Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Fei–Ranis model of economic growth wikipedia , lookup
Nominal rigidity wikipedia , lookup
Monetary policy wikipedia , lookup
Foreign-exchange reserves wikipedia , lookup
Ragnar Nurkse's balanced growth theory wikipedia , lookup
Modern Monetary Theory wikipedia , lookup
Interest rate wikipedia , lookup
Fear of floating wikipedia , lookup
Money supply wikipedia , lookup
Okishio's theorem wikipedia , lookup
1 NEW KEYNESIAN MACROECONOMICS Why monopolistic competition? To model price pre setting one must allow for endogenous goods supply. This requires not assuming endowments of goods. The determinants of the cost of supplying goods are important because the supply costs are a prime determinant of the optimal pricing. Differentiated goods and monopolistic competition among the suppliers of these goods serve as a device to allow for expansion/contraction of output whose price is pre set in response to the realization of demand and supply shocks. See Rothemberg (1982), Mankiw (1985), Svensson (1986), and Blanchard and Kiyotaki(1987) who assume monopolistic competition rather than assuming a single good in perfectly competitive supply. A. Open Economy with Flexible Prices I - Household (The Dixit and Stiglitz setup): M n E o t u (C t ) v t ht ( j ) 2 dj t 0 Pt 2 0 * Pt C t Pt Tt M t M t 1 t B t 1 f t 1,t (1 i t*1 ) B * t Bt 1 (1 i t 1 ) Bt Max s.t. n n 0 0 w t ( j )ht ( j )dj t ( j )dj 2 n = number of domestically produced goods. 1 = number of domestically consumed goods. The term h ( j ) dj represents the disutility (convex) n 2 0 2 t function, of supplying labor of type j. We have written it as if the representative household simultaneously supplies all of the types of labor. Bt = bond holdings at the beginning of date t (denominated in the domestic currency) Bt* = bond holdings at the beginning of date t (denominated in the foreign currency) Mt = money holdings at the end of date t Pt = aggregate domestic price level Ct = consumption index ht(j) = supply of labor of type j by the representative individual wt(j) = wage rate of labor of type j it* = world interest rate t(j) = profit of firm j (domestic) t = exchange rate in period t Tt = government lump-sum transfers Wt+1 = Mt + Bt+1* + Bt+1 = financial wealth f = forward exchange rate (the price paid in the present of foreign currency in terms of domestic currency to be delivered next period) t ,t 1 3 There is a constant elasticity of substitution between any two goods (varieties). Ct is a composite of all these goods. n C t ct ( j ) 0 1 1 dj ct* ( j ) n 1 dj 1 ct = goods produced at home ct* = goods produced abroad (imports) Elasticity of substitution: a ,b c (a) d t c ( b ) t d MRS a ,b ct ( a ) c ( b ) t MRS a,b Pick any two goods “a” and “b” in 1 1 n 1 1 * Ct ct ( j ) dj ct ( j ) dj 0 n MRS a ,b c (a) t ct (b) then : 1 a ,b 1 1 I.1 Prices The corresponding price index (the minimum expenditure that buys one unit of the consumption good composite): n 1 0 n Min Z pt ( j )ct ( j )dj t pt* ( j )ct ( j )dj {ct ( j )} s. t. 1 1 n 1 1 Ct ct ( j ) dj ct* ( j ) dj 1 0 n 4 Pt 0 pt ( j ) n 1 1 dj ( t p ( j )) * t n 1 dj 1 1 pt (j) = price of domestic good j (in domestic currency) pt*(j) = price of foreign good j (in foreign currency) I.2 The Inter and intra-temporal First-Order Conditions Substituting C from the budget constraint into the Max problem yields: t E o t [u{Tt t 0 1 { M t M t 1 t B * t 1 f t 1,t (1 i t*1 ) B * t Bt 1 Pt n n M (1 i t 1 ) Bt w t ( j )ht ( j )dj t ( j )dj )} v t 1 0 0 Pt 2 Differentiating with respect to n 0 ht ( j ) 2 dj ] M t , Bt 1 , B * t 1 , ht ( j ) yields: (1) Interest parity 1 it (1 it* ) f t ,t 1 t As long as there is no constraint on the size of debt (no corner solution), this no-arbitrage condition must hold. 5 (2) First order conditions P u ' (Ct ) (1 it ) Et u' (Ct 1 ) t Pt 1 (1) The Euler Condition (saving rule) M v' t P i t 1 t u ' (C t ) 1 it (2) Money Demand (as a function of ht ( j ) u ' (Ct ) wt ( j ) Pt C t and it ) . (3) Labor supply (of labor category j ) as a function of C t and wt ( j ) Pt (3) Ttransversality conditions (Kuhn Tucker terminal conditions) : BT 1 0 T PT 1 M lim T Et u ' (CT 1 ) T 0 T PT 1 lim T Et u ' (CT 1 ) T lim T Et u' (CT 1 ) T BT* 1 0 PT 1 Derivation of the Dixit-Stiglitz demand Functions Maximize the consumption index subject to a given total income, say, Z n Max C t ct ( j ) * 0 ct ( j ),ct ( j ) 1 1 dj ct* ( j ) n 1 dj 1 6 subject to: n 0 1 pt ( j )ct ( j )dj t pt* ( j )ct* ( j )dj Z n pick any two goods, say good a and good b: First Order Conditions: 1 ... 11 1 ct (a) 1 t p t ( a ) ... 11 1 ct (b) 1 1 => 1 1 pt (b)ct (b) pt (a)ct (a) 1 => t pt (b) pt (b)ct (b) pt (a)ct (a) ct (b) 1 Integrating both sides for all b’s: 1 0 1 1 1 1 * pt ( j )ct ( j )dj t p ( j )ct ( j )dj pt (a)ct (a) ct ( j ) dj ct ( j ) dj n 0 n 1 1 Z pt (a)ct (a) Ct => => 1 * t 1 pt (b) ct (b) pt (a) ct (a) pt (b)ct (b) pt (a) ct (a) pt (b)1 Integrating both sides for all b’s: 1 0 1 n 1 pt ( j )ct ( j )dj t pt* ( j )ct ( j )dj pt (a) ct (a) pt ( j )1 dj ( t pt* ( j ))1 dj 0 n n 7 => Z pt (a) ct (a) Pt 1 1 1 => pt (a) ct (a) Pt => p (a) C t ct (a ) t P t p t ( a )c t ( a ) C t 1 I.3 The government budget 0 Tt M t M t 1 Pt (Seigniorage revenue is rebated to the public in the form of a lump sum transfer Tt) II – Producers Monopolistic competitive firm: Max t ( j ) pt ( j ) yt ( j ) wt ( j )ht ( j ) ht ( j ) That is: Max t ( j ) ( f (ht ( j )) ht ( j ) 1 1 1 * Pt (Yt Yt ) wt ( j )ht ( j ) subject to: yt ( j ) f ht ( j ) The production function p ( j) [Yt Yt* ] yt ( j ) t Pt producer of good j. Dixit-Stiglitz demand, facing 8 1 n 1 Yt yt ( j ) dj . 0 The number of domestically produced goods is n. Returning to the profit maximization problem: Max t ( j ) ( f (ht ( j )) 1 ht ( j ) 1 1 * Pt (Yt Yt ) wt ( j )ht ( j ) A First Order Condition: wt ( j ) 1 ( f ht ( j ) ) Pt (Yt Yt* ) f ' ht ( j ) 1 pt ( j ) 1 wt ( j ) f ' ht ( j ) Demand for hours worked of type j wt ( j ) mc( j ) f ' ht ( j ) 1 the monopolist mark up , III – Targeting Monetary Aggregates (but not the interest rate) IIIa. a closed economy Bt* 0 , , C 0 (Aggregate demand has no foreign demand component) Bt 0 * t 9 Financial wealth = W t 1 Mt Equilibrium Conditions: Ct Yt M t M ts (Goods Demand = Goods supply) (Money Demand = Money supply) The idea is to approximate the behavioral and equilibrium conditions around a fixed point so as to be able to solve for the equilibrium price level. We study equilibrium around a no-shock, constant – money-growth steady state, characterized by: The Purely Deterministic Steady State Mt mt m Pt P t t Pt 1 it i Mt => M t 1 M M P 1 mt t t 1 t t 1 mt 1 t Pt Pt 1 Pt t t (4) In the steady-state: m m In the steady-state: Ct Ct 1 ... C From (1): From (2): 1 i => u (Yt ) it Mt (v) 1 Pt 1 it => i 10 M ts LYt , it Pt => The “LM curve” The steady-state LM Curve: m L Y , , the steady state real money balances III. The dynamic equilibrium in terms of proportional Deviations from the Steady State) Hat overstrikes denote proportional deviations from the steady state for all variables (except for interest rates i and r): X X X Xˆ t t log t X X Let X Yt log t X and take a linear approximation around Xt X => X Yt log X 1 X Xt X X t X => X X X log t t X X Xt X For interest rates i and r, the deviation notation is : Xˆ X X . t t The LM Curve C = equilibrium consumption elasticity of money demand 11 i = equilibrium interest semi-elasticity of money demand ˆ t C Yˆt i iˆt m (5) From (4): ˆt m ˆ t 1 ˆ t ˆ t m (6) We use the fact from consumer theory that in equilibrium the marginal rate of inter-temporal substitution ( u' (C ) / E [u' (C )] ) has to be equal to the price-ratio (1 r ) . Using equation (1): t t t 1 t => 1 rt (1 i t ) Pt E t Pt 1 This is known as the Fisher equation. By log-linearizing it, we obtain: rˆt iˆt E t ˆ t 1 (7) Substituting (5) into (7) to eliminate iˆ , substituting into (6), and then pushed forward by one period, yields: t Et mˆ t 1 mˆ t Et ˆ t 1 rˆt where: Since i 1 i i 0 , 1. C ˆ Yt i 12 Solving for m̂t , in the forward-looking manner ( L1 ) Et mˆ t Et ˆ t 1 rˆt Y Yˆt i 1 (L) 1 1 [(L) 1 (L) 2 (L) 3 ...] 1 1 L 1 (L) L L is an operator which does not affect the timing of the expectation operator. Hence: LE mˆ E mˆ t t 1 t t 1 mˆ t Et ˆ t 1 s rˆt s C Yˆt s s 0 i 1 s We use mˆ Mˆ Pˆ , and get the unique equilibrium value for the Equilibrium Price Level: t s t t 1 1 Pˆt Et ˆ t 1 s rˆt s C Yˆt s Mˆ ts s 0 i s III.b The open economy Equilibrium: Ct Ct* Yt Yt* worldwide goods market equilibrium M t M tS (The domestic money is effectively non- tradable) 13 We study the equilibrium around a steady-state that is characterized by Mt mt m Pt P t t Pt 1 it i Mt M t 1 M M P 1 mt t t 1 t t 1 mt 1 t Pt Pt 1 Pt t t (8) mt m , t t Recall that, 1 n 1 1 Pt pt ( j )1 dj t pt* ( j )1 dj . 0 n * Assume Pt* 1 . Pt 1 Because pt ( j) Pt => t . pt 1 ( j ) Pt 1 t 1 III.b.1 Flexible Exchange Rate pt ( j ) P t t pt 1 ( j ) Pt 1 t 1 (9) Log-linearizing equations (8) and (9) around the steadystate: ˆt ˆt ˆt 1 ˆt m ˆ t 1 ˆ t ˆ t m (10) (11) 14 LM curve, again from (2): u (Yt Yt* Ct* ) it Mt (v) 1 Pt 1 it Proportional deviations from the steady-state: (5’) ˆ t C (Yˆt Yˆt* Cˆ t* ) i iˆt m Log-linearizing the Fisher equation: rˆt iˆt E t ˆ t 1 (12) Substituting (12) into (5’) to eliminate iˆ : t ˆ t C (Yˆt Yˆt* Cˆ t* ) i rˆt iˆ t 1 m Substituting into (11): mˆ t 1 mˆ t ˆ t 1 C ˆ C ˆ * C ˆ * 1 Yt Yt Ct rˆt mˆ t i i i i Solving for m̂t mˆ t (13) as we did before: * * 1 Et ˆ t 1 s rˆt s C Yˆt s C Yˆt s C Cˆ t s s 0 i i i 1 s (14) By using (14) and (11) we can solve for the exchange rate: ˆt ˆ t ˆt ˆ t M t where: * * 1 Et ˆ t 1 s rˆt s C Yˆt s C Yˆt s C Cˆ t s s 0 i i i 1 s ˆt ˆt ˆt 1 15 III.b.2 Fixed Exchange Rate ˆt 0 1 0 E t ˆ t 1 s rˆt s C Yˆt s C Yˆt* s C Cˆ t* s (15) s 0 i i i 1 s the ̂ path is endogenously determined so as to satisfy equation (15). t