International Economics Assignment: Production, Trade, and Welfare

This document presents a comprehensive solution to an International Economics assignment. The assignment explores a Krugman-style model, examining firm pricing strategies, average costs, profit maximization, and equilibrium conditions. The solution details the derivation of optimal pricing, the impact of production levels on pricing, and the calculation of average costs. Furthermore, it addresses profit per firm, imposition of zero-profit conditions to determine equilibrium production, the number of firms, and utility levels. The assignment also delves into production functions, utility maximization problems in home and foreign countries, and the effects of trade and wages on welfare. The analysis includes considerations of population mobility and its relationship to trade shares, offering a thorough exploration of international economic principles. The document also includes references to relevant academic literature.

International Economics
Problem 1
Part 1
The Pricing Strategy
Assume that firm I generates profits as given by :
⫪i=[P(Yi)-C]Yi
You can maximize the profit output with respect to Yi
Let Qj be the profits generated by other firms.
d ⫪ i
dYi =P(Y)-c +yi
dPdY
dQdYi =0
But dY
dYi =1 and dYj
dYi =0
Hence the reaction curve for the firm is P(Y) +Qi
dP
dY =c
Since there equal sizes of firms which in n number,
Yi= Y
n ,Thus
P(Y)= YdP
ndY =c
And P−c
p = 1
n
YdP
PdY = −1
nED
But Ed =nED
Proof of Profit Maximizing Price
Consider either a linear or isoelastic demand curve.
Assume that the demand curve is given by;

Log Yi=σ -b log P +βlog q
But q is the perceived sales that is made i.e demand funcrion.
Yi is the actual total demand
Beta is positive i.e β>0 when the network effect is a negative.
Yi=q at the equilibrium point,Hence;
Log Yi = σ
1−β - b
1−β log P
b
1−β log P= σ
1−β –log Yi
At the equilibrium, Marginal Cost =Marginal revenue.
Hence P= σ
σ−1MC= σ
1−σ Yiβ
Part 2
Consider The total return to be Yi
You will realize that at a positive beta of β>0,the network effect is equal to a
negative, hence the returns Yi would be low.
But B<1 of a kind of an isoelastic demand curve.
Hence at B<1, the elastic demand curve can make sense as the marginal revenue
is equal to the marginal.Yi is at its optimal level
Problem 2.
Average cost of an individual firm is the cost per unit of inputs during production.
For a higher output to be achieved in the short-run,a firm have to use more labor
and more inputs to facilitate the production.

However higher implementation of inputs during production increases the
overall total cost i.e. cost is directly proportional to the level of inputs as
expressed below ;
C= F(L,w,r,e.t.c) i.e. Cost being a function of inputs.
Total Cost=Fixed Cost +Average Cost
Total cost is the sum of all cost of inputs of production.
Suppose the total number of inputs is Q,then it means,
TC
Q = TFC
Q = TAC
Q
And
ATC=AFC +AVC
Problem 3
Profit of a firm is the gain a for firm from the sale of goods and services’ firm will
be more satisfied with a higher profits rather than just a sale.
Profit=Total Revenue-Total Cost.
P=TR-TC
TR –is the total gain for a firm from the sale of a given amount of goods.
TR=pq
Where p is price is quantity sold.
Average Revenue is revenue per unit of goods sold.
AR= TR
q =p

Marginal Revenue (MR) is the extra amount of revenue generated from sale of
additional units of goods.
MR= TR
q =p i.e in every additional unit of q,the marginal cost =p
Problem 4
Assume that the production function is ;
Y=f(L1.L2,,………)where Lf is the function of Lf is the labor function.
f (ʎL1,ʎ2,……….)
Given that L is the labour,
f (L)=aL-b
f(L,K)=aLαKβ
logf(L,K)=aL log L +aklogK +akL(logk-logL]2
The last firm enters the market when entry =σ
⫪I = σ
N 2 =σ
Hence the equilibrium number of firms is given by;

N* =
√ t s
σ
But
F(L,β) =kαLβ= kαLβ n-(α+β)
Fk=αkα-1Lβ =αkα-1Lβn1-(α+β)
FL=Kαβlβ-1 =KαβLβ-1n1-(α+β)
Where α is the share capital and β is the labor share.
U= P
V (n∗−1) [ 2
A + βn∗A
n∗¿ ¿ - V
u ¿¿
Assume θ1 =1,and μ ( n )= n
n−1
U= P∗¿
α + β ¿ (n∗−1)[ 2
A (β +
α + β
n∗2
n∗−1 n∗¿
n8−1−(α + β)
¿ ) n∗n
n∗¿¿
Given that the conditions are sufficient,
2
A (β +
α + β
n∗2
n∗−1 n∗¿
n8−1−(α + β)
¿ ) n∗n
n∗¿¿ <0
Β<(β +
α +β
n∗2
n∗−1 n∗¿
n8−1−(α + β)
¿ ) n∗n
n∗¿¿
Hence
U= L( 1
σ )
α - 1+σβ
σ (1+ β )( α (σ−1)(1+β )
1+σβ ) 1
1+ β ¿ ¿)σ−1
σ
Problem 5

Consider Β<(β +
α +β
n∗2
n∗−1 n∗¿
n8−1−(α + β)
¿ ) n∗n
n∗¿¿
Competition effects is zero with fixed marked ups level i.e (n*<1).Hence the
natural conditions are often interfered with ,that is,β not less zero and the
undershooting can never increase.
This explains the logic that without persistent effect on productivity, then
undershooting of the long term level can never arise. Any effects that weakens
the long-run persistent effect as competition reduces the chances of
undershooting.
SECTION 2
Problem 1
Consider a single firm whose productivity is represented as below:
aYi=Yα1
Where α is the share capital in production.
Α is always assumed be α>0 because productivity is directly proportional to the
share capital. An increase in capital share would often leads to increased
production.
Problem 2
Consider a production return of Yi
P∑
i=1
n
aYi ≤ p ∑
b
m
ya +p∑
c=a
❑
xi + Y’a -∑
b =1
n
yb
≤P∑
c=1
m
Yc +PaY +PaY’ + P∑
a =1
m
wa -P∑
a =1
n
Ya

≤ P ∑
a =1
m
wa-PYa + PY’
But w=1
=1
Problem 3
Consider the utility maximization problem i.e A and B producing goods in Home
and in foreign countries.
The two firms have to make their own unique decisions on whether to remain in
autarky.
You will realize that the maximum utility are subject to X=lx/ax , y= 1
y /ay and lx + ly
=1.
But at equilibrium ,X=α/ax ,y =(1-α)/ay for firm A and X* =α/a* ,y*=(1-α)/a*y
for firm B.
Thus ,the individual utility in home and foreign are given by;
U=(α/ax)α[(1-α)/ay]1-α and U* =(α/a) Type equation here .
=C 12/3 C1/32
Problem 4
Let T→∞ where T is aka life-time budget constraint.
You can then Impose the restriction limT→
∞ Bt +T +1
( 1+r )T =0
(1+r)Bt =∑
i=t
∞
¿¿)i-T(Ci-Yj)

Assume Y1=C1,Y2=C2 And get
(1+r)t=Yt(C-1)∑
i=t
∞ 1+ g
1+r )i-T
(1+r)βt =Yt(C-1)( 1+r
r−g )
c−1
r−g = βt
Yt
From the infinite sum,
∑
i=t
∞ 1+ g
1+r )i-T converges if
g <r.Hence (1+g)/(1+r)<1.
Problem 5
1 home good and 2 foreign goods, with wages w=1 and w*=2,his payoff would
be ;
1X2-2X2=-2.
You will realize that decision makers with even smaller values values of w* will
obtain negative wages. Hence no person should choose a foreign wage.
Problem 6
Let p be the probability of wages where p ∈[ ( 0,1 ) ]
W[Home]=p*1 +(1-p)*2=2-p
W[Foreign]=p*2+(1-p)*1=p+1
This implies that it is better chance to produce both home goods if and only if 2-p
≥p+1

In addition it would be a better chance to produce both foreign goods if and
only if
p+1 ≥2-p
Problem 7
Consider using the relationship of mobility and welfare
V= ( 1
σF ⫪ nn ) 1
σ−1 Ln−¿¿¿ ¿. = v
Where population mobility requires workers to receive the same actual income in all populated
regions.
On rearranging the equation ,we get;
Ln=[ ( 1
σF ⫪nn ) 1
σ −1 H n(1−α)
α ( 1−α
α )1−α
( σ
σ −1 )α
v
]^ σ −1
σ ( 1−α ) −1
You realize that the agglomeration force is sufficiently strong in comparison to the congestion
force from an inelastic supply of land i.e (0<σ(1-α)<1).Hence there is increase in population for
each region for any given trade share for itself.This ensures existence of a stable equilibrium
with a spread population distribution.
You are also to consider;
Var{logLn}=( (1−α )(σ −1)
σ ( 1−α ) −1 ¿ ¿)2Var{Log Hn} + (1−α )(σ −1)
σ ( 1−α ) −1 ¿ ¿)2 Var{log⫪nn}-2( (1−α )(σ −1)
σ ( 1−α ) −1 ¿ ¿)¿
)Covar{logHn,log⫪nn}
From the population mobility,it suggests that the representative agents obtain equal welfare
across all regions.As a result,the location’s trade share is not statistically sufficient to support
the of the representative agent welfare gains from the trade of goods as α→1

REFERENCES
1.McFadden, D. (2011). The mathematical theory of demand models.Toronto: Lexington Books
2.Salanie, B. (2003).The Economics of Taxation.Cambridge:MIT Press .
3.Pigou, A.C. (2008) The Economics of Welfare. 2nd ed.London: Macmillan.
4. Dixon,D. & Rankin, N. (2010).The Economics of Welfare.Cambridge:Cambridge University
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