16. Aiyagari's Model (1994, Quarterly Journal of Economics)

The key assumption is that all individuals in the economy supply labors inelastically and the exogenous idiosyncratic labor supply shocks (e.g. health shocks) are i.i.d. across agents. So, in this model, we have no aggregate shock (by Law of Large Numbers, economy-wide labor supply is constant) and only have idiosyncratic shocks, whereas in Real Business Cycle model we only have aggregate shocks but no idiosyncratic shocks.

16.1 Set-up

16.1.1 Parameters

• Inelastic labor supply with exogenous stochastic shock:
• \(l\in\mathbb L=[\underline l,\bar l]\) with \(\underline l>0\). \(\mathbb L\) is the state space for state variable \(l\).
• \(G(l';l)\) is the transition function with Feller property (see §11.1.2), and \(g(l';l)\) is the density of \(G(l';l)\).
• suppose \(\Phi(l)\) is the unique stationary distribution, and \(\varphi(l)\) is the density of \(\Phi(l)\).
• normalization: the aggregate labor supply is normalized to 1 without uncertainty, i.e. \(\int_{\underline l}^{\bar l}l\cdot\varphi(l)dl=1\).
• Household preferences are assumed to have usual properties:
• utility function \(u(c)\) where \(c\) is consumption. Strictly increasing in \(c\), strictly concave in \(c\), has the Inada condition \(\lim_{c\to0}u'(c)=\infty\), \(\lim_{c\to\infty}u'(c)=0\). Utility discounting parameter \(\beta=\frac{1}{1+\rho}\) where \(\rho>0\).
• Technology has the usual properties:
• production function \(F(K,L)\). The undepreciated asset (capital) \(a\): the depreciation rate for capital is 0. Constant return to scale: define \(k=\frac KL\) and \(f(k)=F(k,1)\). Since aggregate labor supply is deterministic and normalized to 1, \(f(k)\) is the aggregate production of this economy. Aggregate household asset is \(k\). Equilibrium factor price: \(r=f'(k)\) and \(w=f(k)-kf'(k)\).
• other parameters: interest rate \(r\); borrowing limit for household \(B\). \(B\) is the lowest level of asset that a household can have, which implies \(B\le0\). When \(B=0\), it is equivalent to no borrowing constraint.

16.1.2 Assumptions

1. Make sure the household doesn't borrow too much: \(B\ge\hat B=-\dfrac{w\underline l}{r}\), where \(\hat B\) is the natural borrowing constraint that guarantees the household can repay the interest even with the lowest possible labor income.
2. Prevent the household from accumulating wealth to infinity: \(r<\rho\). Recall that for any two adjacent periods, if \(r\ge\rho\), then consider the saving problem \(u((1+r)s_{t-1}+wl_t-s_t)+\frac{1}{1+\rho}u(wl_{t+1}+(1+r)s_t-s_{t+1})\), the f.o.c. of saving \(s_t\) is $$u'\big(\underbrace{a_t+wl_t-s_t}_{c_t}\big)=\frac{1+r}{1+\rho}u'\big(\underbrace{wl_{t+1}+(1+r)s_t-s_{t+1}}_{c_{t+1}}\big)$$ which implies that \(c_t\le c_{t+1}\) is always true regardless of the labor income. Since the labor income has an ergodic distribution, \(c_t\le c_{t+1}\) for \(\forall t\) will make the household accumulate wealth to be consumed in infinite far away future.
3. Need labor supply to be variable enough to make household want to save at least in some period: for some \(\hat l\), \(u'(c^\star(w,\hat l))<\frac{1+r}{1+\rho}\int_{\underline l}^{\bar l}u'(c^\star(w,l'))g(l';\hat l)dl'\). Note that lower curvature (more linear) of \(u\) needs more variable \(l\). This assumption involves \((r,\rho,g,u')\).
4. Not too much risk aversion: there exists \(a^{\max}\) s.t. \(u'(w\bar l+ra^{\max})>\frac{1+r}{1+\rho}u'(w\underline l+ra^{\max})\). By this assumption, once the asset accumulates to \(a^{\max}\), the household doesn't accumulate more asset at all (consumes labor income and asset interest today), and even in the most risk averse scenario (highest today's labor income and lowest tomorrow's labor income) he still wants to consume more today and less tomorrow. So \(a^{\max}\) is the upper limit of asset that may not even be reached.

1. Fix a \((\hat w,\hat r)\) pair, and look at the household problem (Bellman equation).
2. Solve the household's problem to obtain the optimal consumption and saving decisions of a household.
3. (a) use the optimal saving rule to define a transition function of the pair \((a,l)\) where \(a\) is asset (capital, endogenous) and \(l\) is labor supply with exogenous shocks.
4. Characterize the stationary joint distribution of \((a,l)\) and calculate aggregate assets \(k^{agg}\) and aggregate labor supply \(l^{agg}=1\), which are deterministic by i.i.d. property of idiosyncratic shocks and law of large numbers.
5. Check whether the market clear:
6. (a) if \(\hat r=F_k(k^{agg},l^{agg}=1)\) and \(\hat w=F_l(k^{agg},l^{agg}=1)\), we're done.
7. (b) if not, try another pair \((\tilde w,\tilde r)\) and start from step 1 again until the market clears.

16.3 Household's Problem

16.3.1 Sequential problem $$V(a,l)=\max_{\{c_t\}_{t=0}^\infty}\mathbb E\left[\sum_{t=0}^\infty\beta^t u(c_t)\right]\quad\text{s.t.}\quad c_t\le wl_t+a_t(1+r)-a_{t+1}$$

16.3.2 Bellman equation

State variable \((a,l)\): $$V(a,l)=\max_{a'\in\Gamma(a,l)}\left\{u(wl+(1+r)a-a')+\beta\int_{\underline l}^{\bar l}V(a',l')g(l';l)dl'\right\}$$ where \(\Gamma(a,l)=[B,wl+(1+r)a]\).

16.3.3 First-order condition and envelop condition

• The f.o.c. for \(a'\): \(u'(wl+(1+r)a-a')\ge\beta\int_{\underline l}^{\bar l}V_a(a',l')g(l';l)dl'\).
• if \(a'\in\text{int}(\Gamma(a,l))\): \(u'(wl+(1+r)a-a')=\beta\int_{\underline l}^{\bar l}V_a(a',l')g(l';l)dl'\).
• if \(a'=B\): \(u'(wl+(1+r)a-a')>\beta\int_{\underline l}^{\bar l}V_a(a',l')g(l';l)dl'\).
• note that \(a'=wl+(1+r)a\) is ruled out by the Inada condition, so we don't have \(u'(wl+(1+r)a-a')<\beta\int V_a g\,dl'\).
• The envelop condition: \(V_a(a,l)=(1+r)u'(wl+(1+r)a-a')\).

16.3.4 Properties of value function \(V(a,l)\)

• \(V\) is well-defined (existence and uniqueness): for each fixed \(l\), the state space for \(a\) is \([B,a^{\max}]\), which is a convex set; \(\Gamma(a,l)=[B,wl+(1+r)a]\) is a non-empty compact-valued and continuous correspondence; the period return function \(u(wl+(1+r)a-a')\) is bounded and continuous in \(a\) and \(a'\). By the same proof as in the deterministic case in §10.1.2, \(V(a,l)\) is well-defined for each fixed \(l\). The requirements should be satisfied at each fixed \(l\).
• Properties of \(V\) as a function of \(a\):
• (1) Strict monotonicity of \(V\): \(\Gamma(a,l)\) is monotone in \(a\) for each fixed \(l\) (\(a'\ge a\Rightarrow\Gamma(a,l)\subseteq\Gamma(a',l)\)); \(F(a,a',l)=u(wl+(1+r)a-a')\) is strictly increasing in \(a\) for each fixed \(l\). So \(V(a,l)\) is strictly increasing in \(a\) for each fixed \(l\).
• (2) Concavity of \(V\): \(F(a,a',l)=u(wl+(1+r)a-a')\) is weakly (not strictly) concave in \((a,a')\) for each fixed \(l\). But we can go back to the sequence problem to see the strict concavity of \(V(a,l)\) in \(a\) (we can start from \(a_1
• (3) Differentiability of \(V\) at \(a\): \([B,a^{\max}]\) is convex; \(V(a,l)\) is strictly concave in \(a\) for each fixed \(l\); \(F\) is differentiable in \(a\) for each fixed \(l\); suppose \(a\in\text{int}([B,a^{\max}])\) and \(a'\in\text{int}(\Gamma(a,l))\). Note we have a borrowing constraint \(a'\ge B\), so if we decrease \(a\) to some level, then thereon \(a'=B\) is always true — at that specific level of \(a\) corresponding to the shift from interior solution to corner solution of \(a'\), we have a kink in the value function, not differentiable in \(a\). So elsewhere \(V(a,l)\) is differentiable in \(a\).
• Properties of optimal saving \(a'=s(a,l)\) as a function of \(a\):
• for interior solutions of \(a'=s(a,l)\in\text{int}(\Gamma(a,l))\): \(V(a,l)\) is strictly concave in \(a\) for each fixed \(l\), so \(a'=s(a,l)\) is single valued and continuous in \(a\); \(F\) is strictly concave, so \(a'=s(a,l)\) is strictly increasing in \(a\) for each fixed \(l\) with slope less than \(1+r\).
  • Figure 4 (paraphrased): the horizontal axis is next-period asset \(a'\). The RHS of the f.o.c. is decreasing; the LHS (with "smaller \(a\)"/"larger \(a\)") is increasing. The intersection point with higher \(a\) has lower vertical height, i.e. less \(u'(\cdot)\) and thus higher consumption, so consumption is also increasing in \(a\), which makes \(a'\) and \(c\) share the benefit of increase in \(a\), slope \(<1+r\).
  • We only know the slope of \(s(a,l)\) is less than \(1+r\). It is reasonable to impose the further assumption that the slope is always less than 1 to avoid possible problems.
  • Figure 5 (Multiple Ergodic Sets, paraphrased): the 45-degree line with \(s(a,\bar l)\) and \(s(a,\underline l)\). \(\hat a\) is the minimum amount of current asset above which the household would like to start saving given his current labor bar hits the lower bar \(\underline l\). If we allow slope greater than 1 for some points, then \(s\) may have multiple crossing points with the 45-degree line (suppose \(B=0\)): when the optimal saving function crosses the 45-degree line, the optimal asset for next period is the same as current asset, so \([0,a_1]\) and \([a_2,a_3]\) are two ergodic sets and \([a_1,a_2]\) is a collection of transient states (positive probability of leaving but never returning), contradicting the assumption of a unique ergodic set. So we would rather impose the assumption that \(s(a,l)\) has slope w.r.t. \(a\) less than 1.
  • Figure 7 (paraphrased): another good reason — if \(s(a,\bar l)\) has slope greater than 1, then it would never intersect with the 45-degree line, so there is no upper limit for asset accumulation, contradicting bounded asset accumulation.
• for corner solution \(a'=B\): for \(a_1>a_2\), we may have corner solution for both, i.e. \(s(a_1,l)=s(a_2,l)=B\), or equivalently \(u'(wl+(1+r)a_1-B)>\beta\int V_a(B,l')g\,dl'\) and \(u'(wl+(1+r)a_2-B)>\beta\int V_a(B,l')g\,dl'\). So \(a'=s(a,l)\) is only weakly increasing in \(a\) for each fixed \(l\). Figure 6 (corner solution, paraphrased).
• Figure 7 (Optimal Saving, paraphrased): the 45-degree line, \(s(a,\bar l)\), \(s(a,\underline l)\), \(\hat a\), \(\bar a\). \(\bar a\) is the actual maximum amount of asset that the household would like to accumulate to. Assume labor supply shocks are i.i.d. and \(s(a,l)\) is weakly increasing in \(l\); define \(\bar a\) as the crossing point of \(s(a,\bar l)\) and the 45-degree line, which would never be surpassed by the household.
• Property of optimal consumption: Figure 8 (Optimal Consumption, paraphrased): \(c(a,\bar l)\), \(c(a,\underline l)\), \(w\bar l\), \(w\underline l\), \(\hat a\), \(\bar a\). Since we defined \(\hat a\) to be the point starting saving given current labor supply hits bottom \(\underline l\), we have \(s(a,\underline l)=0\) for \(a\le\hat a\), which implies for \(a\le\hat a\), \(c(a,\underline l)=w\underline l+(1+r)a-0\), i.e. slope \(1+r\) in \([0,\hat a]\).
• Property of \(V\) as a function of \(l\): \(F(a,a',l)=u(wl+(1+r)a-a')\) is strictly increasing in \(l\); \(\Gamma(a,l)=[B,wl+(1+r)a]\) is monotone in \(l\) for all \(a\) (\(l'\ge l\Rightarrow\Gamma(a,l)\subseteq\Gamma(a,l')\) for \(\forall a\)). But we don't know whether operator \(M\) maps an increasing (in \(l\)) function \(V(a,l)\) to an increasing (in \(l\)) function \((Mh)(a,l)=\int_{\underline l}^{\bar l}V(a,l')g(l';l)dl'\). So we cannot conclude \(V(a,l)\) is increasing in \(l\) for \(\forall a\).
• Property of optimal saving \(a'=s(a,l)\) as a function of \(l\): If \(l\)'s are i.i.d., then \(g(l';l)=g(l')\). By the same logic as for \(s(a,l)\) as a function of \(a\) in Figure 4 and Figure 5: for interior solutions of \(a'\), \(a'=s(a,l)\) is strictly increasing in \(l\) with slope less than \(w\) (the intersection point with higher \(l\) has lower vertical height, i.e. less \(u'(\cdot)\) and thus higher consumption, so consumption is also increasing in \(l\), which makes \(a'\) and \(c\) share the benefit of increase in \(l\)); for corner solution of \(a'\), \(a'=s(a,l)\) is weakly increasing in \(l\). But if \(l\)'s are not i.i.d., we won't have this property because we don't know how \(\beta\int V_a(a',l')g(l';l)dl'\) changes with \(l\).

Now we want to show that there is a unique stationary joint distribution of \((a,l)\). For simplicity, we will assume that the labor shocks are i.i.d. and assume that \(B=0\). Then, since today's labor supply has no information about tomorrow's labor supply, the optimal saving function \(a'=s(a,l)\) can be reduced to another function $$a'=\sigma((1+r)a+wl)$$ with only one argument. Then, by the argument above for the weakly increasing property of \(a'=s(a,l)\) (strictly for interior and weakly for corner) as a function of \(a\), we have that \(a'\le z\Leftrightarrow(1+r)a+wl\le\sigma^{-1}(z)\). In order to have the function \(\sigma^{-1}(\cdot)\) be well-defined, we require that: - for \(z>0\), by strictly increasing property discussed above, \(\sigma^{-1}(z)\) is a well-defined single-valued function. - for \(z=0\), by possibly weakly increasing property discussed above, \(\sigma^{-1}(0)\) might be a correspondence with multiple images. So we let \(\sigma^{-1}(0)\) be the smallest value in the correspondence so that \(\sigma^{-1}(\cdot)\) is a well-defined function for \(z\in[0,\bar a]\).

Figure 9 (Joint Distribution of \((a,l)\), paraphrased): the vertical axis is \(l\) (\(\underline l\) to \(\bar l\)), the horizontal axis \(a\) (0 to \(\bar a\)). The line \(a'=z\) is \(l=\frac1w[\sigma^{-1}(z)-(1+r)a]\). The blue shaded area corresponds to \((a,l)\) s.t. \(a'=z=0\), i.e. $$(1+r)a+wl\le(1+r)\hat a+w\underline l\Rightarrow l\le-\frac{1+r}{w}a+\left(\frac{1+r}{w}\hat a+\underline l\right)$$ which yields optimal saving zero \(a'=0\), so there is probability mass at point \(a'=0\). The red shaded area corresponds to \((a,l)\) s.t. the optimal saving \(a'\le z\), i.e. \((1+r)a+wl\le\sigma^{-1}(z)\Rightarrow l\le\frac1w[\sigma^{-1}(z)-(1+r)a]\).

Recall that \(\Phi(l)\) is the unique stationary distribution of \(l\) with density \(\varphi(l)\). Let \(M(z)\) denote the (yet unknown) stationary distribution of assets. Then, $$M(z)=\mathbb P(a\le z)=\int_{\underline l}^{\bar l}M\left(\underbrace{\frac{\sigma^{-1}(z)-wl}{1+r}}_{=a}\right)\varphi(l)dl\quad\forall z\in[0,\bar a]$$ which is the sum of probabilities assigned to all points in the red shaded area in Figure 9. Now this becomes a fixed point problem for operator \((T\tilde M)(z)=\int_{\underline l}^{\bar l}M\left(\frac{\sigma^{-1}(z)-wl}{1+r}\right)\varphi(l)dl\). Suppose we have convergence in the iteration and successfully obtain a solution \(M(z)\) to the functional equation \((T\tilde M)(z)=\tilde M(z)\). Then, such \(M(z)\) (with a mass at \(z=0\)) becomes the unique stationary distribution for assets \(a\).

16.5 Check Whether the Market Clears

With \(M(z)\) as the stationary distribution for assets \(a\), we can calculate the average asset \(k^{avg}=\int_0^{\bar a}a\cdot dM(a)\). Since the labor supply is normalized to unit and by the Law of Large Numbers, \(k^{agg}=k^{avg}=\int_0^{\bar a}a\cdot dM(a)\). Note that \(k^{agg}\) is a constant, not a random variable. Then it remains to check market clearing: - If \(\hat r=F_k(k^{agg},l^{agg}=1)\) and \(\hat w=F_l(k^{agg},l^{agg}=1)\), we're done. - if not, try another pair \((\tilde w,\tilde r)\) and start from step 1 again until the market clears.

Once we obtain \(k^{agg}=\int_0^{\bar a}a\cdot dM(a)\): - we can interpret \(k^{agg}\) as a function of \(r\) to make it a supply function of aggregate capital. - we can let the production equilibrium condition \(r=f'(k)\) be a demand function of aggregate capital. - then we can directly get to the point where the market clears by finding an \(r\) that equates demand and supply. See Figure 10 and Figure 11.

Figure 10 (Supply and Demand of Assets, paraphrased): the vertical axis is \(r\), the horizontal axis \(a\) (or \(k\)). \(r=\rho\) is a horizontal line. \(D\) is the demand curve (downward, \(r=f'(k)\)). \(S_{FI}\) is the supply curve when fully insured (horizontal at \(r=\rho\)). \(S_{NI,B=0}\) is the supply curve not insured with \(B=0\). \(S_{NI,B=\hat B}\) is the supply curve not insured with the natural borrowing constraint \(B=\hat B\). - \(D\) (black line): the demand curve of assets (capital), follows \(r=f'(k)\). - \(S_{FI}\) (yellow line): the supply curve when the household is fully insured by some market mechanisms. To see why it is horizontal at \(r=\rho\): consider the f.o.c. of saving \(s_t\), \(u'(\underbrace{wl_t+(1+r)s_{t-1}-s_t}_{c_t})=\frac{1+r}{1+\rho}u'(\underbrace{wl_{t+1}+(1+r)s_t-s_{t+1}}_{c_{t+1}})\). Since each period labor income is completely insured, \(l_t\)'s are constant, so: if \(r<\rho\), \(c_{t+1}\rho\), \(c_{t+1}>c_t\) for \(\forall t\), means infinite saving in steady state, aggregate assets supply infinite; thus aggregate assets supply is perfectly elastic at \(r=\rho\). - \(S_{NI,B=0}\) (blue line): the supply curve not insured at all with borrowing constraint \(B=0\). Theoretically we cannot prove it is upward sloping: higher interest rates means higher return on assets, so the household has more incentive to save (price effect); higher interest rates means lower aggregate capital, means lower labor income, so the household has less incentive to save (income effect); how these two forces compare is unclear; so we have to assume the price effect dominates the income effect to have the upward sloping supply curve. - \(S_{NI,B=\hat B}\) (red line): the supply curve not insured at all with the natural borrowing constraint \(B=\hat B=-\frac{w\underline l}{r}\). Since setting the borrowing constraint \(B=\hat B\) gives the household more latitude to choose negative assets than \(B=0\), intuitively \(S_{NI,B=\hat B}\) is weakly to the left of \(S_{NI,B=0}\).

Each intersection point of the demand curve and supply curve pins down, for each case of economy, the equilibrium interest rate, aggregate capital, household optimal saving function, household optimal consumption function, and the stationary distribution of individual household assets.

Figure 11 (Quick Way to Locate the Intersection Point, paraphrased): the vertical axis is \(r\), the horizontal axis \(a\) (or \(k\)). \(D\) demand, \(S_{NI,B=0}\) supply, \(r_1\) (high), \(r_2\) (low), the intersection. - Figure 11 gives us a quick way to computationally locate the intersection point of the demand and supply curve: - first start with an arbitrary \(r_1\); WLOG suppose \(k^{agg}(r_1)>(f')^{-1}(r_1)\). - then try a sufficiently small \(r_2\) such that \(k^{agg}(r_2)<(f')^{-1}(r_2)\). - the basic idea is to use \(r\in(r_2,r_1)\) to further approximate \(r^\star\) s.t. \(k^{agg}(r^\star)=(f')^{-1}(r^\star)\). E.g. use \(r_3=\frac12(r_1+r_2)\): if \(k^{agg}(r_3)<(f')^{-1}(r_3)\), repeat this process for \(r_4\in(r_3,r_1)\); if \(k^{agg}(r_3)>(f')^{-1}(r_3)\), repeat this process for \(r_4\in(r_2,r_3)\); then the sequence \(r_n\) should converge to \(r^\star\) quickly.