Risk Sharing Tests and Covariate Shocks: Drought, Floods, and Pests in Uganda

This paper identifies and corrects a fundamental flaw in the standard methodology for testing efficient risk-sharing when shocks are covariate (affecting common prices rather than only individual incomes). The standard Townsend (1994) approach infers marginal utilities of expenditure (MUEs) from total expenditures, which implicitly assumes homothetic preferences — specifically Constant Relative Risk Aversion (CRRA) — under which all goods have unitary income elasticities and a single scalar price index captures all price effects. Ligon demonstrates that this assumption causes the standard test to fail when applied to covariate shocks such as droughts, floods, and agricultural pests, because these shocks change relative prices in ways that cannot be captured by a single price index. The perverse consequence is that in Ugandan data, every covariate shock — drought, floods, pests, and adverse prices — appears to improve household welfare under the CRRA specification (significant positive coefficients of 0.046, 0.097, 0.095, and 0.103 respectively, all significant at p<0.01), a result the paper argues is mechanically induced by the mis-specification rather than reflecting reality.

The paper makes two core theoretical contributions. First, it characterizes the complete class of preferences that permit MUE inference from expenditure data alone — specifically, requiring that item-level expenditures be “lambda-separable” (additively separable in the MUE and prices). Solving the resulting functional equations yields exactly two families of semiparametric demand systems: Constant Frisch Elasticity (CFE) demands (a generalization of CRRA) and Generalized Stone-Geary demands. Only CFE demands are tractable for panel estimation. Second, the paper shows that under CFE preferences, log expenditures on each good j follow the system: log x^j_it = a_j(p_t) + g_j(z_it) + beta_j * w_it + epsilon^j_it, where beta_j is the good-specific Frisch elasticity and w_it = -log lambda_it is the negative log MUE. This allows price effects to enter flexibly through good-time fixed effects rather than a single index, and MUEs to be recovered via factor analysis on the residual covariance matrix.

The empirical work uses eight waves of the Ugandan National Panel Surveys (2005–2020), an unbalanced panel of 5,601 distinct households yielding 22,791 usable household-year observations across 41 consumption goods (primarily food items). Uganda is divided into four regional markets, producing 32 market-year cells and 1,312 market-year-good dummies. Estimated Frisch elasticities vary substantially across goods — passion fruit is roughly three times as income elastic as cassava — emphatically rejecting the hypothesis of equal elasticities required by CRRA.

Using CFE-estimated MUEs, the risk-sharing test shows that none of the four covariate shocks has a significant effect on welfare (CFE coefficients: drought 0.010, floods 0.035, pests 0.041, adverse prices -0.043, all insignificant). The pattern holds across all time windows from 0–12 months: 42 of 52 covariate shock coefficients are significant and positive in the CRRA specification, versus only 4 of 52 in the CFE specification — barely above the 2.6 false positives expected under the null. These findings indicate that the welfare impacts of covariate shocks in Uganda operate primarily through the common price channel rather than through idiosyncratic income variation, meaning they are broadly shared within market-regions. Idiosyncratic income shocks, by contrast, show the expected pattern: they reduce welfare significantly in both specifications (CFE: 0.050***, CRRA: 0.071***), and health shocks are significant only in CFE (−0.059**).

Q: Why does the standard CRRA risk-sharing test fail for covariate shocks? A: Under CRRA preferences, MUEs depend on total expenditures only through a single scalar price index pi(p). When a covariate shock raises prices of inelastic goods (primarily food), total food expenditures increase even as actual consumption quantities fall. Because risk-sharing tests based on CRRA total expenditures cannot separate this price effect from a welfare improvement, the shock appears to raise welfare. The disturbance term in the CRRA TWFE regression depends on the very prices affected by covariate shocks, violating the exclusion restriction.

Q: What is the lambda-separability condition, and why does it matter? A: Lambda-separability requires that for each good j, some transformation phi_j of expenditures on that good can be written as the sum of a function of prices and a function of the MUE: phi_j(x_j(p,lambda)) = a_j(p) + b_j(lambda). This property is necessary for time fixed effects to absorb price variation and household fixed effects to absorb Pareto weights, which is the identification strategy behind all TWFE risk-sharing tests. Without it, no panel estimator using only expenditure data can consistently recover MUEs.

Q: What are the two demand families that satisfy lambda-separability, and what distinguishes them? A: Theorem 1 establishes that rationalizable lambda-separable demands must belong to either the Constant Frisch Elasticity (CFE) family or the Generalized Stone-Geary family. In CFE demands, log expenditures on each good equal the log of a price function minus beta_j times log lambda, where beta_j is a good-specific constant Frisch elasticity. The Stone-Geary family has a more complex nonlinear form that does not lend itself to linear estimation of log MUEs, making CFE the tractable choice. Both families nest CRRA as the special case where all beta_j are equal.

Q: How are MUEs estimated from the CFE system in practice? A: Estimation proceeds in two steps. First, log expenditures on each good are regressed on good-time-market effects and household demographic controls to obtain residuals. Second, the covariance matrix of these residuals has the factor structure Sigma = Var(w)betabeta’ + Psi, where beta is the vector of Frisch elasticities; the rank-one matrix beta*beta’ is recovered from the sample covariance matrix via factor analysis, and household-level MUEs are then obtained by regression using the estimated beta as generated regressors.

Q: What do the estimated Frisch elasticities reveal about preferences in Uganda? A: The Frisch elasticities beta_j vary substantially across the 41 goods in the Ugandan sample. Starchy staples and salt are least elastic (lowest beta_j), while fresh milk, sweet bananas, coffee, oranges, and passion fruit exhibit high elasticities — passion fruit is roughly three times as income elastic as cassava. The hypothesis that all elasticities are equal (the CRRA restriction) is easily rejected, providing direct evidence against homothetic preferences in this population.

Q: What direct evidence does the paper provide that droughts, floods, and pests are genuinely covariate and harmful? A: About 39% of Ugandan households reported drought in the 2005–06 round. Among drought reporters, 92% said it affected their production, 80% said it affected their income, and 50% said it affected their consumption. Drought, pests, and adverse prices (but not floods) led to statistically significant increases in local farmgate prices. Among markets experiencing covariate shocks, 82%, 74%, 44%, and 53% of t-tests rejected equality of relative food prices for drought, floods, pests, and adverse prices respectively. Dietary diversity and intake of vitamin B-12 (from animal-source foods) declined significantly following covariate shocks.

Q: How do households cope differently with covariate versus idiosyncratic shocks? A: Households experiencing covariate shocks primarily relied on self-insurance: 51% of drought-affected households reduced consumption and 45% drew on savings, with increased labor supply also reported. In contrast, households experiencing idiosyncratic shocks most often relied on help from friends and family (52%). This behavioral difference is consistent with the finding that covariate shocks affect welfare mainly through common price channels that are not individually insurable through social networks, while idiosyncratic shocks are partially absorbed via informal transfers.

Q: What do the CFE results imply about the nature of insurance against covariate shocks in Uganda? A: The CFE regression finds that none of the four covariate shocks (drought, floods, pests, adverse prices) has a statistically significant effect on household MUEs when time-market fixed effects are included. This implies that the welfare impact of covariate shocks is transmitted primarily through common price changes that affect all households in a market-region symmetrically, rather than through idiosyncratic income variation. Effectively, covariate shocks are “shared” within market-regions — but through price deterioration affecting everyone, not through informal transfers.

Q: How robust are the results across different shock time windows? A: Figure 3 shows that for the CRRA specification, any prior covariate shock 3–12 months earlier has a significant positive effect on log consumption in every month, while for the CFE specification no shock window produces a significant effect on w. In the full tabulation across all shock types and windows (Tables 4 and 5), 42 of 52 covariate shock coefficients are significant and positive in CRRA versus only 4 of 52 in CFE — the latter barely exceeding the 2.6 false positives expected under the null hypothesis of full insurance.

Q: What are the policy implications of these findings for relief program design? A: Because covariate shocks affect welfare mainly through common prices within market-regions, relief programs should target communities rather than individual households, since the burden is broadly shared and not concentrated. Policies that integrate markets across regions of Uganda or connect Ugandan markets to broader African or world markets would reduce the price impact of local covariate shocks. Targeted household transfers would be less effective than interventions that stabilize regional prices or supply.

Q: What broader applicability do CFE MUEs have beyond risk-sharing tests? A: Since MUE construction is independent of the risk-sharing hypothesis, CFE-estimated MUEs can be used to estimate and test any dynamic life-cycle model that puts structure on the evolution of MUEs over time, including consumption Euler equations, intertemporal marginal rates of substitution calculations, and household bargaining models. The CFE approach requires only the same expenditure data used in the standard CRRA approach and therefore serves as a more general drop-in replacement across all settings where CRRA MUEs are currently employed.

Marginal Utility of Expenditure (MUE): The Lagrange multiplier lambda on the household budget constraint in the consumer’s optimization problem; the object whose proportionality across households (log lambda_it = log mu_t - log theta_i) characterizes efficient risk-sharing. It is a function of budget, prices, and household characteristics — not reducible to a scalar function of total expenditure except under special preference restrictions.

Lambda-separability: A property of Frischian expenditures on good j such that some transformation phi_j(x_j) can be written as the sum of a function of prices and a function of the MUE alone — phi_j(x_j(p,lambda)) = a_j(p) + b_j(lambda). This is the necessary and sufficient condition for using time fixed effects to control for prices and household fixed effects to control for Pareto weights in a TWFE risk-sharing regression based solely on expenditure data.

Constant Frisch Elasticity (CFE) expenditure system: The tractable member of the two demand families satisfying lambda-separability, characterized by log x^j_it = a_j(p_t) + g_j(z_it) + beta_j * w_it + epsilon^j_it, where beta_j is a good-specific constant elasticity of expenditures with respect to MUE. Nests CRRA as the special case of equal beta_j across all goods, but admits nonhomothetic preferences and fully flexible relative-price responses.

Frischian demands: Demands expressed as functions of prices and the MUE lambda rather than prices and budget — f(p, lambda). Homogeneous of degree zero in (p, 1/lambda), equivalently written f(p*lambda). This representation is central to the lambda-separability characterization because it separates the role of the budget (via lambda) from the role of prices directly.

Covariate shock: In this paper’s usage, a shock that affects prices common to all households in a market-region — not merely a shock affecting many households simultaneously. The key analytical distinction is that idiosyncratic shocks change individual budgets without changing prices, while covariate shocks change prices, which is what causes the standard CRRA test to fail.

Nonhomothetic preferences: Preferences for which expenditure shares vary with income (budget), so no single scalar price index can fully represent the welfare impact of price changes. The paper confirms nonhomotheticity in the Ugandan data through widely varying Frisch elasticities, and argues this is the root cause of the CRRA test’s failure for covariate shocks — a problem that does not arise when shocks are idiosyncratic and leave prices unchanged.