Time inconsistency and credibility Advanced Political Economics Fall 2011 Riccardo Puglisi

Time inconsistency and credibility Initial work: Kydland-Prescott (JPE 1977) Idea: Patent protection of inventions; exams after a university course Time inconsistency: a problem of timing Example: Taxation of capital/labor, central banking

Capital/labor taxation (Fischer [JEDC 1980]) Announcement Saving Decision Policy Implementation Labor supply Decision (g, τ, θ) k(θ) (g, τ, θ) ℓ Central Banking (Barro-Gordon [JME 1983], Rogoff [QJE 1985]) Announcement Agents forming Shock gets realized Monetary Policy expectations implemented (m or  )  e   Time inconsistency and credibility

Plan  Ex-Ante optimal solution: Commitment  Ex-Post optimal Solution: Discretion Solutions:  Heterogeneous Agents  Delegation (Persson- Tabellini [JPuE 1994])  Reputation (Chari-Kehoe [JPE 1990], Kotlikoff- Persson-Svensson [AER 1988])

Capital and Labor Taxation: the Model  Two period model with heterogeneity among agents Preference: U = W(c 1 ) + c 2 + V(x) Time Constraint: x + ℓ = 1 + e Budget Constraint: c 1 + k = 1 – e (period 1) c 2 = (1 – θ) k + (1 – τ) ℓ (period 2)  Heterogeneity, measured by e, is unidimensional: more labor and less capital or viceversa.

e Distribution of Ability elel eueu eMeM E ( e ) = 0 Capitalists Workers G (e M ) = 1/2 Median Ability e M > 0 median voter is a worker Capital and Labor Taxation: the Model  Fiscal policy: (g, τ, θ) takes place in the second period, and it is balanced budget: g = θE(k) + τE(ℓ)

Agent’s Economic Optimization  Economic decision for an agent e is to maximize w.r.t. k, ℓ given the budget constraints FOC: W' c = 1 – θ  c 1 = W' c -1 (1 – θ) V' x = 1 – τ  x = V' x -1 (1 – τ) ℓ = 1 + e – V' x -1 (1 – τ) = L(τ) + e k = 1 – e – W' c -1 (1 – θ) = K(θ) – e c 2 = (1 – θ) K(θ) + (1 – τ) L(τ) – (τ – θ) e Notice: L(τ) = 1–V' x -1 (1 – τ), K(θ) = 1–W' c -1 (1 – θ) are respectively the average L s and savings in the economy

Ex-Ante Optimal Solution: Commitment  The government (or the median voter) makes a binding announcement to follow a fiscal policy (g, τ, θ)  Voter ex-ante preferences (characterized by e). Indirect utility: H(θ,τ)=W(1–K(θ))+V(1–L(τ))+(1–θ)K(θ)+(1–τ)L(τ)–(τ–θ)e  Recall that g = θE(k) + τE(ℓ) = θK(θ) + τL(τ)

 Political decision for an agent e is to maximize the indirect utility function w.r.t. τ and θ.  We can use the government budget constraint to reduce this voting problem to a unidimensional one. Define τ=τ(θ), then: θK(θ)+τ(θ)L(τ(θ))=g.  Taking partial derivative of the labor tax function: (which is negative if taxes are on the rising portion of the Laffer curve) Ex-Ante Optimal Solution: Commitment

 Max H(θ,τ(θ)) w.r.t. θ and using the envelop theorem H(θ,τ)=W(1–K(θ))+V(1–L(τ))+(1–θ)K(θ)+(1–τ)L(τ)–(τ–θ)e FOC: (for interior solutions) Ex-Ante Optimal Solution: Commitment

Define:  Ex-Ante Optimal Solution: Commitment

 Suppose there is no heterogeneity, e=0  e, then RAMSEY RULE The distortion on the last unit of revenue is equated across the two tax bases  lower tax rate for the more elastic base Ex-Ante Optimal Solution: Commitment

Consider the heterogeneity, look for a political equilibrium of the voting game Notice: 1) The optimal value of θ for a voter e is a continuous and increasing function of e 2) H'' θθ  0 3) H' θ = 0 unique (assume) Then, an equilibrium of this voting game exists and the decisive voter is the voter with median ability e m. Political Equilibrium under Commitment

 No commitment on fiscal policy  Agents still follow the economic optimality conditions, just replacing θ with θ e (the expected value)  However, when they take the political (or voting) decision the capital decision has already been taken (and thus its elasticity is 0). Hence, we talk about ex- post preference  Since ε K,θ =0, the optimality condition becomes: Ex-Post Optimal Solution: Discretion

 With no heterogeneity, e=0  e, then τ=0 and the total revenue should be obtained through the capital taxation: θ = min{1, g  K}, since θK+τL=g.  With heterogeneity: 1) if e m =0  τ=0 2) if ↑e (more labor income)  ↓τ (due to efficiency and redistribution) 3) if e  0 (more capitalist)  redistribution pushes the other way: τ  0 and θ  min{1, g  K} Ex-Post Optimal Solution: Discretion

More formally:  From the govt BC: where is the ex-post labor tax, i.e. for a given capital stock. Ex-Post Optimal Solution: Discretion

 Now let’s maximize the indirect utility function H(θ,τ)=W(1–K)+V(1–L(τ))+(1–θ)K+(1–τ)L(τ)–(τ–θ)e w.r.t. θ and for a given capital stock Ex-Post Optimal Solution: Discretion

 Evaluate this FOC at θ=g/K, with τ=0 for a capital stock large enough.  From we have  therefore, FOC: Ex-Post Optimal Solution: Discretion

 FOC: 1)Iff e = 0 the condition is verified 2) If e > 0 then H θ = e [1+K /L(0)] > 0 corner solution: θ = g / K, τ = 0. Notice: if K ≤ g  θ = 1, τ = (g-K) / L(τ) 3) If e < 0 (capitalist) then H θ < 0  ↓θ, ↑τ Ex-Post Optimal Solution: Discretion

Summarizing: Ex-Post Optimal Solution: Discretion

 To rule out multiple equilibria (driven by expectations) assume: max[τL(τ)] > g > max[θK(θ)]  unique equilibrium of the voting game.  The decisive voter is the median voter, e m > 0, then: Political Equilibrium under Discretion

 Agents perfectly forecast that there will be full appropriation: θ = 1  K(1) = 0  Therefore, the equilibrium implies: θ = 1, K = 0; τ = g/L(τ)  Recall that under commitment: τ,θ > 0; L(τ),K(θ) > 0 Political Equilibrium under Discretion

 Idea: the political agent can elect a representative to carry out the fiscal policy. The representative will choose her most preferred policy (i.e., the one maximizing her utility function)  Timing: 1) If elections take place at the beginning of the second period nothing changes. Every voter would vote for herself (discretionality result). 2) If elections take place in the first period and taxes are decided in the second period. hence, there is a delegation, and every agent will vote for the representative who will give the most preferred ex-ante fiscal policy SOLUTION: Representative Democracy

 Suppose an individual ẽ wins the election. She will implement a policy where is the stock of capital consistent with and thus in circulation when the representative ẽ is called to decide  Clearly: if ẽ > 0  like in direct democracy if ẽ < 0 (delegation to a capitalist)  where  is continuous and increasing in ẽ SOLUTION: Representative Democracy

Who wins the election?  Ex-ante optimum: is increasing in e. We can therefore match ex-ante optimum for an agent ē with the ex-post optimum for an agent ẽ: that is ē votes for ẽ.  Clearly ẽ is increasing in ē (with ẽ ≤ ē  ē), the ranking does not change and we can apply the median voter thm:  Notice: ẽ m < e m  delegation to a representative with more capital than the median voter SOLUTION: Representative Democracy