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Handout 5 - Equilibrium 2

# Handout 5 - Equilibrium 2 - University of Minnesota...

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University of Minnesota Department of Economics Econ 3102: Intermediate Macroeconomics Handout 5 Just for completeness, this handout presents the definition of a competitive equilibrium (CE) for a two-period economy, and characterizes its solution. To ease notation, I’ll use subindices 1 and 2 to identify the current and future periods, respectively. 1 Competitive equilibrium Consider the following: Definition 1.1. A competitive equilibrium is a set of prices ( w 1 , w 2 , r ), current-period vari- ables ( C 1 , ‘ 1 , S, N d 1 , I, T 1 ) and future-period variables ( C 2 , ‘ 2 , N d 2 , T 2 ) such that given the exogenous variables ( G 1 , G 2 , z 1 , z 2 , K 1 ) we have: 1. Given ( w 1 , w 2 , r ), the representative consumer chooses ( C 1 , C 2 , ‘ 1 , ‘ 2 , S ) to solve max C 1 ,C 2 ,‘ 1 ,‘ 2 ,s U ( C 1 , ‘ 1 ) + U ( C 2 , ‘ 2 ) subject to C 1 + S = w 1 ( h - 1 ) + π 1 - T 1 C 2 = w 2 ( h - 2 ) + π 2 - T 2 + (1 + r ) S C 1 , C 2 0 0 1 , ‘ 2 h, where N s 1 = h - 1 , N s 2 = h - 2 . 2. Given ( w 1 , w 2 , r ), the representative firm chooses ( N d 1 , N d 2 , I ) to solve max N d 1 ,N d 2 ,I π 1 + π 2 1 + r subject to π 1 = z 1 F ( K 1 , N d 1 ) - w 1 N d 1 - I π 2 = z 2 F ( K 2 , N d 2 ) - w 2 N d 2 + (1 - d ) K 2 K 2 = (1 - d ) K 1 + I N d 1 , N d 2 , I 0 , where Y 1 = z 1 F ( K 1 , N d 1 ), Y 2 = z 2 F ( K 2 , N d 2 ). 3. The government’s budget constraints are satisfied G 1 = T 1 + B G 2 + (1 + r ) B = T 2 . 1

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4. All markets clear N s 1 = N d 1 N s 2 = N d 2 S = B C 1 + I + G 1 = z 1 F ( K 1 , N d 1 ) C 2 + G 2 = z 2 F ( K 2 , N d 2 ) + (1 - d ) K 2 . 2 Characterizing the competitive equilibrium Our characterization of the CE should give us enough information to obtain the 13 endoge- nous variables which form the equilibrium, or the set ( w 1 , w 2 , r, C 1 , C 2 , ‘ 1 , ‘ 2 , S, N d 1 , N d 2 , I, T 1 , T 2 ) . As we mentioned in class, for the consumer’s problem the following conditions should hold: (i) MRS 1 ,C 1 = w 1 (ii) MRS 2 ,C 2 = w 2 (iii) MRS C 1 ,C 2 = 1 + r .
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