handout 3 - University of Minnesota Department of Economics...

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University of Minnesota Department of Economics Econ 3102 °002: Intermediate Macroeconomics Handout 3 (Last revised: October 10) This handout presents how we can use all the results obtained in handouts 1 and 2 in order to characterize our competitive equilibrium. We° ll do this in two di/erent routes: following the de±nition of a competitive equilibrium, and then using the social planner°s equivalence. 1 Competitive equilibrium In class we gave the de±nition of competitive equilibrium, now we can do it mathsmatically: Def: A competitive equilibrium is a set of ( C; N s ; N d ; T; Y; w ) such that given ( G; z; K; h ) : 1. The representative consumer chooses C; ‘ to solve max C;‘ U ( C; ‘ ) s.t. C = w ( h ° ) + ° ° T C ± 0 0 ² ² h where N s = h ° . 2. The representative ±rm chooses N d to solve max N d zF ( K; N d ) ° wN d s.t. N d ± 0 1
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where Y = zF ( K; N d ) , ° = Y ° wN d = zF ( K; N d ) ° wN d . 3. All markets clear: N d = N s : 4. The government°s budget constraint is satis±ed: G = T: 2 Characterizing the competitive equilibrium To characterize the competitive equilibrium, note that from the consumer°s problem we get: U ( C; ‘ ) ° wU C ( C; ‘ ) = 0 (1) wN s + ° ° T ° C = 0 : (2) From the ±rm°s problem we get zF N ( K; N d ) = w; (3) and ±nally, we need the market clearing condition, plus the government budget constraint: N d = N s (4) G = T: (5) Now let°s try to simplify (1) ²(5). From (4), let N d = N s = N , and substitute in (2), 2
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(3). Then we get: wN + ° ° T ° C = 0 (6) zF N ( K; N ) = w: (7) Combining (1) and (7) gives U (
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