ECON 561 SDSU Ricardian Model and Open Economy Equilibrium Exam

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ECON 561 Exam 1
Instructions: There are 3 questions for a total of 100 points. You have one day to complete the exam. This is an
open book exam – you may use notes, slides, or reference any materials. You may NOT collaborate with anyone
else. I will be watching for suspicious activity. Please answer each question on a new page of paper, and put it in
order before you turn it in. Show your work: even if you don’t get the answer correct, I want to be able to give
you partial credit! Write your name and the number of the question at the top of each piece of paper. Good luck!
1. (10 points) If a Gravity Equation is given as:
ln Xijt = ? + ? 1 ln GDPit + ? 2 ln GDPjt + ? ln distij + eijt ,
where, i =exporter, j =importer, t =year.
• Do you expect the coefficient on ? 1 to be postive or negative, why?
• Do you expect the coefficient on ? to be postive or negative, why?
• If you want to examine the effect of tariffs on exports, which variable should you include in the above
equation? What the sign of the estimated coefficient on that variable would be?
• Why is there no t in the varialbe ln distij ? Does the variable you just included have t?
2. Consumer Theory and Constrained Optimization (20 points)
Solve the following Cobb-Douglas Utility Maximization Problem. Show all your steps!
?
max UZ1 ,Z2 = Z11 Z2?2 , s.t.p1 Z1 + p2 Z2 = 100, 0 < ?1 hard (b) Type 2: Both countries completely specialize in one products. i. Prices –> hard
ii. Market Clearning –> easy
Solve the Free Trade Equilibrium
Suppose workers have the same preferences as before, this is true for both economies (for simplicity):
US
US
US ?
US 1? ?
U (CFB
, CSB
) = (CFB
) (CSB
) , ? = 0.5
LUS = 10, L MEX = 5;
US
MEX
MEX
?US
= 2, ?SB
=1
FB = 1, ?SB = 1, ? FB
I. What is the equilibrium world price
II. What is the equilibrium consumption of workers in the U.S. and Mexico?
First, Let’s solve for the Type 1 case:
• Step 1: if US is producing both products –> then the new world price ratio is the same as US autarkic
price ratio –> pin down price sytem
• Step 2: pin down wages in both economies –> using either numeraire method or directly solve for the
budge constraint –> main goal –> get rid of w wage.
• Step 3: solve for Equilibrium consumptions for both economies’ agents
• Step 4: get the national consumption level
• Step 5: use market clearing condition to solve for optimal production. DONE!
• Type II is hard to solve for. Reverse engineering. [This will be in the homework.]
1
Warning: Math ahead
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I am now going to define things formally using
mathematics.
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Why use math at all?
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Allows us to derive general insights (does not depend on
particular examples).
Sometimes, models get too complicated to give full
insight in a picture.
What do you need to know?
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(This is a general plan for the course: first provide the
intuition, then formalize it).
Definitely understand the intuition.
Work through the math on the problem sets (they are
meant to be hard).
Math is fair game for exams.
Ask questions and slow me down if I go too fast!
The production possibility frontier generalized
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FB
Let QUS
denote the number of footballs produced and
SB
QUS
denote the number of soccer balls produced by the
U.S.
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FB
SB
Let ?US
and ?US
denote how much labor is required to
produce a single football or soccer ball by a U.S. worker,
respectively. We call this the unit labor cost;
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Note that the unit labor cost is the inverse of worker
productivity.
Let LUS be the number of workers in the U.S.
The production possibility frontier generalized
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Then the set of production possibilities is:
FB
SB
FB FB
SB SB
{QUS
, QUS
| ?US
QUS + ?US
QUS ? LUS }
I
And the production possibility frontier is:
SB
FB
SB
FB FB
QUS
QUS
? max Q s.t. ?US
Q + ?US
QUS ? LUS
Q>0
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[Class question]: what is the solution to above equation?
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Answer:
SB
FB
QUS
QUS
=
LUS
SB
?US
FB
?US
Q FB
SB US
?US
More general PPF
More general PPF
More general PPF
Opportunity cost
?FB
US
.
?SB
US
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The slope of the production possibility frontier is
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[Class question]: What is the economic interpretation of
this slope?
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To see this:
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SB soccer balls or 1/?FB
A worker can make 1/?US
US
footballs.
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SB workers to make a soccer ball
Equivalently, it takes ?US
FB
and ?US workers to make a football.
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Hence, for each football made, we could have made
soccer balls.
We call this the opportunity cost of producing a
football.
?FB
US
?SB
US
Preferences
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To determine equilibrium, we need to specify the
preferences of workers.
For simplicity, suppose there is a “representative agent” in
the economy who receives utility:
SB
FB
W = U CUS
, CUS
,
where:
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SB is the quantity of soccer balls consumed in the
CUS
United States.
FB is the quantity of footballs consumed in the United
CUS
States.
U is some (given) function
W is a number that tells you the total utility of the
representative agent.
I will assume that
SB ,C FB
@U (CUS
US )
SB
@CUS
> 0 and
SB ,C FB
@U (CUS
US )
FB
@CUS
> 0.
Indi?erence curves
I
SB
FB
We can use the preferences W = U CUS
, CUS
to rank
any consumption combination.
I
SB
FB
SB
FB
That is, if U CUS
, CUS
> U C?US
, C?US
, then we know
that the representative agent would prefer to consume
SB
FB
SB
FB
CUS
, CUS
rather than C?US
, C?US
.
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The typical way to show this on a diagram is to draw
indi?erence curves.
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An indi?erence curve is a set of all consumption
bundles that yield the same utility. Formally the
indi?erence curve corresponding to utility W1 is:
SB
FB
SB
FB
IC (W1 ) = CUS
, CUS
| U CUS
, CUS
= W1
Indi?erence curves
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[Class question]: Why are the indi?erence curves curved
like they are?
Types of indi?erence curves
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[Class question] Suppose that:
SB
FB
SB
FB
U CUS
, CUS
= CUS
+ CUS
What will the indi?erence curves look like?
I
Answer:
SB
FB
W = CUS
+ CUS
()
SB
CUS
=W
FB
CUS
so that the indi?erence curves will be straight lines with
intercept W and slope
.
Types of indi?erence curves
Types of indi?erence curves
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[Class question] Suppose that:
SB
FB
SB
FB
U CUS
, CUS
= min CUS
, CUS
What will the indi?erence curves look like?
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SB ,
Answer: The key thing to note if you consume CUS
FB or
then the utility will be the same if you consume CUS
FB
CUS + x for any x > 0.
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Hence these preferences (known as “Leontief”
preferences) will have a “kink” at {x, x}.
Types of indi?erence curves
Autarkic equilibrium
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We can now (finally!) define the autarkic equilibrium.
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[Class question]: What are the exogenous model
parameters?
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SB and ?FB , population L ,
Answer: Productivities ?US
US
US
and preferences U (·, ·) (and the same for Mexico).
[Class question]: What are the endogenous model
outcomes?
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SB and Q FB , consumption C SB
Answer: Production QUS
US
US
FB , and relative prices
and CUS
Mexico).
FB
pUS
SB
pUS
(and the same for
Autarkic Equilibrium
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Defining the equilibrium:
SB
FB
“For any set of productivities ?US
and ?US
,
population LUS , and preferences U (·, ·),
equilibrium is defined as a set of Production
SB
FB
SB
FB
QUS
and QUS
, consumption CUS
and CUS
, and
FB
pUS
relative prices pSB such that…”
US
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[Class question]: Any guesses as to what the equilibrium
conditions are?
1. The utility of the representative agent is maximized.
2. Workers maximize their revenue.
3. Consumption is equal to production.
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[Class question]: Which of the three equilibrium
conditions will change when we introduce trade?
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Answer: the last one.
Autarkic Equilibrium
Autarkic Equilibrium
Autarkic Equilibrium
Autarkic Equilibrium
Autarkic Equilibrium Recap
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Equilibrium prices are “pinned-down” by the production
technology:
FB
FB
pUS
?US
=
SB
SB
pUS
?US
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[Class question]: Will this always be the case?
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Answer: No, but if it is not the case, the country will
completely specialize in the production of one good.
Total quantity produced is determined by the point where
the indi?erence curve lies tangent to the production
possibility frontier.
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This depends on preferences.
Consumption is simply equal to production.
“Comparative static” example
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Economists love to derive “comparative statics”
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A comparative static tells us how an equilibrium object
changes as we change model fundamentals.
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These make good exam questions.
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For example: if we decrease the productivity of football
production, what happens to relative prices and the
equilibrium production/consumption of footballs and
soccer balls?
“Comparative static” example
“Comparative static” example
“Comparative static” example
“Comparative static” example
“Comparative static” example
“Comparative static” example
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So reducing the productivity of footballs:
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Reduces the equilibrium consumption and production of
footballs.
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Increases the relative price of footballs to soccer balls.
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Has no a?ect on the equilibrium consumption and
production of soccer balls.
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More generally, the e?ect could go either way depending
on the strength of the income and substitution e?ects.
Mathematical example
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As above:
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US has population of workers LUS .
Workers produce soccer balls and footballs with unit
SB and ?FB , respectively.
labor costs ?US
US
Suppose workers have preferences:
FB
SB
FB
U CUS
, CUS
= CUS
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SB
CUS
1
,
where 2 (0, 1).
These preferences are known as Cobb-Douglas
preferences.
One of my go-to preferences for exams (other go-to:
Leontief).
Question: What is the equilibrium quantity of footballs
and soccer balls consumed per worker?
Mathematical example: Production
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Step #1(a): We calculate the PPF. From above, recall
that labor can be used either to produce footballs or
soccer balls:
FB FB
SB SB
QUS
?US + QUS
?US = LUS
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Can then write the quantity produced of soccer balls as
a function of footballs:
SB
QUS
=
LUS
SB
?US
FB
?US
FB
QUS
SB
?US
Mathematical example: Production (ctd.)
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Step #1(b): We calculate the relative price. In autarky,
the relative price of footballs to soccer balls is equal to
the (negative) of the slope of the PPF:
FB
pUS
=
SB
pUS
SB
@QUS
FB
@QUS
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[Class question: What is the intuition? Is this true with
trade?]
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In this linear case, we then have (as we found above)
that:
FB
FB
pUS
?US
=
SB
SB
pUS
?US
Mathematical example: Consumption
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Step #2(a): We calculate the wage of a worker.
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Assume the price of soccer balls is 1 [Why is this okay?].
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A worker can produce ?1SB soccer balls. Hence her wage
US
is:
1
1
SB
wUS = pUS
? SB = SB
?US
?US
.
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[Class question: what would happen if I had calculated
wages using her football production?].
Mathematical example: Consumption (ctd)
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Step #2(b): We calculate the equilibrium consumption of
a worker.
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Maximize utility subject to the worker’s budget
constraint:
max
FB ,C SB
CUS
US
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FB
CUS
1
FB FB
SB
CUS + CUS
? wUS
s.t. pUS
The Lagrangian is:
FB
L : CUS
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SB
CUS
SB
CUS
1
FB FB
SB
pUS
CUS + CUS
wUS
Yielding first order conditions:
?
SB
CUS
FB
CUS
?1
FB
= pUS
and (1
)
?
FB
CUS
SB
CUS
?
=
Mathematical example: Consumption (ctd)
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Recall first order conditions:
? SB ?1
CUS
FB
= pUS
and (1
FB
CUS
Combining both equations to get rid of
SB
CUS
= (1
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FB
CUS
SB
CUS
?
=
yields:
FB FB
) pUS
CUS
Using the budget constraint, this implies:
SB
CUS
= (1
I
)
?
FB FB
) wUS and pUS
CUS = wUS
Implication: With Cobb-Douglas preferences, always
spend a constant fraction of income on each good, where
fraction pinned down by exponent!
Mathematical example: Equilibrium
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Step #3: Combine production and consumption
equilibrium relationships:
?FB
US
?SB
US
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FB =
Prices: pUS
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Wages: wUS =
I
SB = (1
Consumption: CUS
1
?SB
US
FB =
) wUS and CUS
wUS
FB
pUS
Answer to the question:
SB
CUS
=
I
SB = 1
and pUS
1
FB
and CUS
= FB
SB
?US
?US
[Class questions: whats the intuition for and the unit
costs? Why doesn’t the labor supply a?ect the production
decision?]

Purchase answer to see full
attachment

Tags:
microeconomics

Constrained Optimization

Ricardian Model

tariffs on exports

open economy equilibrium

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