1 The Sidrauski model

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1 The Sdrausk model There are many ways to brng money nto the macroeconomc debate. Among the fundamental ssues n economcs the treatment of money s probably the LESS satsfactory and there s very lttle agreement about what s the rght way to look at monetary ssues. The Sdrausk model assumes we derve drect utlty from holdng money. Ths utlty s due to the lqudty servces we get and not due to the value of money as an asset. In models lke ths there s no drect utlty from holdng total assets, only from money. Assume the representatve agent who lves for ever maxmzes the followng utlty: max V 0 = Z Assumng no populaton growth one wrtes: Because: 0 u (c t,m t ) e θt dt () where : u c,u m > 0, u cc,u mm < 0 (2) s.t. : C t + K + M/P = wn + rk + G (3) K/N = k (4) M/PN = m + πm µ M MPN PNM m = = = M/PN πm PN P 2 N 2 Usng ( 4) one can rewrte the budget constrant n per capta terms: Defne TOTAL assets as a = m + k c t + k + m + πm = w + rk + g a = rk + w + g c πm a = [ra + w + g] [c +(π + r) m] a = [ncome] [ consumpton ]

2 and assume NPG: Hamltonan: n lm t a t exp R o t r 0 vdv =0we can wrte down the current value H = u (c t,m t )+λ [ra + w + g c (π + r) m] where a s the state varable and c, m are control varables. u c (c t,m t ) = λ (5) u m (c t,m t ) = λ (π + r) λ = θλ rλ NPG :... To close the model we assume (compettve) equlbrum n the captal market and CRS n producton + compettve labor market to get: r t = f 0 (k t ), w t = f (k t ) f 0 (k t ) k t. An IMPORTANT addtonal assumpton s: g = G/N =( M/P)N =( M/M)(M/PN) µm. Ths assumpton states that government transfers equal ALL the revenue from prntng money. In the steady state we want to assume that not only a =0but also m = λ =0. The last equalty mples that n the steady state the value of assets does not change. m = 0 = µ = π (6) λ = 0 = θ = f 0 (k ss ) The frst equalty states that the rate of prntng n the steady state equals the rate of nflaton and the second condton mples that n the steady state money s SUPER NEUTRAL [the rate of money growth has no effect on real economc actvty - the economy s at the modfed golden rule ] (Ths condtons are general and could be modfed to account for economc growth etc.). 2

3 Usng (5, 6) one gets the optmum quantty of money n the steady state. Consumpton equals producton (we assumed no deprecaton), producton s at the modfed golden rule level and the demand for money s gven by: u m =(µ + θ) u c Snce money has no effect on real varables n the steady state the best money rule (FRIED- MAN S RULE) s to make the margnal utlty from money = zero (sataton level). Ths happens f the government REDUCE money (have dsnflaton!!) at the rate equal to the tme preference. We want to compensate people for the lqudty servce they need. Snce compensaton n the steady state s free (prntng/absorbng money s free) we should compensate them fully. FREIDMAN0S RULE = µ = θ 3

4 . Non unqueness n monetary models Can the market determne the prce level? In many general cases t cannot do so even f the model s well specfed a la Sdrausk. Ths s an mportant ssue snce f there s more than one representatve agent they have no mechansm to pck a prce level everybody agree upon. The concept of equlbrum s ll n such cases. Let us use a varant of the Sdrausk model. ths tme we wll assume that money s needed for lqudty purposes n frms so the producton functon s a functon of real balances. A very smlar model could use separable nstantaneous utlty between money and consumpton. Assume ndvduals maxmze: max V 0 = s.t. : Z 0 u (c t ) e θt dt (7) M = P [h(m t ) c t ]+G (8) where:h(m t ) s the producton functon and the budget constrant s wrtten n nomnal terms. c t ]+g In real terms the dynamc budget constrant s (here g = G/P ): Snce M/P = µm and M/P = m + π t m then π t = µ m/m. Usng the above we can rewrte the real dynamc constrant as: M/P =[h(m t ) m t = h (m t ) c t π t m t + g t. (9) The Current value Hamltonan s: H = u (c t )+λ t [h (m t ) c t π t m + g t ], and the FOC are: u c (t) = λ t = λ t = u cc (t) c t (0) λ t = λ t θ λ t h 0 (m t )+λ t π t = λ t = λ t [h 0 (m t ) π t θ] 4

5 Usng (7 0) one can wrte the dynamc optmal behavor as: c = u h c h 0 (m t )+ m/m µ θ u cc. () The full employment means output equal consumpton (there s no nvestment here) and equlbrum n the money market mples demand=supply or c t = h (m t ) and m t = m s t. Takng dervatve w.r.t. tme of equlbrum n goods market yelds: c = h 0 (m t ) m. (2) balances: Usng ( to 2) wecanrewrtethedynamcoptmalpathntermsofrealmoney ³ m/m h 0 (m t ) m t h (m t ) h 0 (m t ) m = u h c h 0 (m t )+ m/m µ θ u cc + u c (t) c u cc (t) OR = u c (t) c u cc (t) [h0 (m t ) µ θ] Defne the elastcty of output (producton) w.r.t money as α (m) : α (m t ) h0 (m t ) m t h (m t ) Defne the elastcty of the MARGINAL utlty w.r.t consumpton as β (m) : (t can be wrtten as a functon of m snce c = h (m)) We can rewrte the dynamc equaton as: m/m = β (m) u c (t) c u cc (t) β(m) α (m) [h 0 (m t ) µ θ] β(m) To examne ths equaton n a smple way let us assume smple functonal forms: h (m) =m δ, u(c t )= c γ γ (3) 5

6 so α (m) =δ and β (m) =γ. In ths case In the steady state: m (t) = γ δ γ h δ (m t ) δ µ θ m t defne : = δ = γ m (t) = hδ (m t ) δ (µ + θ) m t γ µ µ + θ m = δ δ. Thus: m steady state = [(δ ) (µ + θ)] = [sgn ( )] m h m, Analyzng ths equaton n the m space depends crucally on the sgn of. If > 0 we get non unqueness. There s no way to determne the value of the prce level n the steady state. Any prce level people agree upon leads them to the steady state BUT how (why) can they all choose the same prce level? If < 0 we have unqueness BUT n the general case the curve may bend back and we can have bubbles n the prce level. Every prce level that s hgher than the unque steady state prce level s an acceptable nflatonary bubble. 6

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