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The Complexity of Models of International Trade Christopher The Complexity of Wilkens Models of International Trade Complexity of Equilibria Models Complexity Results Upside-Down Christopher Wilkens Reduction Example: Ricardo with

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

The Complexity of Models of International Trade

Christopher Wilkens

UC Berkeley

WINE ’09

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

The Big Picture: Computer Science

Computer science questions: (last decade)

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

The Big Picture: Computer Science

Computer science questions: (last decade)

Can we compute market equilibria?

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

The Big Picture: Computer Science

Computer science questions: (last decade)

Can we compute market equilibria? What about special models?

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

The Big Picture: Microeconomics

Welfare economics:

What determines prices?

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

The Big Picture: Microeconomics

Welfare economics:

What determines prices?

International Trade:

What determines the pattern of trade between countries (agents)?

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

The Big Picture: Microeconomics

Trade Example:

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

The Big Picture: Microeconomics

Trade Example:

My advisor is a very good researcher.

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

The Big Picture: Microeconomics

Trade Example:

My advisor is a very good researcher. I am a mediocre researcher compared to my advisor.

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

The Big Picture: Microeconomics

Trade Example:

My advisor is a very good researcher. I am a mediocre researcher compared to my advisor. My advisor pays me to do research. Why?!

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

The Big Picture: Microeconomics

Trade Example:

My advisor is a very good researcher. I am a mediocre researcher compared to my advisor. My advisor pays me to do research. Why?! This should be explained by the model

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

The Big Picture: Microeconomics

The Ricardo model (1800s):

Efficiency in production determines trade pattern

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

The Big Picture: Microeconomics

The Ricardo model (1800s):

Efficiency in production determines trade pattern

The Heckscher-Ohlin model (1930s):

The availability of resources determines trade pattern

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

Ricardo’s Law of Comparative advantage

(Ricardo, 1800’s) Given equilibrium wages, easy to recover equilibrium pattern of trade with 2 countries

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

Ricardo’s Law of Comparative advantage

(Ricardo, 1800’s) Given equilibrium wages, easy to recover equilibrium pattern of trade with 2 countries Order goods and wages wi by relative production efficiencies aij: a11 a21 ≥ · · · ≥ a1k a2k > w1 w2 > a1(k+1) a2(k+1) ≥ · · · ≥ a1m a2m

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

Ricardo’s Law of Comparative advantage

(Ricardo, 1800’s) Given equilibrium wages, easy to recover equilibrium pattern of trade with 2 countries Order goods and wages wi by relative production efficiencies aij: a11 a21 ≥ · · · ≥ a1k a2k > w1 w2 > a1(k+1) a2(k+1) ≥ · · · ≥ a1m a2m Country 1 produces goods on left of w, country 2 produces goods on right

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

Ricardo’s Law of Comparative advantage

(Ricardo, 1800’s) Given equilibrium wages, easy to recover equilibrium pattern of trade with 2 countries Order goods and wages wi by relative production efficiencies aij: a11 a21 ≥ · · · ≥ a1k a2k > w1 w2 > a1(k+1) a2(k+1) ≥ · · · ≥ a1m a2m Country 1 produces goods on left of w, country 2 produces goods on right Problem is back to computing equilibrium

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

The Market

Goods

m tradable goods K non-tradable raw materials

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

The Market

Goods

m tradable goods K non-tradable raw materials

Agents

n agents utility functions ui : Rm → R

Here, ui is linear or Leontief

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

The Market

Goods

m tradable goods K non-tradable raw materials

Agents

n agents utility functions ui : Rm → R

Here, ui is linear or Leontief

endowment ei ∈ RK of raw materials production functions fij : RK → R

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

The Ricardo Model

Labor is the only raw material (K = 1) Production is linear: fij(x) = aij · x aij: how much of good j country i can make with 1 unit of labor

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

The Heckscher-Ohlin Model

Multiple raw materials (K > 1) Different endowments Countries have same production functions: ∀i, i′ : fij = fi′j

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

Complexity Results

The Ricardo Model Production Utilities Complexity Note Linear Leontief NP-hard Linear P Known1 Leontief P “Similar” producers The Heckscher-Ohlin Model Production Utilities Complexity Note Linear Leontief NP-hard Linear Linear P Leontief Leontief P O(1) raw materials

1For example, auction algorithm by Kapoor, Mehta, and Vazirani (TCS

2008)

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

Upside-Down Reduction

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

Upside-Down Reduction

Used by Jain and Mahdian (WINE 2005) for linear Fisher model

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

Upside-Down Reduction

Used by Jain and Mahdian (WINE 2005) for linear Fisher model Reduce production economy to exchange economy

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

Upside-Down Reduction

Used by Jain and Mahdian (WINE 2005) for linear Fisher model Reduce production economy to exchange economy Make raw materials tradable, goods non-tradable

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

Upside-Down Reduction

Agents trade raw materials

Raw materials retain production function of source Upside-down market has (n × K) tradable goods

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

Upside-Down Reduction

Agents trade raw materials

Raw materials retain production function of source Upside-down market has (n × K) tradable goods

Agent produces optimal bundle with acquired raw materials

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

Upside-Down Reduction

Agents trade raw materials

Raw materials retain production function of source Upside-down market has (n × K) tradable goods

Agent produces optimal bundle with acquired raw materials Utilities given by ui(x) = max ui(f11(x11) + · · · + fn1(xn1), . . . , f1m(x1m) + · · · + fnm(xnm)) s.t. ∀i : xi ≥

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

Upside-Down Reduction

Agents trade raw materials

Raw materials retain production function of source Upside-down market has (n × K) tradable goods

Agent produces optimal bundle with acquired raw materials Utilities given by ui(x) = max ui(f11(x11) + · · · + fn1(xn1), . . . , f1m(x1m) + · · · + fnm(xnm)) s.t. ∀i : xi ≥

If fij have constant returns to scale, old and new economies are equivalent!

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

Ricardo: Leontief Utilities and Similar Producers

Theorem Let M be a Ricardo market with Leontief utilities and let ai = (ai1, . . . aim) ∈ S define the production functions of country i. If |S| = O(1) in M (i.e. there are few distinct a vectors so there is similarity in production), then equilibria are computable in polynomial time.

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

Ricardo: Leontief Utilities and Similar Producers

Idea: Devanur and Kannan (FOCS 2008):

Equilibria are efficiently computable for Leontief utilities with O(1) goods

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

Ricardo: Leontief Utilities and Similar Producers

Idea: Devanur and Kannan (FOCS 2008):

Equilibria are efficiently computable for Leontief utilities with O(1) goods

Here:

Similar producers ⇒ few raw materials and few patterns of trade

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

Ricardo: Leontief Utilities and Similar Producers

Proof: Recall ai = (ai1, . . . aim)

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

Ricardo: Leontief Utilities and Similar Producers

Proof: Recall ai = (ai1, . . . aim) O(1) distinct vectors ai in economy implies implies O(1) distinct labor goods

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

Ricardo: Leontief Utilities and Similar Producers

Proof: (contd.) Observation:

Let wi be the price of Country i’s labor

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

Ricardo: Leontief Utilities and Similar Producers

Proof: (contd.) Observation:

Let wi be the price of Country i’s labor In equilibrium, for all i, j: πjaij ≤ wi

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

Ricardo: Leontief Utilities and Similar Producers

Proof: (contd.) Observation:

Let wi be the price of Country i’s labor In equilibrium, for all i, j: πjaij ≤ wi In equilibrium, country i produces good j if and only if πjaij = wi.

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

Ricardo: Leontief Utilities and Similar Producers

Proof: (contd.) Observation:

Let wi be the price of Country i’s labor In equilibrium, for all i, j: πjaij ≤ wi In equilibrium, country i produces good j if and only if πjaij = wi. ⇒ only if for all i′: wi aij ≤ wi′ ai′j ⇒ wi wi′ ≤ aij ai′j

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

Ricardo: Leontief Utilities and Similar Producers

Proof: (contd.) Observation:

Let wi be the price of Country i’s labor In equilibrium, for all i, j: πjaij ≤ wi In equilibrium, country i produces good j if and only if πjaij = wi. ⇒ only if for all i′: wi aij ≤ wi′ ai′j ⇒ wi wi′ ≤ aij ai′j

m goods ⇒ 2m + 1 distinct ranges for wi

wi′

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

Ricardo: Leontief Utilities and Similar Producers

Proof: (contd.) Observation:

Let wi be the price of Country i’s labor In equilibrium, for all i, j: πjaij ≤ wi In equilibrium, country i produces good j if and only if πjaij = wi. ⇒ only if for all i′: wi aij ≤ wi′ ai′j ⇒ wi wi′ ≤ aij ai′j

m goods ⇒ 2m + 1 distinct ranges for wi

wi′

O(1) raw materials ⇒ O(mO(1)) distinct production patterns!

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

Ricardo: Leontief Utilities and Similar Producers

Proof: (contd.) Given production pattern, upside-down reduction collapses to Leontief exchange economy with O(1) goods

Use Devanur and Kannan to compute equilibrium

Utilities may be arbitrary ⇒ production makes equilibrium computation easier! QED.

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

n-Country Comparative advantage

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

n-Country Comparative advantage

Map relative production efficiencies (x, y, and z) and prices (w) to (n − 1)-simplex Country A produces goods in green region, etc.

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage

Further Questions

Reduction with non-constant returns to scale

Increasing returns to scale does not work (Papadimitriou and W) “Decreasing returns to scale” should work

Reduction with more general utilities

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The Complexity of Models of International Trade Christopher Wilkens Complexity of Equilibria

Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers

Comparative Advantage