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Transcript
1 THE NEW KEYNESIAN MACROECONOMICS: AGGREGATE SUPPLY Main feature: The degree to which prices are determined in advance and its consequence for the outputinflation trade-off. I - Households (Differentiated products and Money in the Utility Function): n M E o t u(C t ; t ) ( t 1 ; t ) (ht ( j ); t )dj 0 Pt t 0 * * * Pt Ct Pt Tt M t M t 1 t B t f t 1,t (1 it 1 ) B t 1 Bt (1 it 1 ) Bt 1 Max s.t. n n 0 0 wt ( j )ht ( j )dj t ( j )dj Bt = bond holdings at the end of date t (denominated in the domestic currency) Bt* = bond holdings at the end 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 2 t(j) = profit of firm j (domestic) t = exchange rate in period t Tt = government lump-sum transfers t = preference shock ft,t+1 = forward exchange rate (the price paid in period t in terms of domestic currency, of one unit foreign currency to be delivered in period t+1) Interest parity (can be obtained by obtaining the FOC with respect to Bt and Bt*; without borrowing constraints of any kind) : 1 it (1 it* ) f t ,t 1 t There is a constant elasticity of substitution between any two goods in the economy. Ct is a composite of all these goods. 1 1 n 1 1 Ct ct ( j ) dj ct* ( j ) dj 0 n ct = goods produced at home ct* = goods produced abroad (imports) Corresponding price index (the minimum expenditure that buys one unit of the consumption index): Pt pt ( j ) 0 n 1 1 dj ( t p ( j )) n * t 1 dj 1 1 pt(j) = price of domestic good j (in domestic currency) 3 pt*(j) = price of foreign good j (in foreign currency) [The assumption is that the law of one price (PPP) prevails, Pt pt ( j )1 dj t pt* ( j )1 dj 0 n n 1 1 1 Taking a log approximation (where a hat (^) over a variable indicates log deviation from steady state) yields: 1 1 n1 1 t Pt* t pt ( j )1 dj pt* ( j )1 dj n 0 t P t (1 n) t A one percent movement in the exchange rate will have an effect on domestic consumers prices equal to the share of imports in consumption. Introduction of non-traded goods would allow for deviations from PPP. Alternatively, a fraction of firms set prices in the buyer’s currency, or Local 4 Currency Pricing (LCP). Accordingly, let s represent the fraction of foreign firms who set prices in domestic currency and use to indicate p that a price is fixed, we have: Pt pt ( j ) 0 n 1 dj n (1 n ) s n * t p ( j) 1 1 dj n (1 n ) s t p ( j) * t 1 dj 1 1 1 n 1 1 t Pt t pt* ( j )1 dj pt* ( j )1 dj 0 n * It is useful to compare the case s=0, full Producer’s Currency Pricing (PCP, which amounts to PPP), with LCP. If PPP prevails there is full pass through of exchange rate movements to import prices. P t (1 n) t . Whereas, if LCP prevails, P t (1 (n (1 n) s)) t . 5 That is, the degree of Pass Through is lessened. ] Now let’s go back to the PPP assumption. The First-Order Conditions Labor: vh (ht ( j ); t ) t wt ( j ) Consumption: u c (Ct ; t ) t Pt = Lagrange multiplier Substituting for t: h (ht ( j ); t ) wt ( j ) u c (Ct ; t ) Pt (1) (labor supply) The First-Order Inter-temporal Condition: uc (C t ; t ) Pt (1 i t ) E t uc (C t 1 ; t 1 ) E t Pt 1 choice) (2) (consumption-saving 6 uc (C t ; t ) (1 rt ) => E t uc (C t 1 ; t 1 ) The Fisher equation: 1 rt 1 it Pt (1 i t ) 1 t E t Pt 1 Demand for a variety: p ( j) C t ct ( j ) t Pt (Dixit-Stiglitz demand for good j) Government budget: 0 Tt M t M t 1 Pt (Government income, seigniorage: M t M t 1 , Pt is rebated to the public in the form of a lump sum transfer Tt). II –Producers yt ( j ) At f ht ( j ) (The production function) At = random productivity shock Variable cost of supplying: 7 y ( j) wt ( j )ht ( j ) wt ( j ) f 1 t At Nominal Marginal cost: Real marginal cost: st ( j ) xt ( j ) wt ( j )ht ( j ) ' y ( j) 1 wt ( j ) f 1 t yt ( j ) At At xt ( j ) ' y ( j) 1 wt ( j ) f 1 t Pt A t At Pt (3) Substituting (3) in (1) and assuming a symmetric equilibrium (dropping the index j because of the symmetry assumption): st => h (ht ; t ) 1' y t f uc (C t ; t ) At 1 At h ( f 1 ( y t / At ); t ) 1' y t s t ( y t , C t ; t , At ) f uc (C t ; t ) At 1 At World demand for the firm j product: YtW Yt H Yt F C t C tF An index for all the goods produced around the world. Producer j demand function: p ( j) y t ( j ) Yt t P t W 8 III. The Labor Market The market for each type of goods-specific skill of labor service is characterized by workers as wagetakers and producers as wage-makers, as in the monopsony case. Figure 1 describes equilibrium in one such market. The downward-sloping, marginal-productivity curve, is the demand for labor. implicitly determined by Supply of labor, Sh, is the utility-maximizing condition for h. The upward-sloping marginal factor cost curve is the marginal cost change from the producer point of view. It lies above the supply curve 9 because, in order to elicit more hours of work, the producer has to offer a higher wage not only to that (marginal) hour but also to all the (inra-marginal) existing hours. Equilibrium employment occurs at a point where the marginal factor costs is equal to the marginal productivity. Equilibrium wage is shown at point B, with the worker'’ real wage marked down below her marginal product by a distance AB.1 Full employment obtains because workers are offered a wage according to their supply schedule. This is why our Phillips curve will be stated in terms of excess capacity (product market version) rather than unemployment (labor market version). In fact, the model can also accommodate unemployment by introducing a labor union, which has monopoly power to bargain on behalf of the workers with the 1 In the limiting case where the producers behave perfectly competitive in the labor market, the real wage becomes equal to the marginal productivity of labor and the marginal cost of labor curve is not sensitive to output changes. Thus, with a constant mark-up no relation exists between inflation and excess capacity. , the Phillips curve becomes flat, i.e., 1 10 monopsonistic firms over the equilibrium wage. In such case, the equilibrium wage will lie somewhere between Sh and M Ph, and unemployment can arise – so that the labor market version of the Phillips curve can be derived as well. To simplify the analysis, we assume in this paper that the workers are wage-takers. W/P Figure 1: The Labor Market Equilibrium Marginal Factor Cost Marginal product Mark Down wage Labor Supply Marginal Productivity h Note: wages are perfectly flexible. Price Setting A fraction of the firms set their prices flexibly at p1t, supplying y1t. 11 A fraction 1- of the firms set their prices one period in advance (in period t-1) apt p2t, supplying y2t. The flexible price producer (type-1 firms) sets a constant mark-up, >1 , 1 above the actual marginal cost.: p1t s( y1t Ct ; t , At ) Pt (4) The producer who sets the price one period in advance (type-2 firms), charging p2t , the objective function, expected discounted profit, is: 1 1 W 1 Y W p P p 2t y 2t wt ht Et 1 p 2t Yt Pt wt f 1 t 2t t Et 1 At 1 it 1 1 it 1 . The maximization problem: W 1 1 W 1 Yt p 2 t Pt Max Et 1 p 2t Yt Pt wt f p2 t At 1 it 1 The FOC: 1 (1 ) p 2t YtW Pt wt f Et 1 1 it 1 Substituting 1 ' YtW p 2t Pt At YtW p 2t 1 Pt At 0 (5) 12 s( yt , Ct ; t , At ) h ( f 1 ( yt / At ); t ) 1' yt f u c (Ct ; t ) At 1 w ' y t f 1 t At At Pt At in (5), mwe get 1 (1 ) p 2t YtW Pt s ( y 2t , Ct ; t , At )YtW p 2t 1 Pt1 0 Et 1 1 it 1 => 1 (1 )YtW p 2t Pt s ( y 2t , Ct ; t , At ) Et 1 p 2t 1 Pt1 0 1 1 it 1 => 1 p (1 )YtW p 2t 1 Pt1 2 t s( y 2 t , C t ; t , At ) 0 E t 1 Pt 1 i t 1 (6) A weighted average of the deviation of relative price from the marked up marginal costs is set equal to zero. Where, 1 (1 )Y p P can be viewed as a weight at t 1 i t 1 W t 1 2t 1 t a given state of nature. Aggregate price index: P np 1 1t t (1 ) p 1 2t (1 n) t p 1 *1 1 t Potential Output The potential (or the Natural level of ) output (YtN) is the output level under perfect price flexibility ( = 1). Using (4) and (6) with = 1 we get: np 1 1t pt (1 n) t p 1 *1 1 t s(Yt n , Ctn ; t , At ) 13 If there are no capital flows (closed capital account), then CtN = YtN. In this case the natural output is defined by np pt 1 t (1 n) t p 1 *1 1 t s(Yt n , Yt n ; t , At ) If there are no capital flows and no commodity trade, then the economy is completely closed (A closed capital account and closed current account), then n = 1 and CtN = YtN . The natural output is defined by: 1 s (Yt n , Yt n ; t , At ) The natural output is independent of monetary policy. Note that the efficient output, 1 s(Y , Y ; , A ) is larger * t * t t than the natural output. IV. The Aggregate Supply The aggregate supply is a set of 6 equations: 1 1t Pt n p (1 ) p p1t s( y1t Ct ; t , At ) Pt 1 2t (1 n) t p 1 *1 1 t 1 p (1 )YtW p 2t 1 Pt1 2 t s( y 2 t , C t ; t , At ) 0 E t 1 Pt 1 i t 1 p y1t Yt 1t Pt W t 14 p y 2t Yt 2t Pt W Yt y1t (1 ) y 2t 1 1 1 There are 6 endogenous variables that are determined in the aggregate supply block of the model: THE Quantities-- y , y , Y THE Nominal prices--- p 1t 2t t 1t , p 2t , Pt . The Solution technique: log-linearization of the 6 aggregate-supply equations around the no shock steady state. IVa. The No Shock Steady State (1 r * ) 1 Assume Consider a deterministic steady-state, where A A 1, , p p , C C , Y Y . t t * t * t Log-linearization s t ( y t , C t ; t , At ) state point yields: and t of h ( f ( y t / At ); t ) 1' y t f uc (C t ; t ) At 1 t 0 1 , A t equation (5), around the steady- 15 sˆ t yˆ t 1Cˆ t _ log[ _ h ( f 1 ( y/ A);0) uc (C ;0) _ log[ _ h ( f 1 ( y/ A);0) uc (C ;0) A _ _ ' 1 f 1 ( y/ A) _ ] A t _ _ ' 1 f 1 ( y/ A) _ ] A (7) At ' Where: w p , w vhh f 1 y vh A , p f 1 '' 1 ' f y A , and 1 u cc C uc . The expression for the real marginal cost, evaluated at the natural level of output, is: sˆ tN Yˆt N 1Cˆ tN _ log[ _ h ( f 1 ( y/ A);0) uc (C ;0) _ log[ _ h ( f 1 ( y/ A);0) uc (C ;0) A _ _ ' 1 f 1 ( y/ A) _ ] A t _ _ ' 1 f 1 ( y/ A) _ ] A (7’) At Subtracting (7’) from (7): sˆt sˆtN ( yˆ t Yˆt N ) 1 (Cˆ t Cˆ tN ) Log-linearizing (4), state yields: pˆ 1t Pˆt sˆt (7’’) p1t s( y1t Ct ; t , At ) Pt , around the steady- 16 Subtracting the (log-linearized version of the ) equation evaluated at the natural level of output, substituting p P , and using (7’’) yields: N 1t N t pˆ 1t Pˆt ( yˆ1t Yˆt N ) 1 (Cˆ t Cˆ tN ) (8) We go through a similar procedure for equation (6) p E t 1 t 2 t s( y 2 t , C t ; t , At ) 0 Pt (in this case the relevant part N t t of the equation is the term inside the square brackets) and get: pˆ 2t Et 1[ Pˆt ( yˆ 2t Yˆt N ) 1 (Cˆ t Cˆ tN )] (9) Log-linearizing the price index yields: Pˆt n[pˆ 1t (1 ) pˆ 2t ] (1 n)(ˆt pˆ t* ) (10) Assume now that in steady-state there is zero inflation ; then: pˆ 1t log( p1t ) log( p1t ) log( p1t ) Pˆ log( P ) log( P ) log( P ) t t t t pˆ 2t log( p2t ) log( p2t ) log( p2t ) ˆt pˆ t* log( t pt* ) log( t pt* ) log( t pt* ) The rate of inflation rate is given by: 17 t P Pt Pt 1 log t Pˆt Pˆt 1 Pt 1 Pt 1 => t Et 1 t Pˆt Et 1 Pˆt (the surprise rate of inflation) The real exchange rate is defined as: et t Pt* Pt IV c. Deriving the Aggregate Supply Relationship Log-linearizing the (Dixit-Stiglitz) demand for the good produced by firm j, p ( j) y t ( j ) Yt t Pt W : yˆ jt Yˆt w [ pˆ jt Pˆt ] , with Yˆt w nYˆt N (1 n)Yˆt F With symmetry between firms of type 1 (flexible price firms) and between firms of type 2 (sticky price firms), we have: yˆ 1t Yˆt w [ pˆ 1t Pˆt ] yˆ 2t Yˆt w [ pˆ 2t Pˆt ] Substituting for ( yˆ 1t , yˆ 2 t ) in (8) and (9): 18 pˆ 1t Pˆt 1 ˆ (Yˆt w Yˆt N ) (Ct Cˆ tN ) 1 1 (8’) 1 ˆ w N ˆ ˆ ˆp2t Et 1 Pˆt Et 1 (Yt Yt ) (Ct Cˆ tN ) 1 1 => (9’) (11) pˆ 2t Et 1 pˆ 1t From (10): Pˆt Et 1 Pˆt t Et 1 t n[pˆ 1t (1 ) pˆ 2t ] (1 n)(ˆt pˆ t* ) Et 1 n[pˆ 1t (1 ) pˆ 2t ] (1 n)(ˆt pˆ t* ) Using (11) pˆ 2t Et 1 pˆ 1t and Et 1 pˆ 2t Et 1 pˆ 1t yields: t Et 1 t n [ pˆ 1t pˆ 2t ] (1 n)[(ˆt pˆ t* ) Et 1 (ˆt pˆ t* )] (12) From the definition of the real exchange rate we get: eˆt ˆt Pˆt* Pˆt => ˆt Pˆt* eˆt Pˆt Substituting in (12) yields: t Et 1 t n [ pˆ 1t pˆ 2t ] (1 n)[eˆt Et 1eˆt ] (1 n)[ Pˆt Et 1 Pˆt ] => n( t Et 1 t ) n [ pˆ 1t pˆ 2t ] (1 n)[eˆt Et 1eˆt ] From (10), pˆ 2t is given by: (13) 19 pˆ 2t 1 [nPˆt npˆ 1t (1 n)eˆt ] n(1 ) Substituting t E t 1 t pˆ 2t in (13); we have: 1 n 1 [ pˆ 1t Pˆt ] eˆ t (eˆ t E t 1eˆ t ) 1 n 1 1 Using (8) pˆ Pˆ ( yˆ Yˆ ) (Cˆ Cˆ ) to substitute for pˆ Pˆ in this expression, we obtain the open-economy Aggregate Supply (Phillips) Curve: N 1t t Et 1 t t 1t t 1 t N t 1t t 1 1 n 1 ˆ w Yˆ N ) ( Y (Cˆ t Cˆ tN ) eˆt Et 1eˆt t t 1 1 1 n 1 But because that the world output is divided between the domestic and foreign world as: Yˆt w nYˆt h (1 n)Yˆt f we have 20 => Yˆt w Yˆt N n(Yˆt h Yˆt N ) (1 n)(Yˆt f Yˆt N ) and t Et 1 t 1 n 1 n ˆ h ˆ N (1 n) ˆ f ˆ N 1 ˆ ( Y Y ) ( Y Y ) (Ct Cˆ tN ) eˆt Et 1eˆt t t t t 1 1 1 1 n 1 [If LCP prevails, P t domestic (1 (n (1 n) s)) t . The Pass Through from exchange rate fluctuations to domestic inflation is lessened, and the effect of the real exchange rate on surprise inflation is: 1 (n (1 n) s) 1 eˆt Et 1eˆt n 1 . 21 s = The fraction of foreign producers which preset prices in a domestic buyer’s currency.] IV.1 Perfect Capital Mobility If capital is perfectly mobile, then the domestic agent has a costless access to the international financial market. As a consequence, household can smooth consumption similarly in the rigid price and flexible price cases. 22 => Cˆ t Cˆ tN The Aggregate Supply curve becomes: t E t 1 t n ˆ h ˆ N (1 n) ˆ f ˆ N 1 n 1 (Yt Yt ) (Yt Yt ) eˆt E t 1 eˆt 1 1 1 n 1 IV. 2 Closed Capital Account 23 If the domestic economy does not participate in the international financial market, then there is no possibility of consumption smoothing, and we have that: N N ˆ ˆ ˆ ˆ Ct Yt ; Ct Yt In this case, the Aggregate Supply Curve becomes: t E t 1 t n 1 ˆ h ˆ N (1 n) ˆ f ˆ N 1 n 1 (Yt Yt ) (Yt Yt ) eˆt E t 1 eˆt 1 1 1 n 1 24 IV.4 Closed Economy If both the capital and trade accounts are closed, then the economy is an autarky, completely isolated of the rest of the world. In this case, all the goods in the domestic consumption index are produced domestically, which means that n = 1. The Aggregate Supply Curve becomes: t E t 1 t 1 1 ˆ h ˆ N (Yt Yt ) 1 25 IV.4 Slopes (Sacrifice Ratios) The slope of the Aggregate Supply Curve under each scenario is: (i) 1 n (1 )(1 ) (perfect capital mobility) (n 1 ) (ii) 2 (1 )(1 ) (closed capital account) ( 1 ) (iii) 3 (1 )(1 ) (closed economy) It can be seen that 1 2 3 Successive opening of the economy will flatten the Aggregate Supply Curve. 26 (Note however that the fraction of flex- and fixedprice firms is assumed to be given exogenously. Intuitively, liberalizing trade account transactions may increase the number of flex-price firms. If so, opening may increase the slope.)