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# handoutmoney - A Primer on Equilibrium in Spatial Models of...

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A Primer on Equilibrium in Spatial Models of Money 1 Competitive Equilibrium with Fiat Money on the Townsend Turnpike HH i 0 s problem max { c i t 0 ,m i t +1 0 } t =0 X t =0 β t u ( c i t )( 1 ) s.t.p t c i t + m i t +1 = p t y i t + m i t z i t where z i t are lump sum taxes or transfers of money to agent type i. Given u 0 (0) = , agent will never choose c i t =0 , but may choose m i t +1 , so attach multiplier θ t to non-negativity constraint on m i t +1 . The Lagrangian for this problem is: £ = X t =0 β t · u μ y i t + m i t m i t +1 z i t p t + θ i t m i t +1 ¸ The foc wrt m i t +1 is u 0 ( c i t ) p t = β u 0 ( c i t +1 ) p t +1 + θ i t (2) ⇐⇒ u 0 ( c i t ) βu 0 ( c i t +1 ) = p t p t +1 + p t θ i t 0 ( c i t +1 ) u 0 ( c i t ) 0 ( c i t +1 ) p t p t +1 where the latter inequality holds since θ i t 0 . Thus the MRS 6 =MRT (the rate of return on money between periods t and t + 1) in periods in which θ i t is binding. De f nition 1 A competitive monetary equilibrium is a sequence of f nite positive prices { p t } t =0 and sequences of consumptions { c i t } t =0 , money balances { m i t +1 } t =0 , and lump sum taxes/transfers { z i t } t =0 for each agent type i such that: (i) Given { p t } t =0 and { z i t } t =0 , { c i t } t =0 and { m i t +1 } t =0 solve the house- hold problem (1); (ii) markets clear: c A t + c B t =1 (goods) and m A t + m B t = M t ; and (iii) the government budget constraint is satis f ed z A t + z B t = M t M t 1 . Can we support an optimal allocation without intervention? Proposition 2 An (interior) optimum cannot be supported in a competitive monetary equilibrium without intervention (i.e. when z i t , i, t ). 1

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Proof. By contradiction. Suppose we try to support ( c A t = λ, c B t =1 λ ) t w/o intervention. Since y A even =0and u 0 (0) = , type A hh will enter an even period with money accumulated in the odd period when y A odd = 1 (i.e. m A even > 0). But then the non-negativity constraint is not binding in the odd period (i.e. θ A odd = 0). In that case, from (2) we have u 0 ( λ ) βu 0 ( λ ) = p t p t +1 ,t odd. (3) A similar argument which applies for type B agents implies u 0 (1 λ ) 0 (1 λ ) = p t p t +1 even. (4) But (3) and (4) imply 1 β = p t p t +1 , t (5) ⇐⇒ p t +1 = βp t which implies that the price level should be decreasing (i.e. de f ation) at rate (1 β ) in order to satisfy the optimal allocation. Given this price sequence, what is happening to household money balances. Again take agent type A.
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handoutmoney - A Primer on Equilibrium in Spatial Models of...

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