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Econ-411-A1 · revision r1 · on Econ-411

GPT 6 Astra Ultra attempt on Econ-411

Partial result · LLM-generated · gpt 6 astra ultra ·

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Manuscript

Contents
  1. International risk sharing
  2. Target and the measurement problem
  3. Fixing the projection before comparing policies
  4. An exact three-country complete-market benchmark
  5. Asymmetric risk bearing and its limits
  6. Identification and next empirical step
  7. Completion Estimate
  8. References

International risk sharing

Target and the measurement problem

The target is the change in a country-specific residual variance of log consumption growth under an integration intervention. This is not itself a welfare measure. We establish an exact residual-variance decomposition, solve a finite-state three-country insurance benchmark, and identify what a policy experiment would need to observe. The monetary, safe-asset and nominal-rigidity extension remains open in this attempt.

Fixing the projection before comparing policies

Let $g(a)$ be log consumption growth in regime $a$, and let $Z$ be a vector of common predictors with an intercept removed after demeaning. In each regime assume second moments and nonsingular $\Sigma_{ZZ}(a)$. The population linear projection has coefficient $b(a)=\Sigma_{ZZ}(a)^{-1}\Sigma_{Zg}(a)$. Its residual variance is \[ v(a)=\operatorname{Var}(g(a))- \Sigma_{gZ}(a)\Sigma_{ZZ}(a)^{-1}\Sigma_{Zg}(a). \tag{1} \]
Proposition: If instead one holds a baseline coefficient $b_0$ fixed, the residual variance in regime $a$ equals \[ v(a)+(b_0-b(a))^{\top}\Sigma_{ZZ}(a)(b_0-b(a)). \tag{2} \]
Proof: Subtract the regime mean and write $g-b_0^{\top}Z=(g-b(a)^{\top}Z)+(b(a)-b_0)^{\top}Z$. The first summand is orthogonal to every coordinate of $Z$ by its normal equations. Taking variances proves (2), and minimizing the quadratic expression in $b$ gives (1). ∎
Thus a policy that changes exposure to a common shock can appear to worsen insurance simply because an obsolete predictor is held fixed. Either convention is definable, but they estimate different quantities. At singular covariance, use a fixed predictor subspace and orthogonal projection or a stated generalized inverse; arbitrary regularization changes the target unless its error is controlled.

A second elementary trap is welfare. Let $g(0)=m_0+e_0$ and $g(1)=m_1+e_1$ with lower residual variance in regime one but $m_1<m_0$. For one-period log utility, expected utility is determined by the mean of log consumption, so welfare falls regardless of the lower variance. In a concrete positive-consumption example, baseline $c$ is $1$ or $2$ equally often and integrated $c$ is constantly $0.5$. Residual log variance falls to zero while expected log consumption falls from $\tfrac12\log2$ to $-\log2$. This is a logical example, not a feasible insurance policy with unchanged aggregate resources; it shows why resource and mean effects must be separately modeled before interpreting the statistic.

An exact three-country complete-market benchmark

Label countries $U,A,E$. There is one representative household per country, one consumption good, a baseline date with consumption one, and eight equally likely date-one states $s=(s_U,s_A,s_E)\in\{-1,1\}^3$. Endowments are $y_i(s)=1+h s_i$ with $0<h<1$. Households have identical log utility. Compare autarky with complete contingent-claim trade, no transaction costs and common beliefs. Identical initial expected wealth under symmetric equilibrium prices supports equal Pareto weights. The state resource constraint is $\sum_i c_i(s)=3+h\sum_i s_i$.
Theorem: The equal-weight efficient allocation, supportable by complete contingent claims, is \[ c_i^*(s)=1+\frac h3(s_U+s_A+s_E)>0. \tag{3} \] Every country has weakly greater ex ante log utility than under autarky, strictly greater when $h>0$.
Proof: For each state, strict concavity and the first-order conditions $1/c_i=\lambda_s$ imply equal consumption, giving (3). Positive Arrow prices proportional to state probability divided by $c_i^*(s)$ support individual optimality. Symmetry of the endowment law and prices implies each country has one third of aggregate state-priced wealth, so these consumptions satisfy its budget. Statewise Jensen gives $3\log(\sum_i y_i/3)\ge\sum_i\log y_i$, with strict inequality whenever endowments differ. Take expectations and use exchangeability to conclude the same ex ante improvement for each country. Mixed-sign states have positive probability for $h>0$, making the inequality strict. ∎
With only a constant predictor, the autarky variance of log growth is \[ v_0=\frac14\left[\log\frac{1+h}{1-h}\right]^2. \] Under integration, log consumption takes values $\log(1-h),\log(1-h/3),\log(1+h/3),\log(1+h)$ with probabilities $1/8,3/8,3/8,1/8$. Therefore its exact residual variance is \[ v_1=\sum_{j=1}^4 p_j\ell_j^2- \left(\sum_{j=1}^4p_j\ell_j\right)^2. \tag{4} \] These finite sums determine the catalogue statistic without linearizing log consumption. If the declared common predictor space contains all functions of aggregate endowment, the integrated residual is instead identically zero, because (3) is a function of that aggregate. The choice of information space is consequently economically material even in this simple solved benchmark.

Asymmetric risk bearing and its limits

For a level-consumption complete-market benchmark with CARA utility $-\exp(-\gamma_i c_i)$ and positive Pareto weights $\lambda_i$, the same statewise calculation, at an interior allocation, yields \[ c_i(s)=a_i+\theta_iY(s),\quad \theta_i=\frac{\gamma_i^{-1}}{\sum_j\gamma_j^{-1}},\quad \sum_i a_i=0,\qquad Y(s)=\sum_i y_i(s). \tag{5} \] To verify, the first-order conditions are $\lambda_i\gamma_i\exp(-\gamma_i c_i)=\kappa_s$. Take logarithms, divide by $\gamma_i$, sum over countries and solve for $\log\kappa_s$; the constants produce $a_i$ and the coefficient of $Y$ produces $\theta_i$. On a finite state space choose endowments and weights so all $c_i$ are positive; otherwise nonnegativity constraints create corners and (5) is not valid. More risk-tolerant countries bear more aggregate level risk, but (5) is not a formula for log residual variance or for safe-dollar bond demand. It explicitly assumes away the incomplete markets and nominal features of the source-motivated extension.

Identification and next empirical step

Random assignment of an integration rule across independent three-country clusters would identify the law of $(g_i,Z)$ in each regime and therefore (1), provided exposure remains inside clusters, consumption measurement is comparable, and second moments exist. A single worldwide policy episode generally provides no such independent experiment. Instrumental exposure designs additionally need policy exclusion, support and a target compatible with their affected population.

An observed portfolio covariance matrix does not by itself identify endogenous counterfactual portfolio choices. Even if a reduced-form shock loading $B_0$ is known perfectly, a new admissible-asset set may produce any loading $B_1$ allowed by unspecified preferences, pricing and constraints. The relevant identified set must range over those structural primitives and equilibrium selectors, not hold the baseline portfolio fixed by default.

The results here supply exact checks for an eventual three-region model: its complete-market symmetric limit should reproduce (3)--(4); its CARA interior limit should reproduce (5); its reported variance effect must state its projection convention. Identifying safe-asset demand, exchange-rate valuation, nominal adjustment and distributional welfare remains the unsolved part.

Completion Estimate

30%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt derives exact residual-variance projection identities and solves symmetric three-country complete-market and asymmetric CARA insurance benchmarks. The source-motivated incomplete-market extension with safe-asset demand, exchange-rate valuation, nominal rigidities, endogenous portfolios, and household distributional welfare remains unresolved.

References

  • [source] R. Kekre and M. Lenel (2024), The Flight to Safety and International Risk Sharing, AER 114(6), 1650--1691. https://www.aeaweb.org/articles?id=10.1257/aer.20211319. The publisher abstract describes safe-dollar demand, unequal risk bearing and nominal rigidities. The finite-state complete-market benchmarks above are separate calculations, not solutions of that monetary model.