2  Full risk-sharing under homogeneous preferences

Here, first I review the theoretical result and then derive the likelihood function. Note that the model here estimates parameters under the assumption that the model follows the full risk-sharing model. This is in contrast to many empirical papers which test a full riks-sharing model (such as Townsend (1994)).

2.1 Theory

2.1.1 Settings

Suuposed that there are N households in a village. Each household maximizes its expected lifetime utility: E \left[ \sum_{t = 1}^{\infty} \delta^t u (c_{it}) \right], where \delta is the discount factor, u is instantaneous preference of households (note that here I assume homogeneous preference), and c_{it} is consumption by household i at time t. Assume that there is no saving.

For income, let s^t = (s_1, \dots, s_t) the history of income states from time 1 to t, and let y_{it}(s_t) be household i’s income at time t and state s_t. Assume that the income process is a Markov process and is independent across households.

With these, I find the Pareto-optimal allocations. For this, a weighted sum of households’ expected lifetime utilities is maximized. In other words, a social planner solves the following maximization problem:

\max_{\{c_{it}(s^t)\}} \quad \sum_i \lambda_i \sum_{t = 1}^{\infty} \sum_{s^t} \delta^t \pi(s^t) u(c_{it}(s^t)) subject to the resource constraint \sum_i c_{it}(s^t) \le \sum_i y_{it}(s_t) \quad \forall s^t, \forall t, where \lambda_i is the Pareto weight of i, \pi(s^t) is the probability that the history s^t is realized, and c_{it}(s^t) is i’s consumption under the history s^t.

2.1.2 Optimization condition

Solving this maximization problem, we can get a well-known result that, for any two households i and j in the village, the following holds: \frac{u'(c_{jt}(s^t))}{u'(c_{it}(s^t))} = \frac{\lambda_i}{\lambda_j} \quad \forall s^t, \forall t.

If I assume the utility function to take the CRRA form, that is, u(c_{it}) = \frac{c_{it}^{1 - \sigma} - 1}{1 - \sigma}, then u'(c_{it}) = c_{it}^{- \sigma}, and thus I get \frac{c_{jt}(s^t)^{-\sigma}}{c_{it}(s^t)^{-\sigma}} = \frac{\lambda_i}{\lambda_j} \quad \forall s^t, \forall t. Dropping s^t from the quation for simplicity and taking the logarithms, I obtain \begin{aligned} -\sigma \log(c_{jt}) + \sigma \log(c_{it}) &= \log(\lambda_i) - \log(\lambda_j) \\ \Leftrightarrow \log(c_{it}) &= \frac{1}{\sigma} \left( \log(\lambda_i) - \log(\lambda_j) \right) + \log(c_{jt}) \\ \Leftrightarrow N \log(c_{it}) &= \frac{1}{\sigma} \left( N \log(\lambda_i) - \sum_j \log(\lambda_j) \right) + \sum_j \log(c_{jt}) \\ \Leftrightarrow \log(c_{it}) &= \frac{1}{\sigma} \left( \log(\lambda_i) - \frac{1}{N} \sum_j \log(\lambda_j) \right) + \frac{1}{N} \sum_j \log(c_{jt}). \end{aligned} By defining the village-average consumption, c_{vt}, as \log(c_{vt}) = \frac{1}{N} \sum_j \log(c_{jt}), and taking the first difference from the equation above, I get \log(c_{it}) - \log(c_{i, t - 1}) = \log(c_{vt}) - \log(c_{v, t - 1}).

2.1.3 Introducing measurement errors

Here I introduce the measurement error in consumption, which is assumed to be multiplicative and log-normally distributed. In particular, I denote observed consumption, c_{it}^* as c_{it}^* = c_{it} \exp(\varepsilon_{it}^c), where \varepsilon_{it} \sim N(0, \gamma_c^2) is the measurement error which is i.i.d. across households and time. With this, the observed village-average consumption, c_{vt}^* is so that \begin{aligned} \log(c_{vt}^*) &= \frac{1}{N} \sum_i \log(c_{it}^*) \\ &= \frac{1}{N} \sum_i \log(c_{it}) + \frac{1}{N} \sum_i \varepsilon_{it}^c \\ &= \log(c_{vt}) + \frac{1}{N} \sum_i \varepsilon_{it}^c \\ &= \log(c_{vt}) + \varepsilon_{vt}^c, \end{aligned} where the village consumption measurement error is defined as \varepsilon_{vt}^c \equiv \frac{1}{N} \sum_i \varepsilon_{it}^c. Substituting these into \log(c_{it}) - \log(c_{i, t - 1}) = \log(c_{vt}) - \log(c_{v, t - 1}), I obtain \log(c_{it}^*) - \log(c_{i, t - 1}^*) = \log(c_{vt}^*) - \log(c_{v, t - 1}^*) + \left(\varepsilon_{it}^c - \varepsilon_{vt}^c - (\varepsilon_{i, t - 1}^c - \varepsilon_{v, t - 1}^c) \right).

2.1.4 Likelihood function

Since \varepsilon_{it} \sim N(0, \gamma_C^2), I get \varepsilon_{vt} \sim N(0, \frac{\gamma_C^2}{N}), and note that Cov(\varepsilon_{it}, \varepsilon_{vt}) = \frac{\gamma_C^2}{N}. With the independence across periods and households of measurement errors, I obtain Var(\varepsilon_{it}^c - \varepsilon_{vt}^c - (\varepsilon_{i, t - 1}^c - \varepsilon_{v, t - 1}^c) ) = 2\left(\gamma_C^2 + \frac{\gamma_C^2}{N} - 2\frac{\gamma_C^2}{N} \right) = 2 \gamma_C^2 \left(1 - \frac{1}{N} \right).

Therefore, the likelihood function is

L(\theta) = \prod_{i} \prod_{t = 2}^{T} f \left( \log \left( \frac{c_{it}^*}{c_{i, t - 1}^*} \right) - \log \left( \frac{c_{vt}^*}{c_{v, t - 1}^*} \right), \theta \right) , and the log likelihood function is l(\theta) = \sum_{i} \sum_{t = 2}^{T} \log \left(f \left( \log \left( \frac{c_{it}^*}{c_{i, t - 1}^*} \right) - \log \left( \frac{c_{vt}^*}{c_{v, t - 1}^*} \right), \theta\right) \right), where f(x, \theta) is the density function of x \sim N \left(0, \sqrt{ 2 \gamma_C^2 \left(1 - \frac{1}{N} \right)} \right), and \theta is a parameter to estimate (here, \theta = \gamma_C). In estimation, I find the parameter to maximize l(\theta).

Without assuming that the model is correctly specified but assuming that there is no autocorrelation, the standard error of the parameter is A(\widehat{\theta})^{-1} B(\widehat{\theta}) A(\widehat{\theta})^{-1}, where A(\widehat{\theta}) is the Hessian matrix of the log-likelihood function and B(\widehat{\theta}) is the outer product of scores (= gradient of log likelihood).

2.1.4.1 Notes

• This model considers that any deviation from the full risk-sharing model is due to measurement errors in consumption.
• I derive the unconditional likelihood function, as opposed to Laczó (2015) who used a conditional likelihood function with simulation. The author clarifies her choice to use the simulation method in a footnote as follows: “Note that it is not necessary to use simulation to take into account measurement error in consumption at time t in the perfect risk-sharing case. I do it to be consistent with the LC case.”

2.2 Code for estimation

This R script estimates the full risk sharing model.

pacman::p_load(
  tidyverse,
  kableExtra,
  numDeriv
)

load('IntermediateData/allData.RData')

calculateLogLikelihoodEachObsFullHom <- function(
    param, 
    currentCons,
    previousCons,
    currentVilLogCons,
    previousVilLogCons,
    numHouseholds
    ) {
  
  gammaC2 <- param
  
  logLikelihood <- dnorm(
    (
      log(currentCons / previousCons)
      - (currentVilLogCons - previousVilLogCons)
    ), sd = sqrt(2 * gammaC2 * (1 - 1 / numHouseholds)),
    log = TRUE
  )
  return(logLikelihood)
}

calculateScoreEachObsFullHom <- function(
    param, 
    currentCons,
    previousCons,
    currentVilLogCons,
    previousVilLogCons,
    numHouseholds
) {
  grad(
    calculateLogLikelihoodEachObsFullHom,
    param,
    currentCons = currentCons,
    previousCons = previousCons,
    currentVilLogCons = currentVilLogCons,
    previousVilLogCons = previousVilLogCons,
    numHouseholds = numHouseholds
  )
}

calculateScoreFullHom <- function(
    param, 
    consdatByVillage,
    vilMeanLogConsByVillage,
    numHouseholds,
    .tnum = tnum
) {
  
  numParameters <- length(param)
  
  scoreMatrix <- matrix(0, nrow = numParameters, ncol = numParameters)
  for (householdIndex in seq(1, numHouseholds)) {
    for (periodIndex in seq(2, .tnum)) {
      score <- calculateScoreEachObsFullHom(
          param,
          consdatByVillage[householdIndex, periodIndex],
          consdatByVillage[householdIndex, (periodIndex - 1)],
          vilMeanLogConsByVillage[periodIndex],
          vilMeanLogConsByVillage[(periodIndex - 1)],
          numHouseholds
        )
      scoreMatrix <- (
        scoreMatrix
        + outer(score, score)
      )
    }
  }
  return(scoreMatrix)
}

calculateLogLikelihoodFullHom <- function(
    param, 
    consdatByVillage,
    vilMeanLogConsByVillage,
    numHouseholds,
    .tnum = tnum
    ) {
  
  logLikelihood <- 0
  for (householdIndex in seq(1, numHouseholds)) {
    for (periodIndex in seq(2, .tnum)) {
      logLikelihood <- (
        logLikelihood
        + calculateLogLikelihoodEachObsFullHom(
          param,
          consdatByVillage[householdIndex, periodIndex],
          consdatByVillage[householdIndex, (periodIndex - 1)],
          vilMeanLogConsByVillage[periodIndex],
          vilMeanLogConsByVillage[(periodIndex - 1)],
          numHouseholds
        )
      )
    }
  }
    
  return(- logLikelihood)
}

calculateStandardErrors <- function(
    estimationResult,
    consdatByVillage,
    vilMeanLogConsByVillage,
    numHouseholds
) {
  (
    solve(estimationResult$hessian) %*% 
    calculateScoreFullHom(
      estimationResult$par,
      consdatByVillage, 
      vilMeanLogConsByVillage,
      numHouseholds
    ) %*%
    solve(estimationResult$hessian)
  ) %>% sqrt %>% diag
}

estimateMLFullHom <- function(
    village,
    consdat,
    vilMeanLogCons,
    lowerBound,
    upperBound,
    villageIndicatorMatrix
) {
  
  numHouseholds <- villageIndicatorMatrix[, village] %>% sum
  consdatByVillage <- consdat[villageIndicatorMatrix[, village], ]
  vilMeanLogConsByVillage <- vilMeanLogCons[village, ]
  
  estimationResult <- optim(
    0.1,
    calculateLogLikelihoodFullHom,
    consdatByVillage = consdatByVillage,
    vilMeanLogConsByVillage = vilMeanLogConsByVillage,
    numHouseholds = numHouseholds,
    method = "L-BFGS-B",
    lower = lowerBound,
    upper = upperBound,
    control = list(trace = 0, maxit = 200),
    hessian = TRUE)
  
  standardErrors <- calculateStandardErrors(
    estimationResult,
    consdatByVillage,
    vilMeanLogConsByVillage,
    numHouseholds
  )
  
  return(list(
    parameter = estimationResult$par,
    logLikelihood = - estimationResult$value,
    standardErrors = standardErrors,
    optimizationResult = estimationResult$message
  ))
}

estimationResultFullHom <- map(
  seq(1, numVillages),
  ~ estimateMLFullHom(
      .,
      consdat,
      vilMeanLogCons,
      1e-3,
      10,
      villageIndicatorMatrix
  )
)

2.3 Estimation result

estimationFullHomTable <- do.call(
  cbind,
  map(
    seq(1, numVillages),
    ~ c(
      formatC(estimationResultFullHom[[.]]$parameter, digits = 3, format = "f"),
      str_interp('(${formatC(estimationResultFullHom[[.]]$standardErrors, digits = 3, format = "f")})'),
      formatC(estimationResultFullHom[[.]]$logLikelihood, digits = 3, format = "f")
    )
  )
)
colnames(estimationFullHomTable) <- c("Aurepalle", "Kanzara", "Shirapur")
rownames(estimationFullHomTable) <- c(
  "Variance of consumption measurement errors",
  "",
  "Log likelihood"
  )

estimationFullHomTable %>% 
  kbl(digits = 3) %>% 
  kable_classic()

	Aurepalle	Kanzara	Shirapur
Variance of consumption measurement errors	0.036	0.037	0.048
	(0.004)	(0.010)	(0.007)
Log likelihood	-13.808	-20.166	-36.422

Laczó, Sarolta. 2015. “Risk Sharing with Limited Commitment and Preference Heterogeneity: Structural Estimation and Testing.” Journal of the European Economic Association 13 (2): 265–92.

Townsend, Robert M. 1994. “Risk and Insurance in Village India.” Econometrica 62 (3): 539. https://doi.org/10.2307/2951659.