Changes

Until <math>t=0\,</math> the budget is balanced, with debt . At <math>b_0 \ge t=0\,</math>. At a shock hits reducing tax revenues, which implies with debt <math>t=b \ge 0\,</math> a shock hits reducing tax revenues. 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.

<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>

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

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

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 the time that you pre-designate that you would concede (<math>T_i\,</math>). For notionnotation, <math>H(T)\,</math> will be the distribution of of the opponents opponent's optimal time of concessionand <math>h(T)\,</math> the associated density function. This 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 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 different in present discounted value of the excess taxes that the loser must pay relative to the winner and loser lifetime utilities from stabilization forward: :<math>V^W(T) - V^L(T) = (2 \alpha -1)\overline(b) e^{(1-\gamma)rT}\,</math>

 This is the present 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>

*<math>\underline{\theta} > \alpha - \frac{1}{2}\,</math> : This prevents a groups group's optimum concession time from being infinite, as otherwise a group may prefer to wait indefinatelyindefinitely, 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.

*Ignoring the equilibria in which one group concedes immediately, as the paper wants to examine delay.*Looking and thus looking for a symmetric equilibriaNash equilibrium.*Lemma 1 in the paper give <math>t_iT_i'(\theta_i) < 0\,</math> : The optimal concession time is monotonically decreasing in <math>\theta_i \,</math>

:<math>\underbrace{\underbrace{\left(-\frac{f(\theta)}{F(\theta)}\frac{1}{tT'(\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>

*As time moves forward, so the cutoff for concession moves down the distribution.

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 goverment 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 close to just above <math>\theta^*\,</math>. Fortunately, it can be shown that there is a cuttoff cutoff <math>\tilde({T) } = T(\tilde{\theta})\,</math> above which the mass at <math>T^*\,</math> will not affect the optimum strategy, and furthermore as . 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 behaviour behavior conforms to the standard solution above rises. With <math>y\,</math> high enough, this the time until the solution above holds can hold for an be made arbitrarily long cutoff.

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:

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.

*A higher <math>\gamma\,</math> means more a higher proportion of expenditure, including interest payments, is covered by distortionary taxes

If the utility loss is decreaing in income and if income is unobservable, then an a mean preserving spread in income the that keeps the expected minimum of the <math>y\,</math>'s constant will result in longer times until stabilization.

*Be interpretted interpreted as follows: <math>\theta\,</math> is the existance of institutions that make it relatively more difficult for opposing groups to stop reform