The Complexity of Wilkens Models of International Trade Complexity - PowerPoint PPT Presentation

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 Similar Producers UC Berkeley Comparative Advantage WINE ’09

The Big Picture: Computer Science The Complexity of Models of International Trade Christopher Wilkens Complexity of Computer science questions: (last decade) Equilibria Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers Comparative Advantage

The Big Picture: Computer Science The Complexity of Models of International Trade Christopher Wilkens Complexity of Computer science questions: (last decade) Equilibria Models Can we compute market equilibria? Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers Comparative Advantage

The Big Picture: Computer Science The Complexity of Models of International Trade Christopher Wilkens Complexity of Computer science questions: (last decade) Equilibria Models Can we compute market equilibria? Complexity Results What about special models? Upside-Down Reduction Example: Ricardo with Similar Producers Comparative Advantage

The Big Picture: Microeconomics The Complexity of Models of International Trade Christopher Wilkens Welfare economics: Complexity of Equilibria What determines prices? Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers Comparative Advantage

The Big Picture: Microeconomics The Complexity of Models of International Trade Christopher Wilkens Welfare economics: Complexity of Equilibria What determines prices? Models Complexity International Trade: Results Upside-Down What determines the pattern of trade between countries Reduction Example: (agents)? Ricardo with Similar Producers Comparative Advantage

The Big Picture: Microeconomics The Complexity of Models of International Trade Christopher Wilkens Trade Example: Complexity of Equilibria Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers Comparative Advantage

The Big Picture: Microeconomics The Complexity of Models of International Trade Christopher Wilkens Trade Example: Complexity of Equilibria My advisor is a very good researcher. Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers Comparative Advantage

The Big Picture: Microeconomics The Complexity of Models of International Trade Christopher Wilkens Trade Example: Complexity of Equilibria My advisor is a very good researcher. Models Complexity I am a mediocre researcher compared to my advisor. Results Upside-Down Reduction Example: Ricardo with Similar Producers Comparative Advantage

The Big Picture: Microeconomics The Complexity of Models of International Trade Christopher Wilkens Trade Example: Complexity of Equilibria My advisor is a very good researcher. Models Complexity I am a mediocre researcher compared to my advisor. Results Upside-Down My advisor pays me to do research. Why?! Reduction Example: Ricardo with Similar Producers Comparative Advantage

The Big Picture: Microeconomics The Complexity of Models of International Trade Christopher Wilkens Trade Example: Complexity of Equilibria My advisor is a very good researcher. Models Complexity I am a mediocre researcher compared to my advisor. Results Upside-Down My advisor pays me to do research. Why?! Reduction Example: This should be explained by the model Ricardo with Similar Producers Comparative Advantage

The Big Picture: Microeconomics The Complexity of Models of International Trade Christopher Wilkens Complexity of The Ricardo model (1800s): Equilibria Efficiency in production determines trade pattern Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers Comparative Advantage

The Big Picture: Microeconomics The Complexity of Models of International Trade Christopher Wilkens Complexity of The Ricardo model (1800s): Equilibria Efficiency in production determines trade pattern Models Complexity Results The Heckscher-Ohlin model (1930s): Upside-Down Reduction The availability of resources determines trade pattern Example: Ricardo with Similar Producers Comparative Advantage

Ricardo’s Law of Comparative advantage The Complexity of Models of International Trade (Ricardo, 1800’s) Given equilibrium wages, easy to recover Christopher equilibrium pattern of trade with 2 countries Wilkens Complexity of Equilibria Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers Comparative Advantage

Ricardo’s Law of Comparative advantage The Complexity of Models of International Trade (Ricardo, 1800’s) Given equilibrium wages, easy to recover Christopher equilibrium pattern of trade with 2 countries Wilkens Order goods and wages w i by relative production Complexity of Equilibria efficiencies a ij : Models Complexity Results > a 1( k +1) a 11 ≥ · · · ≥ a 1 k > w 1 ≥ · · · ≥ a 1 m Upside-Down Reduction a 21 a 2 k w 2 a 2( k +1) a 2 m Example: Ricardo with Similar Producers Comparative Advantage

Ricardo’s Law of Comparative advantage The Complexity of Models of International Trade (Ricardo, 1800’s) Given equilibrium wages, easy to recover Christopher equilibrium pattern of trade with 2 countries Wilkens Order goods and wages w i by relative production Complexity of Equilibria efficiencies a ij : Models Complexity Results > a 1( k +1) a 11 ≥ · · · ≥ a 1 k > w 1 ≥ · · · ≥ a 1 m Upside-Down Reduction a 21 a 2 k w 2 a 2( k +1) a 2 m Example: Ricardo with Similar Producers Country 1 produces goods on left of w , country 2 Comparative Advantage produces goods on right

Ricardo’s Law of Comparative advantage The Complexity of Models of International Trade (Ricardo, 1800’s) Given equilibrium wages, easy to recover Christopher equilibrium pattern of trade with 2 countries Wilkens Order goods and wages w i by relative production Complexity of Equilibria efficiencies a ij : Models Complexity Results > a 1( k +1) a 11 ≥ · · · ≥ a 1 k > w 1 ≥ · · · ≥ a 1 m Upside-Down Reduction a 21 a 2 k w 2 a 2( k +1) a 2 m Example: Ricardo with Similar Producers Country 1 produces goods on left of w , country 2 Comparative Advantage produces goods on right Problem is back to computing equilibrium

The Market The Complexity of Goods Models of International m tradable goods Trade K non-tradable raw materials Christopher Wilkens Complexity of Equilibria Models Complexity Results Upside-Down Reduction Example: Ricardo with Similar Producers Comparative Advantage

The Market The Complexity of Goods Models of International m tradable goods Trade K non-tradable raw materials Christopher Wilkens Agents Complexity of n agents Equilibria utility functions Models u i : R m → R Complexity Results Upside-Down Reduction Example: Ricardo with Similar Here, u i is linear or Leontief Producers Comparative Advantage

The Market The Complexity of Goods Models of International m tradable goods Trade K non-tradable raw materials Christopher Wilkens Agents Complexity of n agents Equilibria utility functions Models u i : R m → R Complexity Results Upside-Down Reduction Example: Ricardo with Similar Here, u i is linear or Leontief Producers endowment e i ∈ R K of raw materials Comparative Advantage production functions f ij : R K → R

The Ricardo Model The Complexity of Models of International Trade Christopher Wilkens Labor is the only raw material ( K = 1) Complexity of Production is linear: Equilibria Models Complexity Results f ij ( x ) = a ij · x Upside-Down Reduction Example: Ricardo with a ij : how much of good j country i can make with 1 unit of Similar Producers labor Comparative Advantage

The Heckscher-Ohlin Model The Complexity of Models of International Trade Christopher Wilkens Multiple raw materials ( K > 1) Complexity of Equilibria Different endowments Models Complexity Countries have same production functions: Results Upside-Down Reduction ∀ i , i ′ : f ij = f i ′ j Example: Ricardo with Similar Producers Comparative Advantage

Complexity Results The The Ricardo Model Complexity of Models of International Trade Production Utilities Complexity Note Christopher Wilkens Leontief NP-hard Known 1 Linear Linear P Complexity of Equilibria Leontief P “Similar” producers Models Complexity Results Upside-Down The Heckscher-Ohlin Model Reduction Example: Ricardo with Similar Producers Production Utilities Complexity Note Comparative Advantage Linear Leontief NP-hard Linear Linear P Leontief Leontief P O (1) raw materials 1 For example, auction algorithm by Kapoor, Mehta, and Vazirani (TCS 2008)

Upside-Down Reduction 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