lower.


macro06 mc

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