View source for Alesina Drazen (1991) - Why Are Stabilizations Delayed

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*This page is referenced in [[BPP Field Exam Papers]]

[http://www.edegan.com/repository/Rui%20Notes%20on%20Alsina%20and%20Drazen.pdf Rui's Notes From Class]

==Reference(s)==
Alesina, A. and A. Drazen (1991), Why are Stabilizations Delayed?, American Economic Review 81, 1170-1188. [http://www.edegan.com/pdfs/Alesina%20Drazen%20(1991)%20-%20Why%20are%20Stabilizations%20Delayed.pdf pdf]


==Abstract==
When a stabilization has significant distributional implications (e.g., tax increases to eliminate a large budget deficit), socioeconomic groups may attempt to shift the burden of stabilization onto other groups. The process leading to stabilization becomes a "war of attrition," each group attempting to wait the others out and stabilization occurring only when one group concedes and bears a disproportionate share of the burden. We solve for the expected time of stabilization in a model of "rational" delay and relate it to several political and economic variables. We motivate this approach and its results by comparison to historical and current episodes.

==Stylized Facts==

The paper begins by point out three common features of stabilizations:
#There is agreement on the need for a fiscal change but a political stalemate over how the burden of higher taxes or expenditure cuts should be allocated. In the political debate over stabilization, this distributional question is central.
#When stabilization occurs, it coincides with a political consolidation. Often, one side becomes politically dominant. The burden of stabilization is sometimes quite unequal, with the politically weaker groups bearing a larger burden. Often this means the lower classes, with the burden of a successful stabilization being regressive.
#Successful stabilizations are usually preceded by several failed attempts. Often a previous program appears to be similar to the successful one.


==Underlying Economics==

The following economic assumptions underlie the model:
*The economy has constant ouput
*The government has a positive deficit, inclusive of debt service, (i.e. expenditures, including debt service, are in excess of tax revenue) implying a growing debt burden
**Given the constant output this is equivalent to a rising debt/GNP ratio
*Stabilization requires an increase in taxes, so that debt is constant
*Prior to higher taxes the government is limited to inefficient and distortionary methods of public finance
**"Monetization of deficits" (i.e. quantative easing, etc) that causes inflation is the only tool available
*Welfare losses may differ across socioeconomic groups
*The tax burden may eventually be born disproportionately by some socioeconomic groups and not others
*Different group differ in the political influence and so effort they must exert to fight during the war of attrition
*Stabilization benefits everyone
*Concession is an agreement to bear a disproportionate tax burden

==The Model==

Until <math>t=0\,</math> the budget is balanced. At <math>t=0\,</math> a shock hits reducing tax revenues, which implies with debt <math>b \ge 0\,</math>. After <math>t=0\,</math> until stabilization, <math>(1-\gamma)\,</math> of government expenditure including interest payments is covered by issuing debt, and <math>\gamma\,</math> is covered by distortionary taxation, where <math>\gamma >0\,</math> but not fixed. 

Denoting <math>g_0\,</math> as the level of expenditure, debt <math>b(t)\,</math> evolves according to:

<math>\underbrace{\frac{db}{dt}}_{\mbox{Change in debt}} = \overbrace{(1-\gamma)}^{\mbox{deficit}}\times\underbrace{[rb(t) + g_0]}_{\mbox{Total government spending}}\,</math>


where <math>r\,</math> is a constant interest rate.

This solves to:

:<math>b(t) = b_0 e^{(1-\gamma)rt} + \frac{g_o}{r}(e^{(1-\gamma)rt} - 1)\,</math>


Taxes before stabilization are therefore:

:<math>\underbrace{\tau(t)}_{\mbox{Taxes}} = \overbrace{\gamma}^{\mbox{Taxed percent}}\times\underbrace{(rb(t) +g_o)}_{\mbox{Total expenditures}}\,</math>

:<math>\therefore \tau(t) = \gamma r \bar{b}^{(1-\gamma)rt}\,</math>


where <math>\bar{b} \equiv b_0 + \frac{g_0}{r}\,</math>, which is the present discounted value of future taxes.


Taxes after stabilization <math>T\,</math> are:

:<math>\tau(T) = rb(T) +g_T\,</math>


where <math>g_T\,</math> is the expenditure after stabilization.

If assume <math>g_T = g_0\,</math>, <math>\tau(T) = r \bar{b} e^{(1-\gamma)rT}\,</math>


Putting these together we have that:
*Prior to the shock, taxes are constant and cover debt service (which is constant) and expenditure (which is constant)
*After the shock but prior to stabilization, debt rises exponentially, expenditure stays constant, and distortionary taxes rise exponentially to cover debt service
*After stabilization, the level of debt is constant, as are taxes (again expenditure stays constant throughout).


There are two groups, a winner and a loser. After stabilization the loser pays <math>\alpha > \frac{1}{2}\,</math> of the taxes (at <math>T\,</math>), where as the winner pays <math>(1-\alpha)\,</math>. The fraction is exogenous and not subject to bargaining - it represents the degree of polarization in society. Taxes after stabilization are non-distortionary (or at least less distortionary).

Each group has a loss from distortionary taxes parameterized by <math>\theta_i\,</math>, <math>\theta \sim F(\theta)\,</math> on <math>[\underline{\theta},\overline{\theta}]\,</math>, where <math>\theta_i\,</math> is private information. The utility loss is:

:<math>K_i(t) = \theta_i \tau(t)\,</math>


After stabilization <math>K_i = 0\,</math>. Groups have additively seperable (flow) utility functions:

:<math>u_i(t) = c_i(t) - y - K_i(t)\,</math>


where <math>c_i\,</math> is consumption for an individual and income, <math>y\,</math>, is assumed (for the most part) constant across individuals. 


The expected present discounted utility function for each group <math>j \in \{W,L\}\,</math> is:

:<math>U^j(T) = \int_0^T u(x) e^{-rx} dx + e^{-rT}V^j(T)\,</math>


where <math>V^j(T)\,</math> is the lifetime utility of the winner or loser from the point of stabilization forward.


The expected utility is the utility of winning times the probability that the opponent concedes at <math>X \;\forall X \le T_i\,</math>, plus the utility of losing times the probability that the opponent hasn't conceded by <math>T_i\,</math>. For notation, <math>H(T)\,</math> will be the distribution of the opponent's optimal time of concession and <math>h(T)\,</math> the associated density function.  The expected utility is:

:<math>EU(T_i) =[1-H(T_i)] U^L(T_i) + \int_0^{T_i} U^W(x) h(x) dx\,</math>

This equation can be simplified as shown on equation (8) in the paper on page 1177.

The model must then solve for:
*The path of consumption of a group
*The optimal concession time of a group, <math>T_i\,</math>
*The distribution of optimal concession times of an opponent.


The solution uses:
*With linear utility, any consumption path satisfying the intertemporal budget constraint with equality gives equal utility
*The assumption that both groups pay half of the taxes before the stabilization.

The model then solves out for a feasible consumption path, and computes the flow utility before a stabilization (which is the income effect of taxes plus the welfare loss from the tax distortion). With constant consumption after the stabilization (which was shown to be feasible), one can then compute the present discounted value of the excess taxes that the loser must pay relative to the winner:

:<math>V^W(T) - V^L(T) = (2 \alpha -1)\bar{b} e^{(1-\gamma)rT}\,</math>


This is used to determine the optimum concession time <math>T_i\,</math> of a group with cost <math>\theta_i\,</math>, subject to:
*<math>\underline{\theta} > \alpha - \frac{1}{2}\,</math> : This prevents a group's optimum concession time from being infinite, as otherwise a group may prefer to wait indefinitely, as the cost of living in the unstable economy and bearing half of the tax burden is less than the cost of being the loser.
*<math>F(\theta) = 1-H(T(\theta))\,</math> : as <math>T_i\,</math> is monotonic in <math>\theta_i\,</math> this can (apparently) be derived.
*Ignoring the equilibria in which one group concedes immediately, as the paper wants to examine delay and thus looking for a symmetric Nash equilibrium.
*Lemma 1 in the paper give <math>T_i'(\theta_i) < 0\,</math> : The optimal concession time is monotonically decreasing in <math>\theta_i \,</math>


Proposition 1 states that there exists a symmetric Nash equilbirum with each group's concession function described by <math>T(\theta)\,</math> where <math>T(\theta)\,</math> is implicitly defined by:

:<math>\underbrace{\underbrace{\left(-\frac{f(\theta)}{F(\theta)}\frac{1}{T'(\theta)}\right)}_{\mbox{A1}}\underbrace{\frac{2 \alpha -1}{r}}_{\mbox{A2}}}_{\mbox{A}}= \underbrace{\gamma (\theta + \frac{1}{2} - \alpha)}_{\mbox{B}}\,</math>

and the initial boundary condition <math>T(\bar{\theta})=0\,</math>

Where:

A1 - conditional probability that opponent concedes

A2 - gain if opponent concedes

A - benefit of waiting another instant to concede

B - cost of waiting another instant to concede


The intuition for a group with <math>\theta < \overline{\theta}\,</math> is a follows:
*At <math>t=0\,</math> there is some probability that the opponent has <math>\theta= \overline{\theta}\,</math> and concedes immediately.
*If the opponent didn't concede immediately then he must have <math>\theta < \overline{\theta}\,</math>, and both sides know this
*As time moves forward, the cutoff for concession moves down the distribution. 
*When the conditional probability is such that the equation above holds, then a group should concede.


There is a question of feasibility though. In determining the consumption path there was a constraint that the loser would have a specific end-game (i.e. after stabilization) consumption. If the game goes on too long, this constraint will be breached. Therefore there is a <math>T^* = T(\theta^*)\,</math> at which, in order for the consumption to be feasible, the government must close the budget deficit by a combination of expenditure cuts and distortionary taxes which impose an extreme disutility on both players. Players would prefer to concede and be the loser rather than face this consequence, so at <math>T^*\,</math> concession occurs with probability one. If both players are still in the game at this point a coin-flip tie-break rule is used to determine the loser.


However, this mass point at <math>T^*\,</math> creates a distortion in incentives for players whose <math>\theta\,</math> is just above <math>\theta^*\,</math>. Fortunately, it can be shown that there is a cutoff <math>\tilde{T} = T(\tilde{\theta})\,</math> above which the mass at <math>T^*\,</math> will not affect the optimum strategy.  Since <math>T^*\,</math> is increasing in <math>y\,</math>, and <math>\tilde{T}\,</math> is increasing in <math>T^*\,</math>, then <math>\tilde{T}\,</math> is increasing in <math>y\,</math>.  Thus, as <math>y\,</math> increases the fraction of groups whose behavior conforms to the standard solution above rises. With <math>y\,</math> high enough, the time until the solution above holds can be made arbitrarily long.


Given concession times as a function of <math>\theta\,</math>, the expected date of the stabilization is then the expected minimum <math>T\,</math>. With two players the expected stabilization time is:

:<math>T^{SE} = 2 \int_{\underline{\theta}}^{\overline{\theta}} T(x) F(x) f(x) dx\,</math>


As long as participants believe that someone may have a higher <math>\theta\,</math>, stabilization doesn't occur immediately. The key to the model is that there are multiple parties that do not know the other parties' costs. Heterogeneity of costs is not sufficient; if costs are known stabilization occurs immediately.

==Why Do Some Countries Stabilize Sooner Than Others?==

The purpose of this section is to use observable characteristics of economies to predict (relative) stabilization times. The assumption that <math>\underline{\theta} > \alpha - \frac{1}{2}\,</math> is maintained to prevent infinite waits.


===Distortionary Taxes or Monetization===

A higher <math>\gamma\,</math> implies an earlier stabilization:
*A higher <math>\gamma\,</math> means a higher proportion of expenditure, including interest payments, is covered by distortionary taxes
*There are two effects, and the first dominates:
**A higher <math>\gamma\,</math> gives a greater distortion for a given deficit which induces an earlier concession
**A higher <math>\gamma\,</math> means debt rises more slowly and hence distortions which induce concession grow slower

===Costs of Distortions===

An increase in the costs of living in an unstable economy, for a given <math>\theta\,</math>, will move the stabilization date forward
*Countries that lessen the distortion effects, ceteris paribus, will prolong the stabilization.
*If the utility loss is accelerating (i.e. if utility is convex) then stabilizations will occur sooner


===Political Cohension===

If <math>\alpha = \frac{1}{2}\,</math> then stabilization occurs immediately, whereas the greater the disparity, the longer the period until stabilization.
*If there is politcal cohesion <math>(\alpha = \frac{1}{2})\,</math> then the burden is shared equally.
*When there is no cohesion <math>(\alpha =1)\,</math> the relative burden of stabilization is more unequally borne, and the gain in holding out hoping that your opponent will concede is greater.


===Income Dispersion===

If the utility loss is decreaing in income and if income is unobservable, then a mean preserving spread in income that keeps the expected minimum of the <math>y\,</math>'s constant will result in longer times until stabilization.
*If <math>\theta'(y) < 0\,</math> the poor lose the war, because the rich can hold out longer.
*This could be interpreted as the funds available for political lobbying.

==Extensions==
The authors note that the model could:
*Be applied to trade or financial liberalization or other wars of attrition other than taxes
*With further shocks lead to one side immediately conceding following a shock
*Be extended to show that more political resources are needed as the war progresses
*Be interpreted as follows: <math>\theta\,</math> is the existance of institutions that make it relatively more difficult for opposing groups to stop reform

Note that reform does not necessarily directly follow a shock!