LN1 - CB046/Starr LN1 December 9 2009 11:46 Lecture 1...

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Unformatted text preview: CB046/Starr LN1 December 9, 2009 11:46 Lecture 1: Existence of general equilibrium in an economy with an excess demand function General equilibrium theory focuses on finding market clearing (equilibrium) prices for all goods simultanously. Since there are distinctive interactions across markets the equilibrium concept includes the simultaneous joint de- termination 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 . The economy has an initial endowment of resources r ∈ R N that is also supplied to the economy. 1 CB046/Starr LN1 December 9, 2009 11:46 2Lecture 1: Existence of general equilibrium in an economy with an excess demand function 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 Z k ( p ), the excess demand for good k , is negative, we will say that good k is in excess supply.is in excess supply....
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LN1 - CB046/Starr LN1 December 9 2009 11:46 Lecture 1...

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