Macroeconomics Theory II

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1 Macroeconomics Theory II Francesco Franco Novasbe February 2016 Francesco Franco (Novasbe) Macroeconomics Theory II February / 8

2 The Social Planner Solution Notice no intertemporal issues (Y t = C t )thereforesimpleintratemporal problem for SP where C t 1 0 C t(i) 1 1 e e 1 e di max U (C t, N t ; Z t ) subject to C t (i) =A t N t (i) 1 a, for all i 2 [0, 1] N t = ˆ 1 0 N t (i)di FOC: U C C t (i) 1 e C 1 e t = l t (i),u N = A t (1 a)n t (i) a l t (i) Francesco Franco (Novasbe) Macroeconomics Theory II February / 8

3 The Social Planner Solution The optimality conditions require C t (i) =C t, all i 2 [0, 1] N t (i) =N t, all i 2 [0, 1] where MPN t (1 a)a t N a t U n,t U c,t = MPN t Francesco Franco (Novasbe) Macroeconomics Theory II February / 8

4 Imperfect Competition With flexible prices firms maximize profits P t (i) =P t (i)y t (i) W t N t (i) subject to the demand and the production function. That gives P t (i) = e W t e 1 A t N t (i) a (1 a) where M = e e 1 is the markup. Therefore U n,t U c,t = MPN t M < MPN t Francesco Franco (Novasbe) Macroeconomics Theory II February / 8

5 Imperfect Competition This distortion can be eliminated with a subsidy M(1 t) =1 t = 1 e usually we assum such a subsidy as it is believed that MP should not focus on such distortions. Francesco Franco (Novasbe) Macroeconomics Theory II February / 8

6 Sticky prices Relative price distortions} resulting from staggered price setting: P t (i) 6= P t (j) ) C t (i) 6= C t (j) Optimal policy requires that prices and quantities are equalized across goods: Price stability. Francesco Franco (Novasbe) Macroeconomics Theory II February / 8

7 Optimal Policy y t = y n t ) ey t = 0 p t = 0 i t = r n t W E 0 Â b t Ut t=0 U c C U n t = 1 2 E 0 Â t=0 apple b t s + j + a ey t 2 + e 1 a l p2 t How to implement it? Francesco Franco (Novasbe) Macroeconomics Theory II February / 8

8 Welfare Second order approximation: W E 0 Â b t Ut t=0 U c C L = 1 2 U n t = 1 2 E 0 Â t=0 apple b t s + j + a ey t 2 + e 1 a l p2 t apple s + j + a var(ey t )+e var(p t ) 1 a Francesco Franco (Novasbe) Macroeconomics Theory II February / 8

9 Welfare relevant output Start with the PC: p t = k [ŷ t ŷt n ] + be t p t+1 In the presence of real imperfections: ŷt e 6= ŷt n. Define the welfare relevant output gap: ˆx e = ŷ t Using ˆx t = ˆx e ŷt n + ŷt e in the PC you obtain: p t = k ˆx t e + be t p t+1 + u t, where u t k(ŷ t e ŷt n ) ŷ e t Francesco Franco (FEUNL) Part I: DSGE 26/10 3 / 27

10 Welfare relevant output In the basic NK model, the distance between Yt e Yt n is constant so that ŷt e ŷt n = 0, and back to the world without trade-offs andwhere optimal policy requires stabilizing prive level The presence of other types of real rigidities make the disturbance u t varying u t is exogenous to policy The presence of u t generates a trade-off because it is now impossible to obtain zero inflation and the efficient level of activity simultaneously Francesco Franco (FEUNL) Part I: DSGE 26/10 4 / 27

11 Policy problem In the standard NK model, introducing a subsidy to employment t to eliminate the monopolistic competition distortion, the approximated Loss function is: where ˆx e t = ˆx t and a x = k h. L t = 1 2 E t  h b t p 2 t + a x (ˆx t e ) 2 i, t The problem of the policy maker is to minimize L subject to the economy constraint given by the PC: p t = k ˆx e t + be t p t+1 Francesco Franco (FEUNL) Part I: DSGE 26/10 5 / 27

12 Discretion The CB can choose to respond each period without committing itself to any future actions: the foc: p t = k ˆx e t + v t min p 2 ˆx t w t + a x (ˆx t e ) 2,p t 2p t = l t, 2a x ˆx e t = l t k. combining the two: ˆx e t = k a x p t In the face of inflationary pressures the CB respond by driving output below its efficient level with the objective of dampening the rise in inflation Francesco Franco (FEUNL) Part I: DSGE 26/10 6 / 27

13 Discretion Plug the optimal condition in the PC: p t = k 2 p t + be t p t+1, a x p t = a x a x + k 2 be tp t+1, p t = 0. so that: No trade-off ˆx e t = 0. Francesco Franco (FEUNL) Part I: DSGE 26/10 7 / 27

14 Discretion What if there are real rigidities that make u t varying in time? (with the same loss function this is the case of a cost push shock). Assume: u t = r u u t 1 + # u t. Same optimal condition but now: And p t = k 2 p t + be t p t+1 + u t, a x a x p t = a x + k 2 be tp t+1 + p t = a x Yu t. a x a x + k 2 u t, Trade-off ˆx W t = kyu t Francesco Franco (FEUNL) Part I: DSGE 26/10 8 / 27

15 Discretion To implement the previous equilibrium the CB has to adopt an interest rule. Consider what is the equilibrium interest rate that the equilibrium achieves. Using: ˆx e t = s(î t E t p t+1 ˆr e t )+E t ˆx e t+1, ˆx e t = kyu t, p t = a x Yu t, you get: î t = ˆr e t + Y 2 u t. Francesco Franco (FEUNL) Part I: DSGE 26/10 9 / 27

16 Discretion Is it a desirable interest rate rule? Substituting it in the IS you get the 2 by 2equilibriumdynamicssystemandyoucanshowthatitimpliesa multiplicity of solution. Abetterruleistoadopt î t = ˆr e t + Y 2 a x Y p t, which is consistent with the desired outcome and guarantess uniqueness if Y 2 a x Y > 1. Actually you can always derive a rule that guarantees uniqueness under the optimal policy by adding a term proportional to the deviation between inflation and the equilibrium value under the optimal policy with a coefficient greater than one. î t = ˆr e t + Y 2 u t + f p (p t a x Yu t ). Francesco Franco (FEUNL) Part I: DSGE 26/10 10 / 27

17 Commitment Turn to the case of a CB which is able to commit to a policy plan :{ˆx e t, p t }, credibly: subject to: The FOC: min { ˆx e t,p t } 1 2 E t  h b t p 2 t + a x (ˆx t e ) 2 i t p t = k ˆx e t + be t p t+1 + u t a x ˆx e t = l t k, p t = l t, E t [a x ˆx e t+i ] = E t [l t+i k] for i > 0, E t [p t+i ] = E t [l t+i 1 l t+i ] for i > 0. The inflation chosen from period t on is an effective constraint on future behavior Francesco Franco (FEUNL) Part I: DSGE 26/10 11 / 27

18 Commitment Combine the foc: for t = 0 : ˆx e 0 = p 0 k a x. for t > 0 : generally: E 0 [ˆx 1 e ] = ˆx 0 e k E 0 [p 1 ] or a x E 0 [ˆx 1 e k ] = E 0 [p 1 p 0 ] a x ˆx e t = k a x [p t p t 1 ] Francesco Franco (FEUNL) Part I: DSGE 26/10 12 / 27

19 Commitment Notice that the optimal commitment solution says that you reduce the output gap only if inflation is increasing. Using p 1 p 0 = (p 1 p 0 ) (p 0 p 1 ) you get: ˆx e t = k a x [p t p 1 ], which says that you deflate the economy if the price level icreases wrt to the initial price level. Francesco Franco (FEUNL) Part I: DSGE 26/10 13 / 27

20 Commitment Insert the rule into p t = k ˆx t e + be t p t+1 + u t ˆx t e k = [p t p t 1 ] a x p t = a x a x + k 2 p t 1 + a x a x + k 2 be tp t+1 + a x a x + k 2 u t the solution is: p t = bp t 1 + b 1 bbr u u t, where b = 1 p 1 4ba 2 2ab 2 (0, 1) and a = a x a x + k 2. Francesco Franco (FEUNL) Part I: DSGE 26/10 14 / 27

21 Commitment Now you can get the path of output gaps: p t = t  i=0 ˆx e t = b ˆx e t 1 b i+1 1 bbr u u t i + b t p 1 k a x b 1 bbr u u t. Bot the the inflation and the outputgap exhibit endogenous persistence (think of r u = 0). Francesco Franco (FEUNL) Part I: DSGE 26/10 15 / 27

22 Commitment versus Discretion To summarize: Discretion: ˆx t e = kyu t, p t = a x Yu t. Commitment: b p t = bp t 1 + u t, 1 bbr u ˆx e t = b ˆx e t 1 k a x b 1 bbr u u t Consider a purely transitory shock, with discretion inflation and the output gap return to their zero initial value after one period. In contrast with commitment the deviation persist well beyond the period of the shock. Francesco Franco (FEUNL) Part I: DSGE 26/10 16 / 27

23 Commitment versus Discretion The intuition for this result is: by commiting to a persistent reponse the CB improves the output gap-inflation tradeoff in the initial period. This is possible because of the forward looking nature of inflation: p t = k ˆx t e + Â b i E t ˆx t+i e + u t i=1 The CB offsets the inflationary impact of u t by lowering ˆx t e and all futures ˆx t+i e which decreases the extent to which ˆx t e as to decrease. You can show that commitment is always welfare improving. Francesco Franco (FEUNL) Part I: DSGE 26/10 17 / 27

24 Real rigidities In general the presence of real imperfections lead to a different loss function: L t = 1 2 E t  h b t p 2 t + a x (ˆx t e ) 2 i Lˆx t e t Here the CB wants to bring output closer to the efficient level which in this case is not the natural level. Solving the discretionary policy leads to the optimal response: ˆx e t = L a x k a x u t, which shows that policy is more expansionary than in the case where the natural and the efficient level of output correspond. You can show that the response does not change but that on average the levels of inflation are higher: inflation bias. Francesco Franco (FEUNL) Part I: DSGE 26/10 18 / 27

25 Real rigidities: an example Assume the production function is Y = M a N 1 a, where M is oil. M is in fixed supply. The real marginal cost faced by firms is in logarithms: s = w mpn = w (y n) ln(1 a). Francesco Franco (FEUNL) Part I: DSGE 26/10 19 / 27

26 Real rigidities: an example Households have the following period utility: U(C, N) =ln C exp(x) N1+f 1 + f, where C is a consumption basket with elasticity of subsitution h,n is employment and x is a preference shock. The MRS in logs is: mrs = c + fn + x Francesco Franco (FEUNL) Part I: DSGE 26/10 20 / 27

27 Real rigidities: an example The efficient allocation (the social planner problem) is such that each good is produced in the same quantity, the goods market is in equilibrium c = y, and the labor market is in equilibrium: mrs = mpn, c + fn + x = (y n)+ln(1 a), y + fn + x = y n + ln(1 a), (1 + f)n e = ln(1 a) x where n e is the efficient level of employment. Output is given by: y e = am +(1 a)n e. Francesco Franco (FEUNL) Part I: DSGE 26/10 21 / 27

28 Real rigidities: an example The equilibrium level of output, or natural level of output is found solving the optimal pricing of the firms: mc = µ = ln(h/(h 1)) which gives the demand of labor. Equating demand and supply and using, symmetry and the goods market equilibrium condition we get: The natural level of output is: c + fn + x = (y n)+ln(1 a) µ (1 + f) n n = ln(1 a) µ x y n = am +(1 a)n n. Francesco Franco (FEUNL) Part I: DSGE 26/10 22 / 27

29 Real rigidities: an example Notice that both the efficient and the natural level of output vary over time but their gap remains constant: y e y n = µ(1 a) 1 + f = d. Assuming Calvo pricing we know that the NKPC around a zero inflation steady state is: p t = k(ŷ t ŷ n t )+be t p t+1, where k = 1+f 1 a l, l = q 1 (1 q)(1 bq) where q is the fraction of fims not adjusting their price. Now given y e y n µ(1 a) = 1+f, ŷ e ŷ n = 0. Francesco Franco (FEUNL) Part I: DSGE 26/10 23 / 27

30 Real rigidities: an example Assume that real wages respond sluggishly to labor market conditions: w t = gw t 1 +(1 g)mrs, where g is an index of real rigidities. Can be derived from staggering of real wages. Francesco Franco (FEUNL) Part I: DSGE 26/10 24 / 27

31 Real rigidities: an example What is the equilibrium level of output in this case? From the wage setting we get: From the firm side: w = gw t 1 +(1 g)mrs, = gw t 1 +(1 g) (y + fn + x), = gw t 1 +(1 g) (a(m n)+(1 + f)n + x). w = (y n)+ln(1 a) µ, = a(m n)+ln(1 a) µ. Francesco Franco (FEUNL) Part I: DSGE 26/10 25 / 27

32 Real rigidities: an example Putting together the two and rearranging terms you can solve for y n as a function of y e : y n t y e t d = Q [y n t 1 y e t 1 d] + Q(1 a) Dm +(1 + f) 1 Dx where Q 2 (0, 1). Here the gap between the efficient level and the natural level is not constant and respond to changes in m and x. You can show that Q is increasing in g. Question: how do you respond to an increase in m? Francesco Franco (FEUNL) Part I: DSGE 26/10 26 / 27

33 Conclusions Monetary Policy faces a trade off between the stabilization of inflation and output The trade off appears once we add interesting real imperfections to the model The next part of the course is devoted to the systematic exploration of real imperfections in the different markets of the economy Francesco Franco (FEUNL) Part I: DSGE 26/10 27 / 27

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