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 AiyagariQJE1994.ox

Households face uninsurable idiosyncratic shock; markets are incomplete.

Household Dynamic Program

1. CCP Smoothing: none
2. Clock: Ergodic
3. \(\alpha = (a),\) assets to hold at the end of today to earn interest tomorrow (original: \(a_{t+1}\))
4. \(\theta = (h,A),\)
   \(l\): Tauchen() shock to earnings/labor supply
   \(A\): Assets held, a LaggedAction or, more generally, LiquidAsset
5. Consumption: \(C = A(1+r) + we^l - a\)
6. \(A(\theta) = \{ a: C \ge -\overline{a} \},\) where \(\overline{a}=0\) for all reported values.
7. \(U(\alpha;\theta) = { C^{1-\mu} -1 \over 1-\mu }.\)

Equilibrium Conditions

1. Aggregate production: \(f(K,L) = K^\alpha L^{1-\alpha}\)
   \(K = \) stationary per capita capital \( = \sum_\theta P_\infty(\theta)A.\)
   \(L = \) per capita labour supply \( =\sum_\theta P_\infty(\theta)e = \exp{\sigma}\)
   \(MP = \) vector of marginal products \(= \pmatrix{ f_K \cr f_L }\)
2. Equilibrium price vector \(p^\star = \pmatrix{r^\star \cr w^\star}.\)
3. Depreciation of capital \(\gamma\)
4. Equilibrium System: $$MP - p^\star - \pmatrix{\gamma \cr 0} = \pmatrix{0\cr 0}$$
5. Closed form solution for equilibrium \(w\) in terms of \(r\): $$w^\star = ?? r^\star$$
6. Equilibrium reduces to a OneDimSystem solved using bracket-and-bisect requiring nested solution of the agent's model and computed value of stationary capital stock \(K\).

Author:

© 2020 Nam Pham and Christopher Ferrall

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