LN0404_2011

# LN0404_2011 - CB046/Starr LN0404˙2011 14:32 Lecture Notes...

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Unformatted text preview: CB046/Starr LN0404˙2011 March 31, 2011 14:32 Lecture Notes for April 4, 2011: Existence of general equilibrium in an economy with an excess demand function See Starr’s ”General Equilibrium Theory: An Introduction” Chapter 5. General equilibrium theory focuses on finding market clearing (equilib- rium) prices for all goods simultaneously. Since there are distinctive inter- actions across markets the equilibrium concept includes the simultaneous joint determination of equilibrium prices. N goods in the economy. A typical array of prices is an N-dimensional vector p = ( p 1 , p 2 , p 3 , . . ., p N- 1 , p N ) = (3 , 1 , 5 , ..., . 5 , 10) . Assume only relative prices (price ratios) matter here, not monetary prices. The price space: The unit simplex in R N , is P = braceleftBigg p | p ∈ R N , p i ≥ , i = 1 , . . ., N, N summationdisplay i =1 p i = 1 bracerightBigg . (5.1) The unit simplex is a (generalized) triangle in N-space. For each household i ∈ H , we define a demand function, D i : P → R N . For each firm j ∈ F , S j : P → R N . Positive co-ordinates in S j ( p ) are outputs, negative co-ordinates are in- puts. p · S j ( p ) ≡ ∑ N n =1 p n S j n ( p ) ≡ profits of firm j . 1 CB046/Starr LN0404˙2011 March 31, 2011 14:32 2Lecture Notes for April 4, 2011: Existence of general equilibrium in an economy with an excess demand fu The economy has an initial endowment of resources r ∈ R N that is also supplied to the economy. The market excess demand function is defined as Z ( p ) = summationdisplay i ∈ H D i ( p )- summationdisplay j ∈ F S j ( p )- r, (5.2) Z : P → R N (5.3) Z ( p ) ≡ ( Z 1 ( p ) , Z 2 ( p ) , Z 3 ( p ) , . . ., Z N ( p )), where Z k ( p ) is the excess de- mand for good k . When....
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LN0404_2011 - CB046/Starr LN0404˙2011 14:32 Lecture Notes...

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