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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 dont 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, Lets 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.]

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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 }

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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

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SB

FB

We can use the preferences W = U CUS

, CUS

to rank

any consumption combination.

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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?

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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 workers 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

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)

?

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

I

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 doesnt 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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