Piecewise integration

I am trying to estimate residential demand for electricity in a country where electricity is sold (to all households (HH)) at an increasing two-part tariff. By choosing marginal prices as my key independent variable (I am mostly interested in estimating the price elasticity of demand), I have come to understand that OLS regression cannot be pursued because the price variable would be endogenous. Therefore, literature suggests I use either Maximum Likelihood estimation (MLE) or Generalized Method of Moments (GMM) estimation. However, I am struggling with finding by objective function.

In their study, Reiss and White (2005) pursue a moments-based approach and go on to define the reduced form for the HH's consumption level of electricity $$x^*$$ (as a function of an increasing two-tier price schedule), as such:

$$x^*= \begin{cases} x(p_1,y,z,\varepsilon;\beta) \qquad & \text{if } \varepsilon \le c_1 \\ \bar{x} & \text{if } c_1< \varepsilon < c_2\\ x(p_2,y_2,z,\varepsilon;\beta) \qquad & \text{if } \varepsilon>c_2, \end{cases} \label{eq:opt.con.lvl.ecx} \tag{1}$$

where:

• $$p_1,p_2$$ are the marginal prices on the two price schedule tiers (demand is assumed strictly decreasing in $$p$$);
• $$\bar{x}$$ is the electricity consumption threshold after which the price switches to the higher tier;
• $$z$$ are observable consumer characteristics;
• $$\varepsilon$$ are unobserved consumer characteristics (demand is assumed strictly increasing in $$\varepsilon$$);
• $$y_2=y+\bar{x} \cdot (p_2-p_1)$$ is the virtual income when consumers lie on the second tier (hence, $$y$$ is income); and
• $$c_j$$ (where $$j=1,2$$) is the solution to $$x(p_j,y_j,z,c_j;\beta)=\bar{x}$$ with $$y_1=y$$. In other words, $$c_j$$ is the maximum (for $$j=1$$) or minimum (for $$j=2$$) value of $$\varepsilon$$ for which consumption occurs on tier $$j$$.

Conditional on the observables, the authors integrate \eqref{eq:opt.con.lvl.ecx} piecewise to obtain:

$$E(x^*|\cdot) = E_{\varepsilon}[x(p_2,y_2,z,\varepsilon;\beta)] + h(p_1,p_2,\bar{x},y,z;\beta) \label{eq:pw.int} \tag{2}$$

where $$h(\cdot)\equiv \tau_2 - \tau_1$$ is a 'sorting correction function defined by the truncated moments' (I used quotation marks as I do not grasp what this means):

$$\tau_j=\int_{-\infty}^{c_j(\beta)} [\bar{x}-x(p_j,y_j,z,\varepsilon;\beta)]dF_{\varepsilon}, \label{eq:srt.corr.fcn} \tag{3}$$

with $$c_1,c_2$$ defined in \eqref{eq:opt.con.lvl.ecx} and $$y_1=y$$. The authors go on to evaluate 'the moments' in \eqref{eq:srt.corr.fcn} with the error specification in $$F_{\varepsilon}$$ assumed $$N(0,\sigma^2)$$ and $$\varepsilon$$ entering demand \eqref{eq:opt.con.lvl.ecx} additevely, to find:

$$E(x^*|\cdot) = [x(p_1,y,z;\beta)-\sigma\lambda_1]\Phi_1 + \bar{x}\cdot(\Phi_2 - \Phi_1) \\ + [x(p_2,y_2,z;\beta)+\sigma\lambda_2](1-\Phi_2) \label{eq:estim} \tag{4}$$

where, $$\Phi_j$$ is the standard normal distribution evaluated at $$c_j(\beta)/\sigma$$, $$\phi_j$$ is the normal density at $$c_j(\beta)/\sigma$$, $$\lambda_1=\phi_1/\Phi_1$$, and $$\lambda_2=\phi_2/(1-\Phi_2)$$.

The authors end up using \eqref{eq:estim} because it fits their data well.

Question(s):

1. What math and reasoning did the authors use to move from \eqref{eq:opt.con.lvl.ecx} to \eqref{eq:pw.int}, \eqref{eq:srt.corr.fcn} and \eqref{eq:estim}?
2. Is the authors' approach strictly for GMM estimation (as they chose to pursue a moments-based approach) or could MLE be carried out as well?
3. Can anyone show me what \eqref{eq:pw.int}, \eqref{eq:srt.corr.fcn} and \eqref{eq:estim} would look like, given that my case has \eqref{eq:my.opt.con.lvl.ecx} (below) as reduced form equation for HH consumption?
4. Alternatively to Q.3, could anyone provide me with either textbook/audio/video material which can help me formulate my own objective function and assumptions to find the best fit for my data, for either GMM or MLE?

$$x^*= \begin{cases} x(p_1,y,z,\varepsilon;\beta) \qquad & \text{if } \varepsilon \leq c_1 \\ x(p_2,y_2,z,\varepsilon;\beta) \qquad & \text{if } \varepsilon>c_2, \end{cases} \label{eq:my.opt.con.lvl.ecx} \tag{5}$$

I am already grateful to those who made it thus far in reading my long question. Any help is deeply appreciated. Thanks!

Gabriele

Reiss, P. C., & White, M. W. (2005). Household electricity demand, revisited. The Review of Economic Studies, 72(3), 853-883.