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Basic ModelSummary

International EconomicsLecture 2 # The Ricardian Model I

Weigang Liu

Beijing University of Technology

21st, January, 2019

Weigang Liu International Economics

Basic ModelSummary

Outline

1 Basic ModelAssumptionsProduction Possibility FrontierPrices and SpecializationPreferencesEquilibriumComparative saticsMathematical example


Basic ModelSummary

AssumptionsProduction Possibility FrontierPrices and SpecializationPreferencesEquilibriumComparative saticsMathematical example

David Ricardo (1772-1823)

A British political economist, one of the most influential ofthe classical economists along with Thomas Malthus, AdamSmith and James Mill.Ricardo was the third of 17 children.Died from an ear infection at 51.


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The Ricardian Model of Trade

Ricardo wrote his rst economics article at the age of 37.Ricardo’s theory of international trade was reformulated byJohn Stuart Mill. The term "comparative advantage" wasstarted by J. S. Mill and his contemporaries.


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Outline



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Model assumptions

Two countries

Two goods

One factor of production

Country-specific production technology


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Model assumptions

Two countries

Two goods




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Model assumptions

Two countries

Two goods




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Model assumptions

Two countries

Two goods




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Example

Two countries:China, USA

Two goods:ChipsT-shirts

One factor of production: LaborChina: 100 people.USA: 20 people

Country-specific production technology:China: Each worker can produce 1 chip or 1 T-shirtUSA: Each worker can produce 2 chips or 1 T-shirt.


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Example






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Example






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Example






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The production possibility frontier

Consider the China in autarky. (“Autarky” means in theabsence of trade).We first characterize the production possibilities of the China:

The set of production possibilities of the China is all thedifferent combinations of chips and T-shirts that the China canproduce.The production possibility frontier (PPF) is the most chips theChina can produce for each number of T-shirts produced.


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The production possibility frontier

Consider the China in autarky. (“Autarky” means in theabsence of trade).We first characterize the production possibilities of the China:

The set of production possibilities of the China is all thedifferent combinations of chips and T-shirts that the China canproduce.The production possibility frontier (PPF) is the most chips theChina can produce for each number of T-shirts produced.


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PPF for the example


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PPF for the example


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PPF for the example


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PPF for the example


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Warning: Math ahead

I am now going to define things formally using mathematics:This is a general plan for the course: first provide the intuition,then formalize it.

Why use math at all?Allows us to derive general insights (does not depend onparticular examples).Sometimes, models get too complicated to give full insight in apicture.

What do you need to know?Denitely understand the intuition.Work through the math on the problem sets (they are meantto be hard).Math is fair game for exams.


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Warning: Math ahead





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Warning: Math ahead





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The production possibility frontier generalized

Let QChipCHN denote the number of chips produced and QT−shirt

CHNdenote the number of T-shirts produced by the China.

Let LChipCHN and LT−shirt

CHN denote how much labor is required toproduce a chip or T-shirt by a China worker, respectively. Wecall this the unit labor cost;

Note that the unit labor cost is the inverse of workerproductivity.

Let LCHN be the number of workers in the China.


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Warning: Math ahead

Then the set of production possibilities is:{QChip

CHN ,QT−shirtCHN |LChip

CHNQChipCHN + LT−shirt

CHN QT−shirtCHN ≤ LCHN

}And the production possibility frontier is:

QChipCHN(QT−shirt

CHN ) ≡ maxQ>0Q

s.t.LChipCHNQ + LT−shirt


[Class question]: what is the solution to above equation?Answer:Q = LCHN

LChipCHN− LT−shirt

CHNLChip

CHNQT−shirt

CHN


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Warning: Math ahead






QChipCHN(QT−shirt

CHN ) ≡ maxQ>0Q





CHNLChip

CHNQT−shirt

CHN


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Warning: Math ahead






QChipCHN(QT−shirt

CHN ) ≡ maxQ>0Q





CHNLChip

CHNQT−shirt

CHN


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More general PPF


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Opportunity cost

The slope of the production possibility frontier is LT−shirtCHNLChip

CHN.

[Class question]: What is the economic interpretation of thisslope?To see this:

A worker can make 1LChip

CHNchip or 1

LT−shirtCHN

T-shirt.

Equivalently, it takes LChipCHN workers to make a chip and

LT−shirtCHN workers to make a T-shirt.

Hence, for each chip made, we could have made LT−shirtCHNLChip

CHNT-shirt..

We call this the opportunity cost of producing a football.


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Opportunity cost


CHN.



CHNchip or 1

LT−shirtCHN

T-shirt.




CHNT-shirt..



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Opportunity cost


CHN.



CHNchip or 1

LT−shirtCHN

T-shirt.




CHNT-shirt..



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Opportunity cost


CHN.



CHNchip or 1

LT−shirtCHN

T-shirt.




CHNT-shirt..



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Prices and specialization

Now suppose the price of chip and T-shirt in the China arepChip

CHN and pT−shirtCHN .

Take the perspective of a worker.

[Class question]: Suppose (as in the example) thatLChip

CHN = LT−shirtCHN = 1. Suppose that pChip

CHN = 1 andpT−shirt

CHN = 1. Then what would happen?


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CHN = 2. Then what would happen?Note: The units of the prices don’t actually matter. Whatmatters is the relative price pChip

CHNpT−shirt

CHN.

This is true throughout the course (and in economics moregenerally; it doesn’t matter if something is measured in centsor yuans).


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CHNpT−shirt

CHN.



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CHNpT−shirt

CHN.



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[Class question]: What is the relative price pChipCHN

pT−shirtCHN

at whichworkers are indifferent between producing chips and T-shirts?

Answer: when pChipCHN

pT−shirtCHN

= 1 workers earn the same amountregardless if they produce chips and T-shirts.


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[Class question]: What is the relative price pChipCHN

pT−shirtCHN

at whichworkers are indifferent between producing chips and T-shirts?

Answer: when pChipCHN

pT−shirtCHN

= 1 workers earn the same amountregardless if they produce chips and T-shirts.


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[Class question]: For a general technology LChipCHN and LT−shirt

CHN ,at what relative price are workers indifferent betweenproducing chips and T-shirts?

Answer: Worker revenue from producing chips is pChipCHN

LChipCHN

and

T-shirt is pT−shirtCHN

LT−shirtCHN

, so they are indifferent when:

pChipCHN

LChipCHN

= pT−shirtCHN

LT−shirtCHN

=⇒ pChipCHN

pT−shirtCHN

= LChipCHN

LT−shirtCHN


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[Class question]: For a general technology LChipCHN and LT−shirt

CHN ,at what relative price are workers indifferent betweenproducing chips and T-shirts?

Answer: Worker revenue from producing chips is pChipCHN

LChipCHN

and

T-shirt is pT−shirtCHN

LT−shirtCHN

, so they are indifferent when:

pChipCHN

LChipCHN

= pT−shirtCHN

LT−shirtCHN

=⇒ pChipCHN

pT−shirtCHN

= LChipCHN

LT−shirtCHN


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Preferences

To determine equilibrium, we need to specify the preferencesof workers.For simplicity, suppose there is a “representative agent" in theeconomy who receives utility:

W = U(CChipCHN ,C

T−shirtCHN )

where:CChip

CHN ,CT−shirtCHN are the quantity of chips and T-shirts in China

U is some given functionW is a number that tells you the total utility of therepresentative agent.

I will assume that dU(.)dCChip

CHN> 0 and dU(.)

dCT−shirtCHN

> 0.


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Preferences


W = U(CChipCHN ,C

T−shirtCHN )

where:CChip




CHN> 0 and dU(.)

dCT−shirtCHN

> 0.


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Preferences


W = U(CChipCHN ,C

T−shirtCHN )

where:CChip




CHN> 0 and dU(.)

dCT−shirtCHN

> 0.


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Indifference curves

We can use the preferences W = U(CChipCHN ,C

T−shirtCHN ) to rank

any consumption combination.That is, if U(CChip

CHN ,CT−shirtCHN ) > U(CChip

CHN , CT−shirtCHN ), then we

know that the representative agent would prefer to consume{CChip

CHN ,CT−shirtCHN } rather than {CChip

CHN , CT−shirtCHN }

The typical way to show this on a diagram is to drawindffierence curves.An indffierence curve is a set of all consumption bundles thatyield the same utility. Formally the indffierence curvecorresponding to utility W is:

IC(W ) = {CChipCHN ,C

T−shirtCHN |U(CChip

CHN ,CT−shirtCHN ) = W }


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Indifference curves














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Indifference curves














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Indifference curves














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Indifference curves

[Class question]: Why are the indifference curves curved likethey are?


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Indifference curves

[Class question]: Why are the indifference curves curved likethey are?


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Types of indifference curves

[Class question] Suppose that:

U(CChipCHN ,C

T−shirtCHN ) = CChip

CHN + βCT−shirtCHN

What will the indifference curves look like?Answer:

W = CChipCHN + βCT−shirt

CHN =⇒ CChipCHN = W − βCT−shirt

CHNso that the indifference curves will be straight lines withintercept W and slope −β.


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U(CChipCHN ,C








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U(CChipCHN ,C








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What do the indifference curves like?


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Types of indierence curves

[Class question] Supposethat:U(CChip

CHN ,CT−shirtCHN ) = min{(CChip

CHN , βCT−shirtCHN }

What will the indierence curves look like?Answer:

The key thing to note if you consume CChipCHN , then the utility

will be the same if you consume βCT−shirtCHN or βCT−shirt

CHN + x foranyx > 0Hence these preferences (known as “Leontief” preferences) willhave a “kink” at {x ;βx}


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What do the indifference curves like?


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Autarkic equilibrium

We can now (nally!) define the autarkic equilibrium.[Class question]: What are the exogenous model parameters?

Answer: Productivities LChipCHN and LT−shirt

CHN , population LCHN ,and preferences U(.) (and the same for USA).

[Class question]: What are the endogenous model outcomes?Answer:Production QChip

CHN and QT−shirtCHN , consumption CChip

CHN

and CT−shirtCHN , and relative prices pChip

CHNpT−shirt

CHN(and the same for

USA).


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CHN


CHNpT−shirt


USA).


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CHN


CHNpT−shirt


USA).


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DefinitionFor any set of productivities LChip

CHN and LT−shirtCHN , population LCHN ,

and preferences U(.), equilibrium is defined as a set of productionQChip


CHN and CT−shirtCHN , and

relative prices pChipCHN

pT−shirtCHN

such that...

[Class question]: Any guesses as to what the equilibriumconditions are?

The utility of the representative agent is maximized.Workers maximize their revenue.Consumption is equal to production.

[Class question]: Which of the three equilibrium conditionswill change when we introduce trade?

Answer: the last one.Denfing the equilibrium:Weigang Liu International Economics

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pT−shirtCHN

such that...





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pT−shirtCHN

such that...





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Autarkic Equilibrium Recap

Equilibrium prices are “pinned-down” by the productiontechnology: pChip

CHNpT−shirt

CHN= LChip

CHNLT−shirt

CHN

[Class question]: Will this always be the case?Answer: No, but if it is not the case, the country willcompletely specialize in the production of one good.

Total quantity produced is determined by the point where theindierence curve lies tangent to the production possibilityfrontier.

This depends on preferences.

Consumption is simply equal to production.


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CHNpT−shirt

CHN= LChip

CHNLT−shirt

CHN






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CHNpT−shirt

CHN= LChip

CHNLT−shirt

CHN






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CHNpT−shirt

CHN= LChip

CHNLT−shirt

CHN






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“Comparative static” example

Consider the China in autarky. (“Autarky” means in theabsence of trade).A comparative static tells us how an equilibrium objectchanges as we change model fundamentals.These make good exam questions.

For example: if we decrease the productivity of chip, whathappens to relative prices and the equilibriumproduction/consumption of chip and T-shirt?


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So reducing the productivity of chips:Reduces the equilibrium consumption and production of chips.Increases the relative price of chips to T-shirts.

[Class question]what about the effect on the equilibriumconsumption and production of T-shirt?

It depends.


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So reducing the productivity of chips:Reduces the equilibrium consumption and production of chips.Increases the relative price of chips to T-shirts.

[Class question]what about the effect on the equilibriumconsumption and production of T-shirt?

It depends.


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Outline



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Mathematical example

As above:China has population of workers LCHN .Workers produce chips and T-shirts with unit labor costs LChip

CHNand LT−shirt

CHN , respectively.

Suppose workers havepreferences::U(CChip

CHN ,CT−shirtCHN ) = (CChip

CHN)α(CT−shirtCHN )1−α}

α ∈ (0, 1). These preferences are known as Cobb-Douglaspreferences.One of my go-to preferences for exams (other go-to: Leontief).

Question: What is the equilibrium quantity of chips andT-shirts consumed per worker?


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CHNand LT−shirt

CHN , respectively.







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CHNand LT−shirt

CHN , respectively.







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Mathematical example: Production

Step #1(a): We calculate the PPF. From above, recall thatlabor can be used either to produce chips or T-shirts.

QChipCHNLChip

CHN + QT−shirtCHN LT−shirt

CHN = LCHN

Can then write the quantity produced of chips as a function ofT-shorts.

QChipCHN = LCHN

LChipCHN

− QT−shirtCHN

LT−shirtCHNLChip

CHN


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Step #1(a): We calculate the PPF. From above, recall thatlabor can be used either to produce chips or T-shirts.

QChipCHNLChip

CHN + QT−shirtCHN LT−shirt

CHN = LCHN

Can then write the quantity produced of chips as a function ofT-shorts.

QChipCHN = LCHN

LChipCHN

− QT−shirtCHN

LT−shirtCHNLChip

CHN


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Step #1(b): We calculate the relative price. In autarky, therelative price of chips to T-shirts is equal to the (negative) ofthe slope of the PPF:

pT−shirtCHNpChip

CHN= − dQChip

CHNdQT−shirt

CHN

[Class question]: What is the intuition? Is this true withtrade?In this linear case, we then have (as we found above) that:

pChipCHN

pT−shirtCHN

= − LChipCHN

LT−shirtCHN


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Step #1(b): We calculate the relative price. In autarky, therelative price of chips to T-shirts is equal to the (negative) ofthe slope of the PPF:

pT−shirtCHNpChip

CHN= − dQChip

CHNdQT−shirt

CHN

[Class question]: What is the intuition? Is this true withtrade?In this linear case, we then have (as we found above) that:

pChipCHN

pT−shirtCHN

= − LChipCHN

LT−shirtCHN


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Mathematical example: Consumption

Step #2(a): We calculate the wage of a worker.Assume the price of T-shirts is 1 [Why is this okay?].WA worker can produce 1

LT−shirtCHN

. Hence her wage is:

wCHIN = pT−shirtCHN

LT−shirtCHN

= 1LT−shirt

CHN

[Class question] what would happen if I had calculatedwagOverivew of the trade in China and Worldes using herchips production?


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Step #2(a): We calculate the wage of a worker.Assume the price of T-shirts is 1 [Why is this okay?].WA worker can produce 1

LT−shirtCHN

. Hence her wage is:

wCHIN = pT−shirtCHN

LT−shirtCHN

= 1LT−shirt

CHN

[Class question] what would happen if I had calculatedwagOverivew of the trade in China and Worldes using herchips production?


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Step #2(b): We calculate the equilibrium consumption of aworker.

Maximize utility subject to the worker’s budget constraint:

maxCChipCHN ,C

T−shirtCHN

(CChipCHN)α(CT−shirt

CHN )1−α

s.t.CChipCHNpChip

CHN + CT−shirtCHN = wCHN


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Step #2(b): We calculate the equilibrium consumption of aworker.

Maximize utility subject to the worker’s budget constraint:

maxCChipCHN ,C

T−shirtCHN

(CChipCHN)α(CT−shirt

CHN )1−α

s.t.CChipCHNpChip

CHN + CT−shirtCHN = wCHN


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To solve the problem, we have

CT−shirtCHN = (1− α)wCHN

CChipCHNpChip

CHN = wCHN

Implication: With Cobb-Douglas preferences, always spend aconstant fraction of income on each good, where fractionpinned down by exponent!


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To solve the problem, we have

CT−shirtCHN = (1− α)wCHN

CChipCHNpChip

CHN = wCHN

Implication: With Cobb-Douglas preferences, always spend aconstant fraction of income on each good, where fractionpinned down by exponent!


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Mathematical example: Equilibrium

Step #3: Combine production and consumption equilibriumrelationships:

Prices: pChipCHN = LChip

CHNLT−shirt

CHN,pT−shirt

CHN = 1Wages:wCHIN = 1

LT−shirtCHN

Consumption: CT−shirtCHN = (1− α)wCHN , CChip

CHNpChipCHN = wCHN .

Answer to the question:

CT−shirtCHN = 1− α

LT−shirtCHN

,CChipCHN = α

LChipCHN

[Class questions] whats the intuition for and the unit costs?Why doesn’t the labor supply affect the production decision?


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CHNLT−shirt

CHN,pT−shirt


LT−shirtCHN





LT−shirtCHN

,CChipCHN = α

LChipCHN



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CHNLT−shirt

CHN,pT−shirt


LT−shirtCHN





LT−shirtCHN

,CChipCHN = α

LChipCHN



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Summary

Assumptions of the model is introduced.

PPF is introduced.

How to find the equilibrium in mathematical language.


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Summary


PPF is introduced.



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Summary


PPF is introduced.



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Homework #1

Find the equilibrium if the preference is:U(CChip


CHN , βCT−shirtCHN }.

U(CChipCHN ,C

T−shirtCHN ) = log(CChip

CHNβCT−shirtCHN ).


Appendix For Further Reading

Further Reading I

Krugman, Obstfeld, and Melitz.Chapter 3 in International Economics.Pearson Press, 2018.


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