Note on Time Consistency

Transcription

1 Note on Time Consistency Danyang Xie HKUST 2011 dxie (HKUST) Note on Time Consistency / 13

2 Time Consistency Kydland and Prescott (JPE 1977) Rules Rather than Discretion: the Inconsistency of Optimal Plans The rst paper to introduce the issue of time inconsistency in a general framework Other important contributions Stanley Fischer (JEDC 1980) on Fiscal Policy Barro and Gordon (JME 1983) on Monetary Policy Lucas and Stokey (JME 1983) on Fiscal and Monetary Policy dxie (HKUST) Note on Time Consistency / 13

3 Kydland and Prescott (JPE 1977) Optimal Control Theory (the ones we have been using) Find optimal policy based on current situation Often, such optimal policy will cease to be optimal once a future date arrives. Reason: Bygones are bygones. People are forward looking. They form expectations about what the government will do in the future. Concept of time consistency A Two-Period Example Suppose teh social objective function is given by, S(x 1, x 2, π 1, π 2 ) where x represents actions and π represents policies. dxie (HKUST) Note on Time Consistency / 13

4 Government s Problem Government choose π 1 and π 2 : max S(x 1, x 2, π 1, π 2 ) subject to : x 1 = X 1 (π 1, π 2 ) x 2 = X 2 (x 1, π 1, π 2 ) Namely max S(X 1 (π 1, π 2 ), X 2 (X 1 (π 1, π 2 ), π 1, π 2 ), π 1, π 2 ) dxie (HKUST) Note on Time Consistency / 13

5 At t = 1, when the government plans for π 1 and π 2 S X 1 + S X2 x 1 π 1 x 2 S x 1 X 1 π 2 + S x 2 X 1 + X 2 x 1 π 1 π 1 X2 X 1 + X 2 x 1 π 2 π 2 + S = 0 π 1 + S = 0 (1) π 2 But when the government arrives at time t = 2, the problem becomes: max π 2 S(x 1, X 2 (x 1, π 1, π 2 ), π 1, π 2 ) and the FOC becomes: S x 2 X 2 π 2 + S π 2 = 0 (2) Note that (1) and (2) are usually di erent, so the optimal policy is usally time inconsistent. dxie (HKUST) Note on Time Consistency / 13

6 When will the Policy be Time Consistent? The su cient condition is: S + S X 2 X1 = 0 x 1 x 2 x 1 π 2 In other words, if X 1 π 2 = 0 (Action in Period 1 is independent of π 2 ) or if x S 1 + x S X 2 2 x 1 (direct + indirect e ect = 0) dxie (HKUST) Note on Time Consistency / 13

7 Stanley Fischer (JEDC 1980): Benevolent Dissembling Government A Two-Period Model The representative agent has the following preferences: ln c 1 + δ [ln c 2 + α ln(1 n) + β ln g] (3) where 1 n is leisure (chosen by the individual) and g is public goods (chosen by the government). Social Optimum: maximize (3) subject to: c 1 + S e c 2 + g (1 + r)s + wn where e is the endowment, F (S, n) = (1 + r)s + wn is the production function. The rst best allocation is easy to calculate. dxie (HKUST) Note on Time Consistency / 13

8 Distortionary Taxes Suppose that lump-sum taxes are not available, and the only possible taxes are τ n and τ k on labor and capital, respectively. The competitive equilibrium can be obtained from maximizing: ln(e S) + δ [ln [(1 + r τ k )S + (1 τ n )wn] + α ln(1 n) + β ln g] FOCs yields: c i = C i (r, w, τ k, τ n, g), i = 1, 2 n = N(r, w, τ k, τ n, g) dxie (HKUST) Note on Time Consistency / 13

9 Benevolent Government Assume that the government shares the same objective as the representative agent. The government s problem is then: max ln C 1(r, w, τ k, τ n, g) τ n,τ k,g +δ [ln C 2 (r, w, τ k, τ n, g) + α ln(1 N(r, w, τ k, τ n, g)) + β ln g] subject to: τ k S + τ n wn = g Let the solution be denoted τ n, τ k, and g. dxie (HKUST) Note on Time Consistency / 13

10 Time Inconsistency At Period 2, after consumers made their saving decisions, the new problem for the representative agent becomes: max ln [(1 + r τ k )S + (1 τ n )wn] + α ln(1 n) + β ln g now, the only decision variable is n. What will be the government s optimal choices of τ n, τ k and g now? Simple: setting τ n = 0, which completely removes the distortion. Time inconsistency arises even when the private agents and the government share the same objective function. Suppose that the individuals trusted the government in period 1 and then government decides to reform in period 2 by setting τ n = 0, we denote this outcome the cheating outcome. We use τ n = 0 and τ k to denote the cheating policy. dxie (HKUST) Note on Time Consistency / 13

11 How to nd a time consistent policy This is also called optimal discretionary policy. And it is derived from backward induction. We know that at period 2, τ n = 0 and τ k, g, satisfying τ k S a = g are the only policies that are credible. And the individual will choose: c 2 = (1 + r τ k )S + w 1 + α 1 n = αc 2 w Back to period 1, the representative agent maximizes: ln(e S) + δ ln (1 + r τ k )S + w 1 + α α (1 + r τ + k )S + w α ln + β ln g w 1 + α which leads to S = S(τ k ). The government then maximizes welfare subject to τ k S(τ k ) = g and obtains the consistent policy ˆτ n = 0 and ˆτ k. dxie (HKUST) Note on Time Consistency / 13

12 How to Rank the welfare under di erent policies First Best (FB): U S.O. Rules: U Discretion: Ũ Cheating: Û dxie (HKUST) Note on Time Consistency / 13

13 A Numerical Example Parameter values: e = 3, 1 + r = 1.5, w = 1, α = 0.25, β = 0.5, δ = 0.9. In all cases, the public goods will be g = 0.5c 2 and is omitted from the table. U c 1 S c 2 n FB τ k τ e k τ n τ e n ^ ~ dxie (HKUST) Note on Time Consistency / 13