19. Lucas and Stokey's Model (1983, Journal of Monetary Economics)

We will be studying a general equilibrium model.

19.1 Set-up

Given (exogenous):

• The government consumption (spending) sequence: \(\{g_t\}_{t=0}^{\infty}\) with stochastic shocks.

The representative agent:

• \(1\) unit of time in each period
  • the time can be used either for leisure or for production
  • there is one-to-one relationship between time devoted to production and goods produced, i.e. the resource constraint for the economy is

$$ c_t+x_t+g_t=1 $$

where \(c_t\) is the consumption of the representative household at date \(t\) (endogenous) and \(x_t\) is the leisure of the representative household at date \(t\) (endogenous).

• preferences characterized by

$$ \mathbb{E}_{t=0}\left[\sum_{t=0}^{\infty}\beta^t u(c_t,x_t)\right] \tag{19.1} $$

where \(\beta\) is the utility discounting factor. Notice the utility is evaluated at date 0; government consumption \(g_t\) does not enter the household's utility.

Choice (endogenous, decision made by government):

• The linear tax rate sequence on labor \(\{\tau_t\}_{t=0}^{\infty}\)
  • at date \(t\), the tax revenue of government is \(\tau_t(1-x_t)\).
• The government bond maturity structure \(i\): \(\{{}_i b_t\}_{t=0}^{\infty}\) for \(i=0,1,2,\ldots\)
  • where \({}_i b_t\) is the outstanding zero coupon bond principal due (to be repaid) at date \(t\) which is issued at date \(i\) (\(t\ge i\));
  • \(\{{}_0 b_t\}_{t=0}^{\infty}\) is the initial debt obligation inherited from its predecessor.

The economy has complete markets for state-contingent zero-coupon bonds with different maturities — state-contingency allows the government (bond issuer) to repay at due date a different amount depending on the state of economy at that time (incorporating the idea of possible default in some sense).

Other details:

• relative price \(p_t\): the price of good at date \(t\) in terms of good at date \(0\); normalize \(p_0=1\); embeds interest rates \(p_t=\prod_{s=1}^{t}\frac{1}{1+r_s}\). The government might have incentive to manipulate the \(r_s\)'s (or \(p_t\)'s) to keep down its debt obligation \({}_0 b_t\).
• suppose at date \(0\) the government can make binding commitment about future tax rates, which means the tax rates are determined at date \(0\) and cannot be revised later (used for step 1, relaxed later).

19.2 Solving the problem: deterministic and exogenous government expenditures

19.2.1 Strategies to solve the problem

A future contract = government debt at each maturity. Complete markets \(\Leftrightarrow\) bonds for every maturity, so government can freely determine every term in \(\{{}_0 b_t\}\); incomplete markets at date \(i\) means some term(s) in \(\{{}_i b_t\}_{t=i}^{\infty}\) as a debt of certain maturity cannot be used by government in that period \(i\) (has to be set as \(0\)). For example, if in each period the government only has access to one period bond, or bond with finite \(n\) period maturities, then the markets are incomplete, and government has to continuously roll over debt in each period.

Under the deterministic exogenous expenditure sequence \(\{g_t\}_{t=0}^{\infty}\), the strategies are as follows.

1. Solve the Ramsey problem with government commitment of tax policy set at date 0.
   • (a) assume complete markets at date 0; under deterministic expenditure: (i) every decision and trading are made at date 0; (ii) no sequential trading in goods; (iii) new government debt issue in each later period is not necessary, because future contracts signed at date 0 are enough to pin down everything.
     • A. as the convention in practice, future contract have upfront payment, and it only pins down the price to be paid at delivery date;
     • B. here the future contracts are basically the government debt issued today, so they are all paid upfront (cash flows happen at date 0, no cash flow at delivery dates);
     • C. once such future contracts for all periods are signed, government's budget constraints for each later period are all satisfied automatically, so government would not issue new debt at all and simply carry over the debt from the previous period; the debt obligation is always \(\{{}_0 b_t\}\).
   • (b) specifically: (i) fix a tax rate sequence \(\{\tau_t\}\), \(\{p_t\}\), and solve the representative household's problem to obtain the first order conditions; (ii) use the f.o.c. and the resource feasibility constraint to construct the implementability constraint; (iii) solve the government's social welfare maximization problem by choosing an optimal \(\{c_t,x_t\}\); (iv) back out the required tax rate sequence \(\{\tau_t\}\) and equilibrium price sequence \(\{p_t\}\).
2. Sequential trading and limited markets with government commitment of tax policy at date 0.
   • (a) markets at date 0 incomplete: futures contracts cannot be signed for all future periods at date 0 \(\Rightarrow\) sequential trading of goods \(\Rightarrow\) new debt issue might be necessary.
   • (b) only markets at date 0 incomplete: (i) if complete at date 1, then date 1 is the last period for bond issue, government signs future contracts at all maturities so budget balance in all later periods are insured; (ii) if incomplete at date 1 too, government needs to sign future contracts at date 2 again, which results in further sequential trading.
3. Time consistency in tax policy.
   • (a) the government makes the decision for all future tax rates at date 0. Suppose at date 1 government can re-optimize its tax policy (which it cannot actually do with commitment), then the household will change supply accordingly. Our question: does there exist a plan to be made at \(t=0\) that is still optimal at \(t=1\)? If yes, the tax rates plan is time consistent.
   • (b) we will show there is a unique debt maturity structure \(\{{}_i b_t\}_{t=i}^{\infty}\) for \(i=0,1,2,\ldots\) that gives the unique time consistent tax rates plan.
   • (c) time consistency is a separate notion from market completeness. It is more convenient to have markets completeness in all periods to discuss time consistency: (i) even if markets are complete at date 0 and all decisions are made at date 0 at once, it is still possible that government, once given re-optimization opportunity, might have incentive to change tax rates policy at later dates; (ii) if markets are incomplete, in order to have time consistent tax policy, in each period we need to roll over the debt in a certain way and get the updated government debt to have a certain set of maturities, so at least those (future) markets at those certain maturities have to be complete at the time of roll over.

19.2.2 Step 1: complete markets at date 0, solve the Ramsey problem with government commitment

Representative household's problem. Given \(\{{}_0 b_t,p_t,\tau_t\}_{t=0}^{\infty}\), the household's maximization problem is

$$ \max_{\{c_t,x_t\}_{t=0}^{\infty}}\left\{\sum_{t=0}^{\infty}\beta^t u(c_t,x_t)\right\} $$

$$ \text{s.t.}\quad \sum_{t=0}^{\infty}p_t\left[c_t-{}_0 b_t-(1-\tau_t)(1-x_t)\right]\le 0 $$

where \((1-\tau_t)(1-x_t)\) is the household's net of tax labor income, and \({}_0 b_t\) is the household's income from bond investment. Since in this setting all decisions are made at date 0, there is only one budget constraint for the household (and for government to be discussed), which is a constraint at date 0. Here \(\{{}_0 b_t\}_{t=0}^{\infty}\) can be considered either a portfolio of infinitely many zero coupon bonds of different maturities, or only one bond (a Consol) with perpetuity of different amount in each period. The Lagrangian:

$$ \mathcal{L}=\sum_{t=0}^{\infty}\beta^t u(c_t,x_t)+\lambda\sum_{t=0}^{\infty}p_t\left[{}_0 b_t+(1-\tau_t)(1-x_t)-c_t\right] $$

First order conditions:

$$ \beta^t u_c(c_t,x_t)=\lambda p_t\quad \forall t \tag{19.2} $$

$$ \beta^t u_x(c_t,x_t)=\lambda p_t(1-\tau_t)\quad \forall t \tag{19.3} $$

Get rid of the Lagrangian multiplier \(\lambda\). Since (19.2) is true for all \(t\) and \(u_c(c_0,x_0)=\lambda p_0=\lambda\):

$$ \beta^t\frac{u_c(c_t,x_t)}{u_c(c_0,x_0)}=p_t\quad \forall t \tag{19.4} $$

Dividing (19.3) by (19.2):

$$ \frac{u_x(c_t,x_t)}{u_c(c_t,x_t)}=1-\tau_t\quad \forall t \tag{19.5} $$

(19.4) is an inter-temporal condition for consumption and (19.5) is an intra-temporal condition for substitution.

Market clearing condition (resource feasibility). Must be satisfied by all agents together (household and government):

$$ c_t+x_t+g_t=1\quad \forall t \tag{19.6} $$

Implementability constraint.

Government's problem.

To solve the government's problem, we are picking an allocation \(\{c_t,x_t\}_{t=0}^{\infty}\) from \(\mathcal{F}_I\cap\mathcal{F}_E\) where \(\mathcal{F}_E\) is the set of allocations that satisfy market clearing, so it is natural that we impose implementability constraint (19.7) and market clearing condition (19.8):

$$ \max_{\{c_t,x_t\}_{t=0}^{\infty}}\left\{\sum_{t=0}^{\infty}\beta^t u(c_t,x_t)\right\} $$

$$ \text{s.t.}\quad \sum_{t=0}^{\infty}\beta^t\left[(c_t-{}_0 b_t)u_c(c_t,x_t)-(1-x_t)u_x(c_t,x_t)\right]=0,\qquad c_t+x_t+g_t=1\ \ \forall t $$

A convenient way to construct the Lagrangian is to choose Lagrangian multipliers (\(\lambda_0\) for implementability, \(\mu_t\) for market clearing) such that

$$ \mathcal{L}=\sum_{t=0}^{\infty}\beta^t u(c_t,x_t)+\lambda_0\left\{\sum_{t=0}^{\infty}\beta^t\left[(c_t-{}_0 b_t)u_c-(1-x_t)u_x\right]\right\}+\sum_{t=0}^{\infty}\beta^t\mu_t(1-c_t-x_t-g_t) $$

The f.o.c. for \(c_t\) is

$$ (1+\lambda_0)u_c(c_t,x_t)+\lambda_0\left[(c_t-{}_0 b_t)u_{cc}(c_t,x_t)-(1-x_t)u_{xc}(c_t,x_t)\right]-\mu_t=0 \tag{19.9} $$

and the f.o.c. for \(x_t\) is

$$ (1+\lambda_0)u_x(c_t,x_t)+\lambda_0\left[(c_t-{}_0 b_t)u_{cx}(c_t,x_t)-(1-x_t)u_{xx}(c_t,x_t)\right]-\mu_t=0 \tag{19.10} $$

Combine the two f.o.c. (subtract (19.10) from (19.9)) into one equivalent equation eliminating \(\mu_t\):

$$ (u_c-u_x)+\theta_0\left[(c_t-{}_0 b_t)(u_{cc}-u_{cx})+(1-x_t)(u_{xx}-u_{xc})\right]=0 \tag{19.11} $$

where \(\theta_0=\dfrac{\lambda_0}{1+\lambda_0}\).

Characterize the solution to government's problem.

• Time independence property: the solution \(\{c_t,x_t\}_{t=0}^{\infty}\) has the following time independence property — suppose for some \(t\ne t'\), \(g_t=g_{t'}\) and \({}_0 b_t={}_0 b_{t'}\), then by the market clearing condition \(c_t+x_t=c_{t'}+x_{t'}\), and so, loosely speaking, with mild assumptions on \(u(\cdot)\), usually we can conclude that \(c_t=c_{t'}\) and \(x_t=x_{t'}\). This means the consumption and leisure will be the same in two periods as long as the exogenous government expenditure and bond repayment are the same in those two periods, independent of which two periods they are.
• Existence and uniqueness: we assume there exists a unique solution \(\{c_t,x_t\}_{t=0}^{\infty}\) to the government's problem (under certain conditions there is no solution, for example if \(\{g_t,{}_0 b_t\}\) is too big, since there is a cap for possible total tax revenue in each period).

• The sign of the Lagrangian multiplier \(\lambda_0\) for the implementability constraint:

  • suppose the government has access to and actually uses lump-sum taxes: then there is no distortion of prices, i.e. \(\tau_t=0\) \(\forall t\); the household's f.o.c. (19.4) pins down the optimal consumption \(c_t^*\) and leisure \(x_t^*\) for each period by \(u_x(c_t^*,x_t^*)=u_c(c_t^*,x_t^*)\) and \(c_t^*+x_t^*=1-g_t\), and the household's f.o.c. (19.5) pins down the equilibrium price \(p_t^*=\beta^t\frac{u_c(c_t^*,x_t^*)}{u_c(c_0^*,x_0^*)}\). The government's net obligation at \(t=0\) at efficient (equilibrium) price \(p_t^*\) is

  $$ G_0^*=\sum_{t=0}^{\infty}p_t^*\left(g_t+{}_0 b_t\right) $$

  and the government collects a lump-sum tax of exactly \(G_0^*\) at \(t=0\) with some future contracts signed with household at \(\{p_t^*\}\) in the complete future markets for goods (transfers in later periods if necessary). - if \(G_0^*=0\), then government doesn't raise any revenue at date 0 at all. Recall that (19.11) and \(u_x(c_0^*,x_0^*)=u_c(c_0^*,x_0^*)\) give \(\theta_0[\cdots]=0\Rightarrow\lambda_0=0\). Intuitively, when \(G_0^*=0\), government doesn't tax the household and government expenditure doesn't enter household's utility function, so it is equivalent to having no government in the economy; then one less good for government expenditure has no shadow value for the household at all. - if \(G_0^*>0\), then government needs to raise a lump-sum revenue at \(t=0\), so one less good for government expenditure is one more extra good for private consumption, which has positive shadow value for the household, i.e. \(\lambda_0>0\). - suppose price distortion is inevitable: then there is positive tax on labor income, which distorts household's choice away from consumption to leisure as leisure is not taxed, so we have \(u_c(c_0^*,x_0^*)-u_x(c_0^*,x_0^*)>0\). By (19.11), where \(c_t^*-{}_0 b_t>0\) is true because \(c_t^*={}_0 b_t+(1-\tau_t)(1-x_t^*)\), \(u_{cc}<0\), and \(u_{xc}(c_t,x_t)=0\) because we can assume that \(u(c_t,x_t)\) is additively separable, i.e. \(u(c_t,x_t)=v(c_t)+m(x_t)\); thus \(\theta_0>0\Rightarrow\lambda_0>0\).

19.2.3 Step 2: incomplete markets at date 0, solve the Ramsey problem with government commitment

Necessary for government to trade sequentially (roll over debt in each period). Suppose there are incomplete markets at date 0 and also at later dates, which means that in each period there is no future market for at least one of periods (i.e. at least one term of \(\{{}_i b_t\}_{t=i}^{\infty}\) has to be set zero in each period \(i\)), and therefore government has to issue (positively or negatively) new debt of total discounted value \(d_t\) in every period (cannot be done at once) where

$$ d_t=\sum_{s=t+1}^{\infty}p_s\left({}_{t+1}b_s-{}_t b_s\right) $$

in each period \(t\) to close the gap between government expenditure and revenue in that period \(t\):

$$ \underbrace{p_t\big(g_t+{}_t b_t-\tau_t(1-x_t)\big)}_{\text{government deficit at date }t}=d_t=\underbrace{\sum_{s=t+1}^{\infty}p_s\left({}_{t+1}b_s-{}_t b_s\right)}_{\text{revenue from new debt issue}} \tag{19.12} $$

i.e. to make the government break even in period \(t\) and the budget constraint still hold in period \(t+1\) without changing the tax rates commitment. To see why such \(d_t\) has this property, consider the government's budget constraint at date \(t\):

$$ \sum_{s=t}^{\infty}p_s\left[g_s+{}_t b_s-\tau_s(1-x_s)\right]=0 \tag{19.13} $$

Plug in (19.12) and rewrite (19.13):

$$ 0=p_t\big(g_t+{}_t b_t-\tau_t(1-x_t)\big)+\sum_{s=t+1}^{\infty}p_s\left[g_s+{}_t b_s-\tau_s(1-x_s)\right]=\sum_{s=t+1}^{\infty}p_s\left[g_s+{}_{t+1}b_s-\tau_s(1-x_s)\right] $$

which means the government budget constraint in period \(t+1\) with updated debt structure \(\{{}_{t+1}b_s\}_{s=t+1}^{\infty}\) still holds without changing \(\tau_s\)'s. So as long as government is rolling over its debt to meet each period's deficit in this way, it can keep doing this in all future periods and still keep its budget constraint satisfied in the next period and the tax rate commitment followed.

Same optimal \(\{c_t,x_t\}\) as in step 1 (complete markets). Note that the amount \(d_t\) for each \(t\) is exactly the amount of goods transfer at date \(t\) specified by the future contracts in step 1 with complete future markets at date 0. So the solution \(\{c_t,x_t\}_{t=0}^{\infty}\) doesn't change at all if government has to keep to its date 0 tax rate policy commitment.

Pattern of change in total discounted government debt. Denote government's total discounted debt obligation in each period as

$$ b_t\equiv\sum_{s=t}^{\infty}p_s\left({}_t b_s\right) $$

which is generally changing over time in the incomplete future markets case as a result of dynamic debt issuing in each period to keep the government breaking even.

Figure 13 (Government Debt, paraphrased): government debt \(b_t\) starts to grow from \(0\) over periods \(1,2,\ldots,T\) (no expenditure but positive tax revenue), peaks, and then declines, ending up negative around \(t=T+n\) because all later infinite periods' expenditures are zero, so the tax revenue in those periods (by optimality) must be useful to pay for debt coupons; it finally returns to \(0\).

19.2.4 Step 3: time consistency with sequential trading

• In step 1, complete markets at date 0, no sequential trading: all decisions are made at date 0 and everything is fixed at date 0 is only true when the government keeps to its commitment of \(\{\tau_t\}_{t=0}^{\infty}\) set at date 0. If the government can break this commitment, or the current government retires and its successor can re-optimize and announce a new tax rates policy, then the tax rates policy might change! So complete markets at all dates and no sequential trading don't guarantee time consistency.
• In step 2, incomplete markets at date 0, government has to issue debt at least at date 1 to make itself break even; in particular, we proved that the net new debt issue \(d_t=p_t(g_t+{}_t b_t-\tau_t(1-x_t))\) can make the government break even at date \(t\) while still satisfying the budget constraint at date \(t+1\).
  • in this incomplete markets case, we do have latitude for maturities, i.e. there are infinitely many ways to net issue new debt with total value of \(d_t\) because the different mixture of maturity structures can bring about the same total discounted present value.
  • consumer doesn't care about the maturity structure of government debt: the resource that is really taken away by the government is \(g_t\), so the resource constraint in each period \(c_t+x_t+g_t=1\) implies that \(c_t+x_t\) is exogenously determined by \(g_t\); as long as the relative price distortion \(\tau_t\) is fixed by government commitment, the choice of \(c_t\) and \(x_t\) for the household would be fixed as well, which is not changing with the maturity structure of new government debt.
  • in step 2, we imposed tax rates policy commitment on government, which means that government cannot re-optimize later by choosing another tax rate sequence.
  • but if we give the government the opportunity to re-optimize later, then it may have the incentive to choose a different tax rates sequence of \(\tau_t\)'s based on its debt structure at that time:
    • once the tax rates are fixed, government debt structure cannot affect consumer's first order conditions;
    • but the government budget constraint is dependent on the debt structure;
    • so the implementability constraint, which is a combination of consumer's f.o.c. and government's budget constraint in the decision period, might be affected by the debt structure in that decision period;
    • therefore, the government's solution \(\{c_t,x_t\}_{t=s}^{\infty}\) and its implied \(\{\tau_t\}_{t=s}^{\infty}\) might be different depending on the debt structure at date \(s\), i.e. \(\{{}_s b_t\}_{t=s}^{\infty}\);
    • how the government issue new debt (with some possible maturity structures) in each period does affect whether or not the government wants to re-optimize \(\{\tau_t\}_{t=s}^{\infty}\) at a later date \(s\).
• We want to have a time consistent policy by finding a way of issuing new debt such that the government doesn't want to make any change to \(\{\tau_t\}_{t=s}^{\infty}\) at date \(s\) when it has re-optimization opportunity.
  • for incomplete markets case, we need the wanted specific roll over to be doable for the government in each period, so the markets must be complete for the maturities needed in that wanted specific roll over, which in general requires markets to be complete: suppose markets are incomplete at date 0, then by (19.17) we know that the maturities needed for \(\{{}_1 b_t\}_{t=1}^{\infty}\) are all maturities from \(t=1\) to \(\infty\), so the markets need to be complete at date 0 for the government to carry out this specific debt roll over at date 0, which contradicts that markets are incomplete at date 0.
  • for complete market case, sequential trading is not necessary, but if it wants to never have incentive to change \(\{\tau_t\}_{t=s}^{\infty}\), then even in complete markets case the government still need to do sequential debt roll over (i.e. sequential trading of future contracts).

For simplicity, we may base the following discussion on the complete markets case. Recall that at date 0, the government has the implementability constraint as \(\sum_{t=0}^{\infty}\beta^t[(c_t-{}_0 b_t)u_c-(1-x_t)u_x]=0\) and the government problem's f.o.c. are summarized by

$$ (u_c-u_x)+\theta_0\left[(c_t-{}_0 b_t)(u_{cc}-u_{cx})+(1-x_t)(u_{xx}-u_{xc})\right]=0 \tag{19.14} $$

where \(\theta_0=\frac{\lambda_0}{1+\lambda_0}\) and \(\lambda_0\) is the Lagrangian multiplier of implementability constraint at date 0. At date 1, the government has the implementability constraint as \(\sum_{t=1}^{\infty}\beta^t[(c_t-{}_1 b_t)u_c-(1-x_t)u_x]=0\) and the government problem's f.o.c. are summarized by

$$ (u_c-u_x)+\theta_1\left[(c_t-{}_1 b_t)(u_{cc}-u_{cx})+(1-x_t)(u_{xx}-u_{xc})\right]=0 \tag{19.15} $$

where \(\theta_1=\frac{\lambda_1}{1+\lambda_1}\). For each \(s\ge 1\), denote

$$ \hat a_s\equiv c_s+(1-x_s)\frac{u_{xx}(c_s,x_s)-u_{xc}(c_s,x_s)}{u_{cc}(c_s,x_s)-u_{cx}(c_s,x_s)} $$

Then, if (19.14) and (19.15) have the same solution of \(c_t,x_t\) for each \(t=s\ge1\), then it must be that \(\forall s\ge1\)

$$ \theta_0\left(\hat a_s-{}_0 b_s\right)=\theta_1\left(\hat a_s-{}_1 b_s\right) \tag{19.16} $$

By the discussion in step 1, as long as the government needs to raise tax revenue in period 0 and 1, we have that \(\theta_1>0\) and \(\theta_0>0\). (19.16) implies that if \({}_1 b_s\) satisfies

$$ {}_1 b_s=\hat a_s-\frac{\theta_0}{\theta_1}(\hat a_s-{}_0 b_s)={}_0 b_s+\left(1-\frac{\theta_0}{\theta_1}\right)(\hat a_s-{}_0 b_s) \tag{19.17} $$

for all \(s\ge1\), then the solution \(\{c_t,x_t\}_{t=1}^{\infty}\) won't change at re-optimization date 1, and thus the implied government's tax rates policy is time consistent at date 1. Note \({}_0 b_s\) and \(\hat a_s\) are all known numbers based on government's solution at date 0, so we only need to solve for \(\frac{\theta_0}{\theta_1}\) in order to pin down \({}_1 b_s\): plug (19.17) into the government's budget constraint at date 1 \(\sum_{s=1}^{\infty}p_s[g_s+{}_1 b_s-\tau_s(1-x_s)]=0\), where everything is based on government's solution at date 0, and then the only unknown \(\frac{\theta_0}{\theta_1}\) can be uniquely solved. So with \(\frac{\theta_0}{\theta_1}\) just solved, we have obtained a unique maturity structure \(\{{}_1 b_t\}_{t=1}^{\infty}\) at date 1 that guides the issuing of new debt at date 0 such that the tax rates policy is time consistent. We can repeat this exercise for period \(1\to2\), \(2\to3\), and so on to obtain a path of maturity structures such that the tax rates policy is time consistent in every period.

• Notice that we started with complete markets for all maturity debts, but it seems unnecessary up to this point: even if the markets are not complete for all maturities, as long as this consol is available for trade in all periods, the government can roll over its debt such that the tax rates policy is time consistent.
• To draw a conclusion, even when the government has complete markets at date 0, it might still be equal to do sequential trading as opposed to making all decisions at date 0 because of the incentive to have time consistency (i.e. no incentive to re-optimize \(\{\tau_t\}_{t=s}^{\infty}\) in later period \(s\)).

19.3 Stochastic shocks to government expenditure

Suppose we have stochastic shocks (Markov chain) to \(\{g_t\}_{t=0}^{\infty}\), every step is the same as the deterministic case except:

• for household's problem, we use conditional expectation of the discounted sum of future utility.
• for government's problem, since the government's budget constraint is stochastic, it has to set state-contingent tax rates policy for all periods at date 0 and issue state-contingent debt (coupon payments are based on expenditure shock realization) in each period to insure itself against the stochastic expenditure shocks.
• such state-contingent debt payoff will appear in household's budget constraint to replace the debt payoff in the deterministic case.

With such changes (which are simply changes of notations), we can easily adopt the same method for deterministic case to analyze the case with stochastic expenditure shocks.