handoutgcp08 - Intro to Government Commitment Problems This...

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Intro to Government Commitment Problems This draws on the simple framework in Stokey (1989, AER). 1 General Environment Continuum (unit measure) of identical households and one “big” govern- ment Each household (hh) chooses x X . Government (govt.) chooses y Y to maximize household utility. If X is chosen by all hh, y is feasible for the govt if ( X,y ) Z Y Hh preferences represented by w ( x, X, y ) which is the utility of a hh if it chooses x , all other hhs choose X ,andthegovtchooses y. 2S t a t i c F i r s t b e s t ( X f ,y f ) = arg max ( X,y ) Z w ( X, X, y ) . Characterization by foc: X : w 1 + w 2 =0 (1) y : w 3 =0 3 Static Commitment Equilibrium (RE) Timing: govt chooses y Y before hhs choose x X . Work backwards: Given y ,hhchooses x (which yields a decision rule η ( y ) ). Consistency and optimality requires: η ( y ) = arg max x X w ( x, η ( y ) ,y ) (2) Given η ( y ) , govt chooses y : y r =a r gm a x y Y w ( η ( y ) ( y ) ,y ) (3) s.t. ( η ( y ) ,y ) Z In general X f 6 = X r η ( y ) , so Ramsey outcomes are not necessarily f rst best since x : w 1 =0 (4) y :( w 1 + w 2 ) η 0 ( y )+ w 3 =0 .
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Static No Commitment Equilibrium (NCE) Timing: hh chooses x X before govt chooses y Y . Work backwards: Given X n ,govtchooses y . ϕ ( X n ) = arg max y Y w ( X n ,X n ,y ) (5) s.t. ( X n ,y ) Z Since each hh is small, it realizes that its impact on the govt is in f nites- imal. It expects that if all other hhs choose X n , the govt will choose ϕ ( X n ) . Consistency and optimality requires X n =argmax x X w ( x, X n ( X n )) (6) In general X r 6 = X n since in the RE, the government takes into account its in F uence on hh, while in the NCE it does not: x : w 1 =0 (7) y : w 3 =0 . Note that if the govt moves f rst (i.e. commits), it can always choose y n = ϕ ( X n ) . HH then chooses x = X n ,sotha ti ti sa lway spo s s ib leto implement the NCE as a RE. Hence, the utility of the RE is always at least as high as the utility of a NCE. 5A S p e c i f cExamp le 5.1 Environment HH receives endowment ω Two storage technologies: “productive" technology yields gross return R> 1 (i.e. net return is r> 0 ) and “pillow" yields gross return equal to 1 (i.e. zero net return). Hence e cient thing to do is to “invest” in the productive technology rather than “hoard” under the pillow. HH prefs: U ( c, g ) where c is a private good, g is a public good, U is strictly increasing, concave, and di f erentiable with U c ( ω, ( R 1) ω ) <U g ( ω, ( R
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This note was uploaded on 08/06/2008 for the course ECON 387 taught by Professor Corbae during the Spring '07 term at University of Texas at Austin.

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handoutgcp08 - Intro to Government Commitment Problems This...

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