Since Phaneuf, Kling, and Herriges (2000) proposed the first strategy to solve for CSH conditional on the simulated unobserved heterogeneity, numerous refinements to their approach have been developed that reduce the computational burden of the task significantly. To place these refinements in context, we first describe the Phaneuf, Kling and Herriges approach. Their strategy exploits the implicit, indirect utility-based definition of CSH in equation (2.4) and uses a numerical bisection routine to iteratively solve for the income compensation that equates utility before and after the price and quality changes. At each step of the numerical bisection routine the analyst must analytically solve the consumer's constrained optimization problem conditional on (p1, Q1, fit, Et) and an arbitrary income level to determine the individual's utility. Phaneuf, Kling and Herriges proposed doing this by calculating each of the 2m regime-specific conditional indirect utility functions and determining which one generated the highest utility.
Although their approach generates consistent estimates of CSH, several factors limit its practical appeal. Perhaps the most significant is that analytically calculating 2M regime-specific conditional indirect utility functions for several observations and simulated values of (fit, Et) is not practical when M is large. Phaneuf and Herriges (1999) have shown that the approach is feasible for recreation applications with as many as 15 sites, but with the addition of only a few more sites, the approach becomes intractable. Moreover, the approach is limiting in the sense that the analyst must choose preference specifications with closed form conditional indirect utility functions. Since the number of preference specifications that satisfy this condition is relatively small, this constraint is notable. In addition, the approach is inefficient because it is based on the implicit, indirect utility-based function definition of CSH in (2.4) instead of its explicit, expenditure-based function definition in (2.5). As a result, the analyst must use a numerical bisection algorithm to solve for the income compensation that equates utility.
von Haefen, Phaneuf, and Parsons (2004) significantly refined Phaneuf, Kling, and Herriges' approach by introducing an iterative algorithm that numerically solves the consumer's constrained optimization problem at each iteration of the numerical bisection routine. By using a numerical approach based on the Kuhn-Tucker conditions to solve for the consumer's optimal consumption bundle and utility, von Haefen, Phaneuf, and Parsons' approach can be applied to preference specifications without closed form indirect utility functions (such as our three specifications in this chapter) as well as data with many goods. Like Phaneuf, Kling, and Herriges' approach, however, their approach exploits the implicit definition of CSH in (2.4) and therefore is not fully efficient. Nonetheless, experience has shown it to be surprisingly fast in practice regardless of the number of goods.
von Haefen, Phaneuf, and Parsons' approach can be best appreciated by inspecting the generic Kuhn-Tucker conditions when preferences are additively separable (i.e., «(•) = ^k Uk(xk) + uz(z)):
Notably in (3.12) only Xj and z enter the jth Kuhn-Tucker equation. This simple structure suggests that if the analyst knew the optimal value for z, she could use equation (3.12) to solve for each Xj. Therefore under additive separability, solving the consumer's problem reduces to solving for the optimal value of z. Based on this insight the following numerical bisection algorithm can be used to solve the consumer's problem conditional on values for the exogenous variables and the simulated unobserved heterogeneity:
1 At iteration i, set z%a = (zj-i + zU-i)/2. To initialize the algorithm, set z1 = 0 and zU = y.
2 Conditional on z%a, solve for xj using (3.12). Use (3.13) and x to construct z%.
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