Estimating discount factors for public and private goods and testing competing discounting hypotheses

Abstract

The observation of declining discount rates in experimental settings has led many to promote hyperbolic discounting over standard exponential discounting as the preferred descriptive model of intertemporal choice. I develop a new framework, consistent with the random utility model, which directly models the intertemporal utility function and produces explicit maximum likelihood estimates of discounting parameters. I apply this estimation method to a stated-preference survey of river basin cleanup options and revealed-preference lottery payment choices. Formal statistical tests fail to find evidence in support of hyperbolic or quasi-hyperbolic discounting. Annual discount rates range from ten to fourteen percent across the data sets and empirical specifications.

Appendices

Appendix A: Alternative error-difference variances

Denote the alternative error-difference terms as $\overset {\symbol {126}}{ \epsilon _{ikj}}=\epsilon _{ik}-\epsilon _{ij}.$ Recalling that, for choice set s, $\epsilon _{ij}=\sum _{t=0}^{T_{s}}\psi _{t}\eta _{ijt}$, I have

$$ \overset{\symbol{126}}{\epsilon _{ikj}}=\sum\limits_{t=0}^{T_{s}}\psi _{t}\eta _{ikt}-\sum\limits_{t=0}^{T_{s}}\psi _{t}\eta _{ijt}.$$

(19)

Then, note that $\overset {\symbol {126}}{\epsilon _{ikj}}$ is heteroskedastic because the number of terms in the summations is determined by the length of the intertemporal alternative. $\overset {\symbol {126}}{\epsilon _{ikj}}$ is a normal error term with mean zero and variance given by

$$ V(\overset{\symbol{126}}{\epsilon _{ikj}})=V\left(\sum\limits_{t=0}^{T_{s}}\psi _{t}\eta _{ikt}-\sum\limits_{t=0}^{Ts}\psi _{t}\eta _{ijt}\right). $$

(20)

$$\begin{array}{rll} &=&\psi _{0}^{2}V(\eta _{ik0})+\psi _{1}^{2}V(\eta _{ik1})+...+\psi _{T_{s}}^{2}V(\eta _{ikT_{s}}) \\ &&+\psi _{0}^{2}V(\eta _{ij0})+\psi _{1}^{2}V(\eta _{ij1})+...+\psi _{T_{s}}^{2}V(\eta _{ijT_{s}}) \end{array}$$

(21)

since the instantaneous errors are independent. With the assumption that $\eta _{ijt}$ $i.i.d$ $N(0,\sigma _{\eta })$, this leads to

$$ V(\overset{\symbol{126}}{\epsilon _{ikj}})=\sum\limits_{t=0}^{T_{s}}\psi _{t}^{2}\sigma _{\eta }+\sum\limits_{t=0}^{T_{s}}\psi _{t}^{2}\sigma _{\eta }.$$

(22)

It is well known that a probit model needs to be normalized for scale so set $\sigma _{\eta }=1$ and I have

$$ V(\overset{\symbol{126}}{\epsilon _{ikj}})=\sum\limits_{t=0}^{T_{s}}\psi _{t}^{2}+\sum\limits_{t=0}^{T_{s}}\psi _{t}^{2}=2\ast \sum\limits_{t=0}^{T_{s}}\psi _{t}^{2} $$

(23)

Appendix B: Random discount factors simulated log likelihood equation

Here, I develop the simulated log likelihood equation for the random discount factors specification. For clarity, I present the exponential discounting case. All other discounting models are easily derived with a few substitutions. This section loosely follows the exposition of Train (2003).

Recall the probability of a single choice for the non-stochastic discounting parameters case, $P_{ij}=F\left ( \frac {\sum _{t=0}^{T_{s}}\delta ^{t}v_{ijt}-\sum _{t=0}^{T_{s}}\delta ^{t}v_{ikt}}{\sqrt {2\ast \sum _{t=0}^{T_{s}}\delta ^{2t}}}\right ) .$ In the case of random discounting parameters, I focus on the sequence of choices by individual i. Denote the choice situation as h and a sequence of alternatives as j $=\{j_{1},...,j_{H}\}.$ Then, conditional on $\delta$, the probability that individual i makes a sequence of choices is the product over all h of the single choice probabilities. I have

$$ \mathbf{P}_{i\mathrm{\mathbf{j}}}(\delta )=\prod\limits_{h=1}^{H}F\left( \frac{\sum_{t=0}^{T_{s,h}}\delta ^{t}v_{ijth}-\sum_{t=0}^{T_{s,h}}\delta ^{t}v_{ikth}}{\sqrt{2\ast \sum_{t=0}^{T_{s,h}}\delta ^{2t}}}\right) . $$

(24)

Since the $\delta$ are random, I integrate out over all values of $\delta$ to get the unconditional choice probability

$$ P_{i\mathrm{\mathbf{j}}}=\int \mathbf{P}_{i\mathrm{\mathbf{j}}}(\delta )f(\delta )d\delta. $$

(25)

I draw R values of $\delta$ from $f(\delta )$ and denote them $\delta _{r}.$ The simulated choice probability is $\widetilde {P}_{i\mathrm {\mathbf {j}} }=\frac {1}{R}\sum \limits _{r=1}^{R}\mathbf {P}_{i\mathrm {\mathbf {j}}}(\delta _{r}).$ In this application, I set $R=200.$ Finally, I insert these simulated choice probabilities into the log-likelihood function to get the simulated log likelihood (SLL)

$$ \mathrm{SLL}=\sum_{i}\sum_{\mathrm{\mathbf{j}}}y_{i\mathrm{\mathbf{j}}}\ln \widetilde{P}_{i\mathrm{\mathbf{j}}}, $$

(26)

where $y_{i\mathrm {\mathbf {j}}}=1$ if i chose sequence j and zero otherwise.

Appendix C: Random cefficients ($\alpha$ and $\gamma$) simulated log likelihood equation

Recall the probability of a single choice for the non-stochastic discounting parameters case, $P_{ij}=F\left (\frac {\sum _{t=0}^{T_{s}}\psi _{t}v_{ijt}-\sum _{t=0}^{T_{s}}\psi _{t}v_{ikt}}{\sqrt {2\ast \sum _{t=0}^{T_{s}}\psi _{t}^{2}}}\right ) .$ In the case of random benefit and cost coefficients, $v_{ijt}=\alpha _{i}q_{ijt}+\gamma _{i}(Y_{it}-c_{ijt})$. Denote the vector for individual i containing both $\alpha _{i}$ and $\gamma _{i}$ as $\theta _{i}.$ $\theta _{i}$ is fixed for an individual across choice occasions, but varies across individuals. Assume $\theta _{i}$ is normally distributed in the population with mean H and covariance W: $\theta _{i}\sim N(H,W).$ I focus on the sequence of choices by individual i. Denote the choice situation as h and a sequence of alternatives as j $=\{j_{1},...,j_{H}\}$ Then, conditional on $\theta$, the probability that individual i makes a sequence of choices is the product over all h of the single choice probabilities. I have

$$ \mathbf{P}_{i\mathrm{\mathbf{j}}}(\theta )=\prod\limits_{h=1}^{H}F\left( \frac{\sum_{t=0}^{T_{s,h}}\psi _{t}v_{ijth}-\sum_{t=0}^{T_{s,h}}\psi _{t}v_{ikth}}{\sqrt{2\ast \sum_{t=0}^{T_{s}}\psi _{t}^{2}}}\right) . $$

(27)

Since the $\theta$ are random, I integrate out over all values of $\theta$ to get the unconditional choice probability

$$ P_{i\mathrm{\mathbf{j}}}=\int \mathbf{P}_{i\mathrm{\mathbf{j}}}(\theta )f(\theta )d\theta . $$

(28)

I draw R values of $\theta$ from $f(\theta )$ and denote them $\theta _{r}.$ The simulated choice probability is $\widetilde {P}_{i\mathrm {\mathbf {j}} }=\frac {1}{R}\sum \limits _{r=1}^{R}\mathbf {P}_{i\mathrm {\mathbf {j}}}(\theta _{r}).$ In this application, I set $R=200.$ Finally, I insert these simulated choice probabilities into the log-likelihood function to get the simulated log likelihood (SLL)

$$ \mathrm{SLL}=\sum\limits_{i}\sum\limits_{\mathrm{\mathbf{j}}}y_{i\mathrm{\mathbf{j}}}\ln \widetilde{P}_{i\mathrm{\mathbf{j}}}, $$

(29)

where $y_{i\mathrm {\mathbf {j}}}=1$ if i chose sequence j and zero otherwise.

Appendix D: Random coefficients (discount factors and $\alpha$ and $\gamma$) simulated log likelihood equation

Here, I show the simulated log likelihood equation when discount factors as well as benefit and cost coefficients are assumed to vary among individuals. Again, in the interest of clarity, I present the exponential discounting case. Retaining the notation and distributional assumptions from the two preceding sections of the appendix, denote the vector for individual i containing both $\delta _{i}$ and $\theta _{i}$ as $\zeta _{i}$. Conditional on $\zeta$, the probability that individual i makes a sequence of choices is the product over all h of the single choice probabilities. I have

$$ \mathbf{P}_{i\mathrm{\mathbf{j}}}(\zeta )=\prod\limits_{s=1}^{H}F\left( \frac{ \sum_{t=0}^{T_{s,h}}\delta _{i}^{t}v_{ijth}-\sum_{t=0}^{T_{s,h}}\delta _{i}^{t}v_{ikth}}{\sqrt{2\ast \sum_{t=0}^{T_{s,h}}\delta ^{2t}}}\right) $$

(30)

with $v_{ijt}=\alpha _{i}q_{ijt}+\gamma _{i}(Y_{it}-c_{ijt})$.

Since the $\zeta$ are random, I integrate out over all values of $\zeta$ to get the unconditional choice probability

$$ P_{i\mathrm{\mathbf{j}}}=\int \mathbf{P}_{i\mathrm{\mathbf{j}}}(\zeta )f(\zeta)d(\zeta ). $$

(31)

I draw R values of $\zeta$ from $f(\zeta )$ and denote them $\zeta _{r}.$ The simulated choice probability is $\widetilde {P}_{i\mathrm {\mathbf {j}}}=\frac {1}{R}\sum \limits _{r=1}^{R}\mathbf {P}_{i\mathrm {\mathbf {j}}}(\zeta _{r}).$ In this application, I set $R=200.$ Finally, I insert these simulated choice probabilities into the log-likelihood function to get the simulated log likelihood (SLL)

$$ \mathrm{SLL}=\sum_{i}\sum_{\mathrm{\mathbf{j}}}y_{i\mathrm{\mathbf{j}}}\ln \widetilde{P}_{i\mathrm{\mathbf{j}}}, $$

(32)

where $y_{i\mathrm {\mathbf {j}}}=1$ if i chose sequence j and zero otherwise.