1. University of Siegen
Faculty III
Department of economics
UNIV.PROF. DR. JAN FRANKE-VIEBACH
Seminar ”International Economics”
SUMMER SEMESTER 2015
TOPIC 9:
Imperfect Competition with Internal Scale Economies
and Homogeneous Goods
Name: Zhang Yi
Deadline: May 28th
3. List of Symbols
Y competitive goods
X increasing return to scale goods
l labor
tc total cost
mc marginal cost
fc fixed cost
ac average cost
R revenue
p price
mc marginal revenue
mr market share
U utility
I income
η elasticity of demand
Π profit
2
4. 1 Introduction
In the previous studies, as shown in Paul R. Krugman, Maurice Obst-
feld (2008) 1
we known that there is a crucial rule of international trade:
it makes it possible for each country to produce a restricted range of
goods and to take advantage of economies of scale without sacrificing
variety in consumption. Indeed, the international trade leads to an in-
crease in the variety of goods available. Here we are going to discuss
a special case of international trade.
The first part is the basic contents about internal increasing return
to scale and the reason why it leads to an imperfect competition. The
second part provides the model assumptions and the autarky equilib-
rium. The most important part here is the third part; it discusses the
general pro-competitive gains from trade with two different economy
structures, one with fixed number of firms, the other is under free entry
and exit condition. Then, the fourth part and the fifth part discuss two
special preferences of consumers separately, which will lead to more
rigorous analyses about the gains from trade.
2 Internal increasing returns to scale with
imperfect competition
2.1 Increasing return to scale
With the background knowledge about international trade, we realize
that the essential point in this field is about the different market struc-
ture which could lead to different outcome after open for trade. In the
previous studies, we have already discussed several trade patterns,
such as the trade under perfect competition or trade between external
increasing return to scale goods. Now, we are going to further discuss
the trade between internal increasing return to scale goods which will
1
Paul R. Krugman, Maurice Obstfeld. (2008) International Economics: Theory and
Policy, 7th Ed., Pearson Education Asia Ltd., and Tsinghua University press copy-
right.
3
5. lead to an imperfect competition.
In the beginning, we can assume that if inputs to an industry are
doubled, industry output would also double with the constant returns
to scale. However, in reality, many industries are characterized by
increasing returns to scale, so that production is much more efficient.
If industry double its inputs, it will get more than double output. So, we
realize that the reason why increasing economies of scale are able to
presume firms to produce more for international trade by using their
advantages.
2.2 Internal V.S. External
Refer to the economies of scale, one important definition has to be
mentioned here, namely the market structure. There are two possible
situations lead to an increasing industry’s output. The one represents
the increasing production supplied by existing firms in the industry;
and the other means new firms entry in the industry that increasing
the number of firms in the industry.
The difference between them is about what kind of production in-
crease is necessary to reduce average cost. Internal economies of
scale occur when the cost per unit depends only on the size of an in-
dividual firm; a firm is more efficient if its output is larger. External
economies of scale occur when the cost per unit depends on the size
of the industry; the efficiency of firms is increased by having a larger
industry, even though each firm is the same size as before.
An industry where economies of scale are internal will typically give
large firms a cost advantage over small and lead to an imperfectly
competitive market structure. By contrast, external economies of scale
consist of many small firms and lead to perfect competition.
2.3 Imperfect competition
Here we are going to analyse why the presence of internal economies
of scale at the level of the firm implies that price-taking behaviour is in
4
6. consistent with non-negative profits and profits and thus that markets
cannot be perfectly competitive.[Elhanan Helpman, Paul R. Krugman
(1985) ]2
At first, we look at the cost curves of a single firm in the
increasing returns to scale industry, which is base on the cost function
of this industry. See equation (1). The total cost of this industry do
not only include constant marginal cost times output but also fixed cost
initially.
tc = mcX + fc (1)
FIGURE 1 COST CURVE
The marginal cost is constant, at the same time fixed cost is falling
with increasing output. Then we can see average cost is falling with
increasing output as well but always higher than marginal cost.
ac =
tc
X
= mc +
fc
X
(2)
Now, we assume firms are price taker in this industry. The ex-
ogenous market price has to be equal to the marginal cost under the
perfectly competitive condition. For every firm, the profit is negative
because the price is below average cost; they are facing a marketing
loss now. So, there will be no equilibrium.
Then, we can also assume another kind of situation in which the
market price exceeds marginal cost. So, every firm realizes that if they
2
Elhanan Helpman, Paul R. Krugman. (1985)Market Structure and Foreign Trade,
Cambridge: MIT press.
5
7. make more production, market price will exceed average cost and with-
out any limitations. They can get infinitely positive profit from infinitely
increasing production. There will also be no equilibrium here.
The consequence of the discussion above shows why there is only
imperfect competition between firms in the increasing returns to scale
industry.
3 The model assumption and autarky equi-
librium
3.1 Assumption
There are two industries in this economy. One producing constant
returns to scale goods under perfect competition. We use Y to denote
the competitive goods. The other producing increasing returns to scale
goods, we assume the IRS goods are homogeneous and denote them
as X. The cost function of IRS goods which we have mentioned in
equation (1) of last part.
Furthermore, we assume labour as a single factor for production,
which must be divided between Y and X sectors and among firms
in each sector. we use the same assumption as in Markusen, J. R.
/ Maskus, K. E (2011) 3
and assume a very simple technology for Y ,
such that one unit of L produces exactly one unit of Y : Y = Ly. Y
and L must have same price and we will use L and therefore Y as
numeraire, giving them a price of one. L is the economy’s total endow-
ment of L.
3.2 Autarky equilibrium
Basis on information we have assumed above, we can draw the pro-
duction frontier in the following figure 2.
3
Markusen, J. R. / Maskus, K. E (2011) International Trade Theory, Unpublished
Manuscript, University of Colorado, Boulder, Chapter 11, pp 1 8, 10.
6
8. FIGURE 2 AUTARKY EQUILIBRIUM
We can see that the production frontier runs from Y to Y 1
to X1
where Y to Y 1
is the fixed cost measured in units of Y or L in this
case.
Suppose that the autarky equilibrium is at point A, where X0
and
Y 0
show the output of the two industries separately. There is a dash
line running from point A to the end of the production frontier Y , the
slope of this line present the average cost of X industry, which we can
get the clue of equation(3).
ac =
L − L0
X0
=
Y − Y 0
X0
(3)
In order to achieve positive profit, the price line through point A of
the monopoly industry has to be steeper than the dash line. Only in
this way, could the price exceed average cost. The slope of price line
shows the relative price of X in terms of Y .
Through the equilibrium point A, we can also draw an indifference
curve which tangency to the price line. However, we can see that there
is a market failure because the indifference curve is not tangency to the
production frontier, which means market price is not equal to marginal
cost. As we have discussed in last part, internal increasing returns to
scale lead to imperfect competition. X industry in this case produce
less but charge much higher price for its goods.
In the coming parts, we are going to discuss the pro-competitive
gains from trade, which will lead to more production, lower price and
7
9. lower average cost, because of increasing competition after open for
trade.
4 Pro-competitive gains from trade: the gen-
eral case
4.1 Overview
We are going to analyse two market structure cases of this economy
which includes IRS industry and perfect competitive industry. The first
one, there are fixed number of firms before trade. The second one,
there is free entry and exit of firms in this market structure before and
after trade. As we have mentioned in the last part, with increasing re-
turns to scale, the large country has a cost advantage in producing and
exporting the monopolized good. This cost-side condition thus tends
to work in the opposite direction to the demand-side condition. This
unfortunately leaves us with a Cournot-Nash equilibrium in which both
the direction of trade and the distribution of gains from trade are inde-
terminate.[Markusen, James R (1981)]4
Cournot-Nash strategy shows
a kind of reaction between each of them, in which each firm is making
best response to others. For mathematical use, we hold the output of
rivals as constant in order to discuss how each firm decide its output
quantity.
Parameters are going to be used as follows:
Revenue: Rij, is represent the revenue of a firm i and selling goods in
country j.
Price: pj, is a function of all firm’s sales like pj(Xj).
Output: Xij, is the quantity of firm i’s sales in market j.
Total output:Xj, is total sales in the market j. Xj = i Xij.
So we can get the revenue function:
Rij = pj(Xj)Xij (4)
4
Markusen, James R. (1981) “Trade and the Gains from trade with imperfect compe-
tition”,Journal of International Economics 11, 531-551.
8
10. Then we can get the marginal revenues with the respect to the
sales of firm i in market j, see equation (5). In addition, as the impli-
cation of cournot competition, one-unit increase in the firm i’s output
contribute to one-unit increase in market total output.
∂Rij
∂Xij
= pij + Xij
∂pj
∂Xj
∂Xj
∂Xij
= pj + Xij
∂pj
∂Xj
(5)
since
∂Xj
∂Xij
= 1 (6)
After calculation, here we can get several important parameters
which are going to influence trade.
The price elasticity of demand is about how much the consumption
changes when production changes by one-unit. See equation (7).
ηj ≡ −[
pj
Xj
∂Xj
∂pj
] (7)
The term
Xij
Xj
represent the market share, which is about how much
the firm i’s output of the total X provided in market j, and we denote it
as sij.
So as to achieve optimal output of firm i. marginal revenue has to
be equal to marginal cost. See equation (8).
mrij = p[1 −
sij
ηj
] = mci (8)
The term
sij
ηj
is called ”markup” here. It shows the reduction of
the market price, when the firm increases its output. If the ”markup”
is small, means only small falls in price with increasing output. That
presumes firms to produce more.
After the general discussions above, the further discussion is di-
vided by two cases.
9
11. 4.2 First case: fixed number of firms
As we have briefly mentioned before. There are fixed number of firms
in each economy. Here we assume each having a single X producer.
Just after the trade between them, they are going to act as what they
do in autarky. However, they realize that if they increase output by
one-unit simultaneously, the total output in this market is doubled by
two-unit. In addition, the total consumption doubled, but that the price
will fall by only half as much as it would if a single firm increases its
output by one-unit in autarky.
Additionally, we assume the firm now perceives demand as more
elastic. That means the term ηj is high. At the same time, the mar-
ket share goes half after trade. The markup definitely falls, thus the
marginal revenue will increase, and that presume firms to produce
more.
FIGURE 3 TRADE EQUILIBRIUM
Figure 3 shows the consequence of this. In last part, we conclude
the autarky equilibrium in point A. After trade, the equilibrium goes to
point T. Obviously, trade release the endogenous distortion in autarky.
We can get the clues from higher welfare gain (Ua to Ut), lower price
(Pa to Pt), higher output (A to T), and lower average cost.
10
12. 4.3 Second case: free entry and exit
Next, we are going to discuss the second case, that there are more
than one firm in each economy initially, and they can entry or exit freely
until reach the zero-profit condition. So the autarky equilibrium condi-
tion in this case is quite different as we have concluded in last part. For
zero-profit, the price equals average cost. The slope of price line and
average cost curve must be equal.
FIGURE 2* AUTARKY EQUILIBRIUM
FIGURE 4 TRADE EQUILIBRIUM
Then, when the two identical economies trade with each other, they
are incentive to produce more as before. However, after they increase
their production after trade, profits of them are going to become nega-
tive. This will lead to the exit of some firms until re-establish zero-profit
11
13. condition, because the reduction of price is much more than the reduc-
tion of average cost. In figure 4, there is a reduction of fixed cost (Y Y0
to Y Y1); Y Y0X0 to Y Y1X1 shows an outward shift of production frontier.
The new equilibrium is in point T now, it shows a higher welfare gain
(Ua to Ut), higher output (A to T), lower price (Pa to Pt) and lower av-
erage cost. Last but not least, even some firms exit from this market,
there will be more competitive in the market because of larger market
size.
From above discussion, we can get the general outcomes about
pro-competitive gains from trade with increasing returns to scale goods
under imperfect competition. However, it is not rigorous enough. For
example, we set the price elasticity of demand as constant in the above
discussion. But generally speaking, it will change when prices and
sales change in reality. In the next two parts, we are going to discuss
two special cases that will give us more rigorous solution of the trade
effects.
5 Gains from trade under quasi-linear pref-
erence condition
Here, we are going to analyses the first special case which we assume
the preferences of a representative consumer in each economy are
given by the following equation (9).
U = αX − (
β
2
)X2
+ Y (9)
5.1 First case: fixed number of firms
In above equation, X represents both total output of IRS good, the
production per firm, and the consumption per capita. Because, we
assume there is only one monopoly firm in each economy, so n = 1,
and normalize the population L to 1.
Under this assumption, we further denote p as the relative price of
12
14. X, and let Π denote profits of the firm. For one representative con-
sumer, his or her income in terms of L has to equal expenditure. Thus,
we get the budget constraint, see equation (10).
L + Π = pX + Y (10)
Then, maximizing one’s utility subject to his or her budget con-
straint. We can get the consumer’s best choice which is calculated
by the following equations.
Max(X)Ui = αX − (
β
2
)X2
+ L + Π − pX (11)
First-order condition, namely the derivation with respect to X, shows
the demand function of X.
∂U
∂X
= α − βX − p = 0 ⇒ p = α − βX (12)
From the demand function, we find that the demand of X is only
depending on its price, and has nothing to do with income and profits.
On the one hand, that truly makes the model much easier to solve.
Nevertheless, it means there is a zero income-elasticity of demand
for X. The increasing of income do not present any attractive to the
consumption of X.
5.1.1 Autarky equilibrium
At first, we can reach the autarky equilibrium by the equation of the
profits function as follows. Where c denotes the marginal cost and f
denotes the fixed cost.
Π = pX − cX − f = [α − βX]X − cX − f (13)
To maximize profit, we draw the first-order condition with respect to
X.
∂Π
∂X
= α − 2βX − c = 0 (14)
13
15. The optimal production of X in one firm is X = α−c
2β
5.1.2 Trade outcome
Secondly, two identical firms trade with each other freely. Same as in
part 2, we denote the two countries and their firms as i and j. Now,
there are twice as many as consumers in the common market, while
the demand price only depends on per capita consumption. So, it
means the world price does not change, and L doubled after trade.
We can get the new demand function:
p = α −
β(Xi + Xj)
L
= α −
β(Xi + Xj)
2
(15)
Then the new profit function in country i.
Πi = [α −
β(Xi + Xj)
2
]Xi − cXi − f (16)
We use the same calculation way with autarky equilibrium, in addi-
tion to absolutely identical economies assumption, we get the symmet-
ric solutions of both firms. They produce same amount in equilibrium
under cournot competition which we have assumed before. Where
Xi = Xj.
X∗
i = X∗
j = 2
(α − c)
3β
>
α − c
2β
= Xa
(17)
Obviously, output of each firm increases by one-third compared to
what they did in autarky. This calculation result definitely proves that
we have discussed in figure 3, there is a mutual gains from trade for
the two firms in each economy.
5.2 Second case: free entry and exit
Suppose there is free entry or exit before trade now. The only dif-
ference here in this case is that there is more than one firm in the
economy, so n > 1, and n is endogenous. In addition, we assume that
there are L workers or consumers, which can again be normalized to
14
16. 1 in autarky. Then, we can get the consumption per capita in this case
as follows:
n
i
Xi
L
(18)
Demand and profit functions are also as follows:
p = α − β[
j
Xj
L
] (19)
Πi = piXi − cXi − f = [α − β[
j Xj
L
]]Xi − cXi − f (20)
5.2.1 Autarky equilibrium
In one economy, the first-order condition of one firm is the derivation
with respect to Xi, holding the output of all the other firms constant.
Because of free entry or exit in this economy, marginal revenue has
to equal to marginal cost, which makes zero-profit for this firm. See
equation (21).
mr − mc = α − 2(
β
L
)Xi − (
β
L
)
j=i
Xj − c = 0 (21)
Under the assumption of cournot competition, every firm in this
economy choice the best output response to others. There will also
be symmetric solution, which means all firm in this economy will pro-
duce same amount output in autarky equilibrium. We denote X as the
output of a representative firm, and n the number of firms.
mr − mc = α − (
β
L
)(n + 1)X − c = 0 (22)
Additionally, under the assumption of free entry or exit, every firm
in this economy has zero profits in autarky equilibrium.
Πi = αX − (
β
L
)nX2
− cX − f = 0 (23)
And also from equation (22), we can get:
αX − (
β
L
)(n + 1)X2
− cX = 0 (24)
15
17. Here, we have two equations and two unknown variables, n and X.
solving them we get output per firm and the number of firm as follows:
X = [
Lf
β
]
1
2 (25)
n = (α − c)[
L
βf
]
1
2 − 1 (26)
5.2.2 Trade outcome
As we know, L represents the size of the economy. After trade, L will
double, meaning increasing common market size for both economies.
From the above two equations, there is a positive relationship between
L and output per firm X, as well there is a positive relationship between
L and the number of firms. Especially, equation (26) shows us that
the number of firms less than doubles when L doubles, because fixed
cost f will fall simultaneously. This means there are some firms exit
from the economy after free trade, which just proves what we have
discussed in figure 3.
6 Gains from trade under Cobb-Douglas pref-
erence condition
As what we have mentioned in last part, there is one limitation of the
quasi-linear case; one of the assumptions in that case is that there
is zero income elasticity of demand for X. However, in reality, the
situation is totally reverse. There are lots of increasing returns to scale
industries are producing goods with high income elasticity of demand
under imperfect competition. It shows that there is a higher degree of
consumption’s changes depends on the change of one representative
consumer’s income.
16
18. 6.1 First case: fixed number of firms
Now, take the case with Cobb-Douglas preference, we again denote
X as the output of an individual X firm, Y as the total output of Y and
let the price of Y numeraire, equal to 1. Firstly, we assume the number
of firm in each economy is fixed. We can see the utility function and
budget constraint in the following equations.
U = (nX)α
Y 1−α
(27)
Subject to:
I = L + Π = pnX + Y (28)
Where I is constant.
Equation (29) and (30) is the general demand function in this case.
nX =
αI
p
(29)
Or:
Y = (1 − α)I (30)
In this case, the price elasticity of demand is equal to one. In addi-
tion, we also assume there is more than one firm in this economy, so
the market share of each firm in this economy is less than one in au-
tarky. Because of what we have calculated in part 3, the ”markup”
which now is larger than zero and less than one, leads to positive
marginal revenue. In order to reach equilibrium in autarky, marginal
revenue has to be equal to marginal cost. See equation (31) and (32).
p(1 −
1
n
) = c (31)
Or:
p =
n
n − 1
c (32)
We must pay much attention here that there is not zero income
elasticity. Then, substituting the budget constrain to the general de-
mand function, we get the aggregate demand for X as follows:
nX = α
(L + Π)
p
= α
(L + n(pX − cX − f))
p
(33)
17
19. Rearranging the demand function by using equation (31) and (32),
we get the solution of X:
X = [
α(L − nf)
n(c)
]
n − 1
n − α
(34)
If we put two identical economies, each of them has same fixed
number of firms, trade together. The economy size L doubles, and
the total number of firms in the common market doubles as well. After
trade, the first part in the right-hand side of equation (34) does not
change, only the number of firms n doubles. Consequently, there is an
increase in output per firm. It is exactly the solution of the discussion
in figure 3.
6.2 Second case: free entry and exit
Here comes to free entry and exit case. The marginal revenue and
marginal cost equations are not changed.
p(1 −
1
n
) = c (35)
Because of free entry, there is zero profit for each firm in this econ-
omy. The following is the equation of it. Furthermore, we can draw the
demand function for X with no profit income.
pX = cX + f (36)
nX = α
L
p
(37)
Rearranging the above equations, we can get the output per firm
and the number of firms as follows:
n =
αL
f
(38)
X = [
αL
f
− 1]
f
c
(39)
18
20. sij =
Xij
Xj
(40)
Under the free entry or exit condition, the size of the economy L
increases which lead to an increase in the number of firms and the
output per firm. Again, we introduce the trade between the two iden-
tical countries. The market size will definitely double, leading some
firms exit the market in each economy. The calculation result here just
matches what we have discussed in figure 4.
7 Conclusion
Internal increasing return to scale industry leads to an imperfect com-
petition in autarky. It causes a distortion when reach the autarky equi-
librium. There is a market failure here because the outcome is not
optimal. The market price of this increasing return to scale goods ex-
ceeds its marginal cost. That means consumers pay much on this kind
of good, while at the same time producers can only produce little of
them.
Pro-competitive gains will be generated when we put this two iden-
tical economies together for trade. Trade introduces more competi-
tion into the common market, where the demand becomes more elas-
tic, and the market share of a single firm decreases. Thus those
firms would expand their outputs, leads to decreasing price, decreas-
ing”markup”, and decreasing average cost. There will be mutual ben-
efits for both firms and consumers. The social welfare improves with
lower price and larger market; while firms have higher revenue at the
same time because of lower cost.
We have also discussed two versions of market structure in this
model. One in which there is fixed number of firms before trade. Trade
releases the endogenous distortion in which it provides an incentive of
firms to produce more. However, there is free entry and exit of firms in
the other version. In this case, the more efficient production leads to a
negative profit in the market, because market price lower than average
19
21. cost. So, some firms will exit until re-establish zero-profit equilibrium.
In addition, more rigorous discussions with two specific preferences
cases as well as proved the general analyses what we have done be-
fore.
References
Markusen, J. R. / Maskus, K. E (2011) International Trade Theory,
Unpublished Manuscript, University of Colorado, Boulder, Chapter
11, pp 1 8, 10.
Paul R. Krugman, Maurice Obstfeld. (2008) International Economics:
Theory and Policy, 7th Ed., Pearson Education Asia Ltd., and Ts-
inghua University press copyright.
Elhanan Helpman, Paul R. Krugman. (1985)Market Structure and For-
eign Trade, Cambridge: MIT press.
Brander,James A. (1981) “Intra-industry trade in identical commodi-
ties”,Journal of International Economics 11, 1 - 14.
Helpman, Elhanan, and Paul A. Krugman(1985) Market Structure and
Foreign Trade, Cambridge: MIT press.
Horstmann, Ignatius J. and James R. Markusen (1986) “Up the aver-
age cost cureve: inefficient entry and the new protectionism”, Jour-
nal of International Economics 20, 225 - 228.
Markusen, James R. (1981) “Trade and the Gains from trade with
imperfect competition”,Journal of International Economics 11, 531-
551.
Melitz, Mark J. and Gianmarco I.P. Ottaviano (2008) “Market size,
trade, and productivity”,Review of Economic Studies 75, 295 - 298.
Venables, Anthony J. (1985) “Trade and trade-policy with imperfect
competition - the case of identical products and free trade”,Journal
of International Economics 19, 1 - 19.
20
22. A Appendices
5.1 quasi-linear preference with fix number of firms condition
Utility function :
U = αX − (
β
2
)X2
+ Y
subject to :
L + Π = pX + Y
Max(X)Ui = αX − (
β
2
)X2
+ L + Π − px
First order condition :
∂U
∂X
= α − βX − p = 0
Demand function :
p = α − βX
Profit function :
Π = pX − cX − f = [α − βX]X − cX − f
First order condition :
Π
X
= α − 2βX − c = 0
Autarky equilibrium :
X =
α − c
2β
World demand function after trade :
p = α −
β(Xi + Xj)
2
New profit function in country i :
Πi = [α −
β(Xi + Xj)
2
]Xi − cXi − f
First order condition :
Πi
Xi
= α −
β
2
(Xi + Xj) −
β
2
Xi − c = 0
α − c =
β
2
(2Xi + Xj)
α − c = βXi +
β
2
Xj
2(α − c) = 2βXi + βXj
21
23. where Xi = Xj
Trade equilibrium :
X∗
i = X∗
j =
2(α − c)
3β
5.2 quasi-linear preference under free entry and exit condition
Utility function :
U = αX − (
β
2
)X2
+ Y
Consumption per capita :
n
i
Xi
L
Demand function :
p = α − β[
j
Xj
L
]
Profit function :
Πi = piXi − cXi − f = [α − β[
j
Xj
L
]]Xi − cXi − f
Zero profit under free entry and exit :
mr = mc
First order condition :
∂Πi
Xi
= α − a(
β
L
)Xi − (
β
L
)
j=i
Xj − c = 0
Denote n as the number of firms :
mr − mc = α − (
β
L
)(n + 1)X − c = 0
The equation would not change if we times X to both sides :
αX − (
β
L
)(n + 1)X2
− cX = 0
Rearrange the profit function :
Πi = αX − (
β
L
)nX2
− cX − f = 0
Put the above two equations together :
αX − (
β
L
)nX2
− cX − f = αX − (
β
L
)(n + 1)X2
− cX = 0
(
β
L
)X2
[−n + (n + 1)] = f
X = [
Lf
β
]
1
2
22
24. Slug X = [Lf
β
]
1
2 into profit function :
α[
Lf
β
]
1
2 − (
β
L
n
Lf
β
− c[
Lf
β
]
1
2 = 0
(α − c)[
Lf
β
]
1
2 − fn − f = 0
(α − c)[
Lf
β
]
1
2 − f = fn
(α − c)[Lf
β
]
1
2
f
− 1 = n
n = (α − c)[
L
βf
]
1
2
6.1 Cobb-Douglas preference with fixed number of firms
Utility function :
U = (nX)α
Y 1−α
Subject to :
I = L + Π = pnX + Y
Use Lagrange equation :
L = (nX)α
Y 1−α
+ λ[I − pnX − Y ]
First order conditions :
∂L
∂X
= αnα
Xα−1
Y 1−α
− λpn = 0
∂L
∂Y
= (nX)α
(1 − α)Y −α
− λ = 0
∂L
∂λ
= I − pnX − Y = 0
Calculate λ from the above equations :
λ = (1 − α)nα
(
X
Y
)α
slug λ into the first order condition with respect to X :
αnα
(
X
Y
)α−1
= (1 − α)nα
(
X
Y
)α
pn
α = (1 − α)(
X
Y
)pn
Slug Y = I − pnX into the above equation :
α(I − pnX) = (1 − α)Xpn
αI − αpnX = pnX − αpnX
αI = pnX
demand function :
nx =
αI
p 23
25. Also slug X = I−Y
pn
into that equation :
αY = (1 − α)(
I − Y
pn
)pn
αY = I − Y − αI + αY
demand function :
Y = I − αI = (1 − α)I
Price elasticity of demand :
η = −[
p
X
∂X
∂p
]
In this case :
X =
1
n
αI
p
η = −[
p
X
(−1)(
αI
n
p−2
]
η =
1
X
αI
np
η = 1
Autarky equilibrium mr = mc :
p(1 −
1
n
) = c
p =
n
n − 1
c
Aggregate demand function :
nX = α
L + Π
p
= α
[L + n(pX − cX − f)]
p
npX = αL + αnpX − αncX − αnf
[(1 − α)np + αnc]X = α(L − nf)
slug p = n
n−1
c into it :
[(1 − α)n
n
n − 1
c + αnc]X = α(L − nf)
[(1 − α)
n
n − 1
+ α]X =
α(L − nf)
nc
[
n
n − 1
−
αn
n − 1
+
αn − α
n − 1
]X =
α(L − nf)
nc
[
n − α
n − 1
]X =
α(L − nf)
nc
X =
α(L − nf)
nc
(
n − α
n − 1
)
6.2 Cobb-Douglas preference under free entry and exit
24
26. Autarky equilibrium mr = mc :
p(1 −
1
n
) = c
p =
n
n − 1
c
Zero Profit condition Π = 0 :
pX = cX + f
Aggregate demand function :
nX = α
L
p
Rearrange the 1th and 2nd equations :
1
n
= 1 −
c
p
X =
f
p − c
Slug the above two equations to the demand function :
f
p − c
= α
L
p
(1 −
c
p
)
replace all p by c :
f
n
n−1
c − c
=
αL
n
n−1
c
[1 −
c
n
n−1
c
]
f
c
n−1
=
αL
n
n−1
c
[1 −
n − 1
n
]
f =
αL
n
1
n
f =
αL
n2
n =
αL
f
Slug p = n
n−1
c in the demand function :
nX = α
L
nc
n−1
X =
n − 1
n2
αL
c
Slug n = αL
f
into it as well :
X =
αL
f
− 1
αL
f
αL
c
X = [
αL
f
− 1]
f
c
25
