The science of monetary policy

Transcription

1 Macroeconomic dynamics PhD School of Economics, Lectures 2018/19 The science of monetary policy Giovanni Di Bartolomeo Doctoral School of Economics Sapienza University of Rome

2 Refresh: Assumptions Main assumptions 1. Monopolistic competition 2. Sticky prices (staggered price setting) 3. Competitive labor markets 4. No capital accumulation/no financial intermediaries 5. No fiscal sector 6. Closed economy A1 and A2 are crucial. Even if prices adjust, A1 prevents Pareto optimality! A3-A6 are simplifications.

3 The New Keynesian model New Keynesian IS curve x t = E t x t+1 1 i t E t t+1 r t n + g t New Keynesian Phillips curve: π t = E t π t+1 + kx t + u t where g t and s t are stochastic processes, e.g. g t = g g t 1 + v t, u t = u u t 1 + t, 4 equation for 5 unknowns!!! v t N 0, v2 t N 0, 2 Monetary policy should be defined Instrumental rules, e.g., Taylor rule: i t = f π π t + f y x t Welfare-based criterion.

4 How to specify monetary policy? Fed Funds Rate Taylor rule

5 The Taylor Principle Taylor principle: To stabilize inflation, central banks must raise nominal interest rates by more than any rise in expected inflation, so that the real interest rate rises when inflation rises. Blanchard Kahn conditions: The solution of the rational expectations model is unique if the number of unstable eigenvectors of the system is exactly equal to the number of forward-looking (control) variables.

6 Welfare-based criterion Welfare-based criterion Discretion: the central bank takes private sector s expectations as given and chooses its instrument optimally period by period. Commitment (two kinds) Commit to hire a central banker with different preferences (delegation), who operates under discretion. The central bank can credibly commit to a certain future path of the output gap at some generic time, and thereby is able to affect private sector s expectations.

7 The efficient allocation Social Planner s problem maxu C t, N t Subject to Optimality conditions: C t (i) = A t N t (i) 1 for all i 0,1 1 N t = න N t (i) di 0 U n,t /U c,t = MPN t C t i = C t for all i [0,1] N t i = N t for all i [0,1] where MPN t = 1 A t N t

8 Sources of sub-optimality of equilibrium We focus on Monopolistic competition Markup variations resulting from sticky prices Others Transactions friction (economy with valued money, see the slides on the Friedman rule); wages (imperfect competition in labor markets); sticky wages; search in labor markets; real wage rigidities; oil shocks; financial frictions; open economy

9 Distortions unrelated to nominal rigidities Optimal price with monopolistic competition implies: P t = M W t, where M = ε > 1, it follows MPN t ε 1 U n,t = W t = MPN t U c,t P t M < MPN t Solution: employment subsidy τ. Under flexible prices, P t = M 1 τ W t MPN t Optimal subsidy: 1 τ M = 1 or, equivalently, τ = ε 1, it follows Optimal allocation. U n,t U c,t = W t P t = MPN t 1 τ M = MPN t

10 Distortions associated with nominal rigidities Prices Price increase Relatively high price (sales low) Relatively high price (sales low) Price of firm i Price level Price of firm i Relatively low price (sales high) t 0 t 1 t 2 Time

11 Distortions associated with nominal rigidities Markup variations resulting from sticky prices (assuming optimal subsidy, 1 τ = 1/M): M t = P t 1 τ W t /MPN t = MP t W t /MPN t W t P t = MPN t M M t U n,t U c,t = W t P t = MPN t M M t MPN t Optimality requires that the average markup be stabilized at its frictionless level. Relative price distortions resulting from staggered price setting: C t i C t j if P t i P t j Optimal policy requires that prices and quantities (and hence marginal costs) are equalized across goods.

12 Optimal policy Optimal employment subsidy flexible price equilibrium allocation is efficient. No inherited relative price distortions, i.e. P i t 1 = P t 1 for all i [0,1] the efficient allocation can be attained by a policy that stabilizes marginal costs at a level consistent with firms desired markup, given existing prices: No firm has an incentive to adjust its price, i.e. P t = P t 1 and, hence, P t = P t 1 for t = 0,1,2, (aggregate price stability). Equilibrium output and employment match their counterparts in the (undistorted) flexible price equilibrium allocation.

13 Policy trade-offs and the Phillips Curve New Keynesian Phillips curve: π t = E t π t+1 + k y t y t n Criticism: no policy trade-offs (Divine coincidence), optimality of strict inflation targeting Implicit assumption: y t e y t n = δ Alternative: time-varying y t e y t n gap: π t = E t π t+1 + kx t + u t where x t = y t y t n and u t = k y t e y t n, and u t = u u t 1 + t, t N 0, 2

14 Welfare-based criterion The central bank loss function L = 1 E 2 0 σ t=0 t (π 2 t + α x x 2 t ) Under some condition it can be derived as a secondorder approximation of the consumer preferences. Subsidy financed by lump-sum taxes and only price stickiness are considered. Then α x is a function of the deep parameters, i.e., α x = κ/ε where ε is the elasticity of substitution between goods in the monopolistic good market, while κ is a function of model parameters. It increases in the inverse Fisch labor supply elasticity (φ), in the slope of the Phillips curve, and in the risk aversion parameter ( ).

15 Welfare-based criterion The relative weight of output gap fluctuations in the loss function is increasing in, φ, and α. The reason is that larger values of those curvature parameters amplify the effect of any given deviation of output from its natural level on the size of the gap between the marginal rate of substitution and the marginal product of labor, which is a measure of the economy s aggregate inefficiency. Instead, the weight of inflation fluctuations is increasing in the elasticity of substitution among goods ε because the latter amplifies the welfare losses caused by any given price dispersion and the degree of price stickiness, which amplifies the degree of price dispersion resulting from any given deviation from zero inflation.

16 Some observations Welfare losses depend on the variability of both the output gap and the rate of inflation (noticeably not on the variability of the price level). The time variance of the output gap matters for welfare because the household wishes to keep output at the efficient level (which corresponds here to the flexible price level of output). The cross-sectional dispersion of output matters as well, due to the presence of relative price distortion. This is proportional to the cross-sectional variation in prices, which in turn is proportional to squared inflation.

17 The monetary policy problem The central bank preference function min L = 1 (π 2 t 2 + α x x 2 t ) + 1 E 2 0 σ t=1 t (π 2 t + α x x 2 t ) Subject to π t = E t π t+1 + kx t + u t π t+1 = E t+1 π t+2 + kx t+1 + u t+1 where u t = u u t 1 + t, t N 0, 2 In addition x t = E t x t+1 1 i t E t t+1 r t n x t+1 = E t+1 x t+2 1 i t+1 E t+1 t+2 r t+1 n

18 Transmission mechanism Inflation 4 Phillips curve short run E 2 t Output gap x t Fixing i t The central bank fixes the nominal interest rate (i t ) By fixing the interest rate, the output gap (x t ) is determined (by IS curve) Given the gap the inflation ( t ) is determined by the Phillips curve

19 Optimal discretionary policy Each period the Central Bank chooses (π t, x t ) to min L = 1 2 π t 2 + α x x t 2 + L 0 Subject to π t = kx t + v t, where v t = E t π t+1 + u t Optimality condition: π t = α x k x t Lean against the wind policy: Whenever inflation is above target, contract demand below capacity by raising the interest rate, and vice versa when it is below target. How aggressively the central bank should reduce x t depends positively on the gain in reduced inflation per unit of output loss, k, and inversely on the relative weight placed on output losses, α x.

20 Each period the Central Bank chooses (π t, x t ) to min L = 1 2 π t 2 + α x x t 2 + L 0 Subject to π t = kx t + v t, where v t = E t π t+1 + u t Optimality condition: Equilibrium Optimal discretionary policy π t = α x k x t x t = kψu t π t = α x ψu t where ψ = k 2 + α x (1 β u ). Equilibrium values are obtained by combining the optimality condition with the Phillips curve.

21 Optimal policy (discretion) π t = (α x /k)x t Policy rule Inflation 4 B Phillips curve short run E 2 Phillips curve long run Output gap C A

22 Optimal policy (discretion) Policy rule Inflation 4 Phillips curve short run Output gap t=1 t=2 t= A Phillips curve long run Inflation Output gap

23 Interest rate rule At the equilibrium, the central bank desire x t = kψu t π t = α x ψu t By IS curve: x t = E t x t+1 1 i t E t t+1 r t n + g t : kψu t = kψe t u t+1 1 i t α x ψe t u t+1 r t n + g t Solving by i t (note that E t u t+1 = u u t ): i t = γ u α x ψu t + g t + r t n where γ = 1 + k 1+ u u α x ψ > 1, since uα x ψu t = E t π t+1 : n i t = γ E t π t+1 + g t + r t What can you say about determinacy? What about demand shocks?

24 Optimal policy rule: π t = α x k x t Equilibrium conditional to the shock u t, implies a trade off between stabilizing the output gap or inflation: x t = kψu t π t = α x ψu t However, shocks can be positive or negative. Unconditional outcomes: The Phillips trade-off var(x t ) = kψ 2 2 var(π t ) = α x ψ 2 2 The trade off is on the variability. var x t α x < 0 var(π t ) α x > 0

25 Demand shock Inflation Demand shock (fall) 4 2 IS curve Phillips curve B A Output gap Expansionary monetary policy (interest rate cuts)

26 Result summary: Discretion Fact A (variance trade-off): To the extent cost push inflation is present, there exists a short run trade-off between inflation and output volatilities. Fact B (Taylor policy and REE stability): Under the optimal policy, in response to a rise in expected inflation, nominal rates should rise sufficiently to increase real rates, assuring REE stability. Fact C (Demand stabilization is a free lunch): If there is a demand shock, the central bank react by varying the interest rate of g t, this in turn offsets the shock that has not effects on the output gap (and thus on inflation). But, what about if the ZLB is binding?

27 The conservative central banker (delegation) Policy rule (discretion) Conservative central banker E B Inflation 4 2 C A Output gap Phillips curve Phillips curve with lower inflation expectations Commit to hire a central banker with different preferences (delegation), who operates under discretion Lesson: in forward-looking models, it s important to manage expectations

28 Result summary: Inflation targeting Fact D (optimal delegation): If there is persistence ( u >0), inflation targeting implies an higher (but positive) weight on inflation stabilization compared to discretion, i.e. flexible inflation targeting is optimal. As a result, the central bank stabilizes inflation more and output gap less, improving welfare. The rationale is that the promise of stabilizing inflation in the future reduces expectations and improves the current trade off between inflation and output gap. It does not work if there is not persistence ( u =0). In this case, expectations are always zero are the expected shock is zero, too.

29 Commitment We saw a government that commits to hire a central banker with different preferences (delegation), who operates under discretion. Now instead we assume that the government (central bank) can commit to follow a state contingent policy rule. We assume timeless perspective.

30 Optimal policy with commitment State-contingent policy {π t, x t } 0 that min L = 1 E 2 0 σ t=0 t (π 2 t + α x x 2 t ) Subject to the sequence of constraints: π t = E t π t+1 + kx t + u t Lagrange (maximize): E 0 σ t=0 t π 2 t +α x x 2 t 2 First order conditions α x x t kγ t = 0 π t + γ t γ t 1 = 0 For t = 0,1,2, and γ 1 = 0. + γ t π t π t+1 kx t u t

31 First order conditions α x x t kγ t = 0 Timeless perspective π t + γ t γ t 1 = 0 For t = 0,1,2, and where γ 1 = 0. We ignore the first order conditions at time zero, then γ t = (α x /k)x t γ t 1 = (α x /k)x t 1 Combining, we get the optimal policy rule π t = α x k x t x t 1

32 Commitment versus discretion The stabilization bias

33 Price targeting The optimal policy rule is π t = α x k x t x t 1 Alternative representation is based on price targeting [p 1 target]): x t = k α x p t p 1 For t = 0,1,2, Note that p 1 is not p t 1. Derive the above expression from the first order condition as exercise.

34 Transitory cost-push shock ( u = 0)

35 Permanent cost-push shock ( u (0,1))

36 Result summary: Commitment Fact E (timeless perspective): Although zero output gap outcome is feasible once a with noise shock is vanished, under commitment (timeless perspective), the central bank find optimal to maintain a persistently negative output gap and inflation. Commitment creates artificial persistence, but it also lows expectations of future inflation improving the current trade-off between inflation and output gap. It implies a current improvement at the cost of future needs of stabilization (but commitment can dominate discretion because the convexity of the loss).