theory - Lange Visits Minnesota A Note on Competitive...

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Unformatted text preview: Lange Visits Minnesota: A Note on Competitive Equilibrium Theory Piotr Evdokimov October 2, 2009 1 1 Introduction In this note, aimed at non-economists, I present one of the most basic macroeconomic modeling tools, the Arrow-Debreu Competitive Equilibrium (ADCE for short) and relate it to an early paper by Oskar Lange. The central question is: “What does modern macroeconomic theory have to say about Lange’s On the Economic Theory of Som ?” 2 The World of Arrow and Debreu Though the ADCE model has some unrealistic components (e.g. time 0 trading), at least a somewhat more realistic model (e.g. sequential instead of time 0 markets) will produce identical results. More to the point, economists use models to answer questions about the economy, and not to provide a complete description of the real world. In a heterogeneous agent environment, the ADCE is defined as prices { p ( t ) ,w ( t ) ,r ( t ) } ∞ t =0 . and alloca- tions { c * i ( t ) ,x * i ( t ) ,k * i ( t ) ,l * i ( t ) ,n * i ( t ) } ∞ t =0 ∀ i ∈ { 1 , 2 , 3 ,...,N } , { y * f ( t ) ,n * f ( t ) ,k * f ( t ) } ∞ t =0 such that consumers maximize utility, the representative firm maximizes profits, and markets clear. The symbol ∈ means “element of” or “in the set.” The curly brackets { } denote sets, and the notation “ } ∞ t =0 ” means that a given set has infinitely many elements: starting with element 0 and going up without bound. Hence, { p ( t ) ,w ( t ) ,r ( t ) } ∞ t =0 . can be written out as { p(0),w(0),r(0), p(1),w(1),r(1),p(2),w(2),r(2),... } . Where “...” reads as “and so on.” The upside down A means “for all.” Hence, the expression ∀ i ∈ { 1 , 2 , 3 ,...,N } means ”for every possible agent,” since we index agents by integers (agent 1, agent 2, ..., agent N). t indexes time, i.e. there are infinitely many time periods starting with the initial time period t = 0 (the point at which we start our analysis) and increasing without bound. You can think of t as years, months, seconds, or any other time unit. Hence, t = 5 can be the fifth month or the fifth millisecond. Notice that there are infinitely many prices in equilibrium: ∀ t, p ( t ) denotes the price of consumption and investment, w ( t ) denotes the wage, and r ( t ) denotes the rental rate for capital in period t . Similarly, the agent makes infinitely many choices: ∀ t , c * i ( t ) denotes agent i ’s optimal choice of consumption in period t (economists like to use stars to denote optimal quantities). Similarly, ∀ t , x * i ( t ) denotes investment in new capital stock, k * i ( t ) how much capital and n * i ( t ) how much labor the agent rents to the representative firm. The hours spent not working are leisure, and that’s denoted by l * i ( t ) ....
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This note was uploaded on 12/29/2010 for the course ECON 4313 taught by Professor Staff during the Fall '08 term at Minnesota.

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theory - Lange Visits Minnesota A Note on Competitive...

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