The Joint identification of utility and discount functions from stated choice data: An application to durable goods adoption

Abstract

We present a survey design that generalizes static conjoint experiments to elicit inter-temporal adoption decisions for durable goods. We show that consumers’ utility and discount functions in a dynamic discrete choice model are jointly identified using data generated by this specific design. In contrast, based on revealed preference data, the utility and discount functions are generally not jointly identified even if consumers’ expectations are known. The separation of current-period preferences from discounting is necessary to forecast the diffusion of a durable good under alternative marketing strategies. We illustrate the approach using two surveys eliciting Blu-ray player adoption decisions. Both model-free evidence and the estimates based on a dynamic discrete choice model indicate that consumers make forward-looking adoption decisions. In both surveys the average discount rate is 43 percent, corresponding to a substantially higher degree of impatience than the rate implied by aggregate asset returns. The estimates also reveal a large degree of heterogeneity in the discount rates across consumers, but only little evidence for hyperbolic discounting.

Appendices

Appendix A: Proofs and further results

Proof of Proposition 1

The proof follows Bajari et al. (2009). Define the expected value function:

$$v(x)\equiv\int\max\limits_{k\in\mathcal{A}}\{v_{k}(x)+\epsilon_{k}\}p(\epsilon)d\epsilon. $$

Using the definition of the expected value function, the choice-specific value functions can be written in simpler form as

$$v_{j}(x_{t})=u_{j}(x_{t})+\delta\int v(x^{\prime})p(x^{\prime}|x_{t},j)dx^{\prime}. $$

Furthermore, the expected value function satisfies the recursive relationship

$$v(x_{t})=\int\max\limits_{k\in\mathcal{A}}\{u_{k}(x_{t})+\epsilon_{k}+\delta\int v(x^{\prime})p(x^{\prime}|x_{t},k)dx^{\prime}\}p(\epsilon)d\epsilon. $$

Recall that based on the results in Hotz and Miller (1993) and Norets and Takahashi (2013) we can obtain the choice-specific value function differences

$$\bar{v}_{j}(x)=v_{j}(x)-v_{0}(x) $$

as a function of the CCPs. □

Now re-write the expected value function:

$$\begin{array}{@{}rcl@{}} v(x) & = & \int\max\limits_{k\in\mathcal{A}}\{v_{k}(x)+\epsilon_{k}\}p(\epsilon)d\epsilon \\ & = & \int\max\limits_{k\in\mathcal{A}}\{\bar{v}_{k}(x)+\epsilon_{k}\}p(\epsilon)d\epsilon+v_{0}(x). \end{array} $$

(8)

Using Eq. 8 and the choice-specific value function differences $\bar {v}_{j}(x)$ we can express v ₀(x) in the following form (remember that u ₀(x) ≡ 0):

$$\begin{array}{@{}rcl@{}} v_{0}(x) & = & u_{0}(x)+\delta\int v(x^{\prime})p(x^{\prime}|x,0)dx^{\prime} \\ & = & \delta\int\max\limits_{k\in\mathcal{A}}\{\bar{v}_{k}(x^{\prime})+\epsilon_{k}\}p(\epsilon)p(x^{\prime}|x,0)d\epsilon dx^{\prime}+\delta\int v_{0}(x^{\prime})p(x^{\prime}|x,0)dx^{\prime} \end{array} $$

(9)

Note that the first term on the right-hand side of the equation above is known given p(𝜖),p(x ^′|x, j), and the CCPs. It is straightforward to show that Eq. 9 satisfies Blackwell’s conditions and thus defines a contraction mapping with a unique solution v ₀(x). Hence, we can infer v ₀(x) along with all the other choice-specific value functions:

$$v_{j}(x)=\bar{v}_{j}(x)+v_{0}(x). $$

Given the choice-specific value functions, we can calculate the expected value function and then infer the utility functions from the equation defining the choice-specific value functions:

$$u_{j}(x)=v_{j}(x)-\delta\int v(x^{\prime})p(x^{\prime}|x,j)dx. $$

1.1 Identification for the case of geometric discounting

Consider the specific case of geometric discounting, ρ(t) = δ ^t,0 < δ < 1. Assume that the planing horizon is infinite, and that the continuation value from owning product j is given by

$$\omega_{j}(x)=\mathbb{E}\left[\sum\limits_{s=T+1}^{\infty}\delta^{s}u_{j}^{*}(x_{s})|x\right]. $$

Then the value function from adopting product j in period t is

$$ v_{jt}(\boldsymbol{x})=\delta^{t}u_{j}(x_{t})+\sum\limits_{s=t+1}^{T}\delta^{s}u_{j}^{*}(x_{s})+\mathbb{E}\left[\sum\limits_{s=T+1}^{\infty}\delta^{s}u_{j}^{*}(x_{s})|x_{T}\right]. $$

(10)

Let u _j (x) and $u_{j}^{*}(x)$ be utility functions such that Eq. 10 holds. Choose some arbitrary number $a\in \mathbb {R}$. Define $\tilde {u}_{j}(x)=u_{j}(x)+\delta a/(1-\delta )$ and $\tilde {u}_{j}^{*}(x)=u_{j}^{*}(x)-a$ . Then the value function corresponding to $\tilde {u}_{j}(x)$ and $\tilde {u}_{j}^{*}(x)$ is

$$\begin{array}{@{}rcl@{}} \tilde{v}_{jt}(\boldsymbol{x}) & = & v_{jt}(\boldsymbol{x})+\delta^{t}\frac{\delta a}{1-\delta}-\sum\limits_{s=t+1}^{\infty}\delta^{s}a\\ & = & v_{jt}(\boldsymbol{x})+\delta^{t}\frac{\delta a}{1-\delta}-\delta^{t+1}\frac{a}{1-\delta}\\ & = & v_{jt}(\boldsymbol{x}). \end{array} $$

We see that the adoption value is identical regardless of whether the present value from a usage utility intercept is received at the time of adoption or spread out over all future usage occasions. This proves the following result:

Proposition 5

Under geometric discounting and an infinite planning horizon the utility functions u _j (x) and $u_{j}^{*}(x)$ are not separately identified.

Now suppose that the planning horizon of the subject is finite, T<∞. Then ω _j (x) = ω ₀(x) ≡ 0. Consider a sequence x where $x_{t}=\bar {x}$ and $x_{s}=\tilde {x}$ for s>t. We can then infer the value

$$\begin{array}{@{}rcl@{}} v_{jt}(\boldsymbol{x}) & = & \delta^{t}u_{j}(\bar{x})+\sum\limits_{s=t+1}^{T}\delta^{s}u_{j}^{*}(\tilde{x})\\ & = & \delta^{t}u_{j}(\bar{x})+\frac{\delta^{t+1}(1-\delta^{T-t})}{1-\delta}u_{j}^{*}(\tilde{x}). \end{array} $$

Similarly, let v _{j τ} (x ^′) be the value corresponding to a sequence x ^′ where $x^{\prime }_{\tau }=\bar {x}$ and $x^{\prime }_{s}=\tilde {x}$ for s>τ, τ ≠ t. Then

$$\delta^{-t}v_{jt}(\boldsymbol{x})-\delta^{-\tau}v_{j\tau}(\boldsymbol{x})=\frac{\delta^{T+1}}{1-\delta}(\delta^{-\tau}-\delta^{-t})u_{j}^{*}(\tilde{x}). $$

Because δ ^−τ−δ ^−t≠0 for t ≠ τ, $u_{j}^{*}(x)$ is identified. Identification of u _j (x) then follows using the same argument used in the proof of Proposition 3. We thus see that although identification of the utility functions is not generally possible under geometric discounting, identification can be achieved if there is some prior knowledge of ω _j (x) and ω ₀(x), here given the finite horizon assumption.

1.2 Generalization of the identification conditions

Consider a sequences x such that $x_{s}=\bar {x}$ for all t ≤ s ≤ t + k. Then

$${\Delta}_{tk}(j,\boldsymbol{x},\boldsymbol{x})=(\rho(t)-\rho(t+k))u_{j}(\bar{x})+{\sum}_{s=t+1}^{t+k}\rho(s)u_{j}^{*}(\bar{x}). $$

Similarly, define a sequence x ^′ such that $x_{s}^{\prime }=\bar {x}$ for all t ≤ s ≤ t + l, l ≠ k. Define the matrix

$$\boldsymbol{A}_{t}(k,l)=\left[\begin{array}{cc} \rho(t)-\rho(t+k) & {\sum}_{s=t+1}^{t+k}\rho(s)\\ \rho(t)-\rho(t+l) & {\sum}_{s=t+1}^{t+l}\rho(s) \end{array}\right]. $$

If A _t (k, l) is invertible, then $u_{j}(\bar {x})$ and $u_{j}^{*}(\bar {x})$ are identified from the following system of equations:

$$\left[\begin{array}{cc} \rho(t)-\rho(t+k) & {\sum}_{s=t+1}^{t+k}\rho(s)\\ \rho(t)-\rho(t+l) & {\sum}_{s=t+1}^{t+l}\rho(s) \end{array}\right]\left[\begin{array}{c} u_{j}(\bar{x})\\ u_{j}^{*}(\bar{x}) \end{array}\right]=\left[\begin{array}{c} {\Delta}_{tk}(j,\boldsymbol{x},\boldsymbol{x})\\ {\Delta}_{tl}(j,\boldsymbol{x}^{\prime},\boldsymbol{x}^{\prime}) \end{array}\right]. $$

In the case of geometric discounting

$$\boldsymbol{A}_{t}(k,l)=\delta^{t}\left[\begin{array}{cc} 1-\delta^{k} & \frac{\delta(1-\delta^{k})}{1-\delta}\\ 1-\delta^{l} & \frac{\delta(1-\delta^{l})}{1-\delta} \end{array}\right], $$

and we see that A _t (k, l) is not invertible. This is expected based on the non-identification result for the case of geometric discounting (Proposition 5). In general, however, A _t (k, l) will be invertible. For example consider hyperbolic discounting with (β, δ)-preferences, ρ (0) = 1 and ρ(t) = β δ ^t for all t ≥ 1. Then

$$\boldsymbol{A}_{0}(1,2)=\left[\begin{array}{cc} 1-\beta\delta & \beta\delta\\ 1-\beta\delta^{2} & \beta\delta+\beta\delta^{2} \end{array}\right]. $$

The determinant of this matrix is given by:

$$\begin{array}{@{}rcl@{}} \det(\boldsymbol{A}_{0}(1,2)) & = & \left(\beta\delta+\beta\delta^{2}-\beta^{2}\delta^{2}-\beta^{2}\delta^{3}\right)-\left(\beta\delta-\beta^{2}\delta^{3}\right)\\ & = & \beta\delta^{2}-\beta^{2}\delta^{2}\\ & = & \beta\delta^{2}(1-\beta). \end{array} $$

Hence, if 0 < β < 1 and δ > 0, then det(A ₀(1, 2)) ≠0 and A _t (k, l) is invertible.

Appendix B: Survey description

In several introductory screens, we provide subjects information about the Blu-ray player and the design of the study. The objective of these screens is to educate the subjects about Blu-ray players and familiarize them with the survey design and the choice tasks they will encounter. First, we show subjects differences in picture quality when using a Blu-ray player versus a traditional DVD. We then provide an in-depth description of the Blu-ray technology. Specifically, we provide the subjects information on how the Blu-ray discs and players work, the leading manufacturers of Blu-ray technology, and similarities between a Blu-ray disc and a traditional DVD. Subjects are then shown the Blu-ray player they will have the option of choosing across different choice tasks. In the first survey, the subjects are told that they have the option of choosing a mid-high quality Blu-ray player, while in the second survey, they have a choice between a medium quality player (Samsung) and a state of the art top quality Blu-ray player (Sony).

In the subsequent screens, we provide subjects details on how to respond to the choice tasks and navigate the survey. We ask the subjects about their current Blu-ray player ownership status and test their understanding of the price (and number of titles) path over time using graphs from a typical choice task. Based on their responses, subjects are provided feedback about the graphs and the choice tasks to make sure they understand the trade-offs involved across different choices. Figures 1 and 3 show typical choice tasks in the first and second survey, respectively. In each choice task, subjects are provided with the prices (and number of titles) of the Blu-ray player(s) over four (six) periods. Subjects are told that the prices and number of titles shown are expert predictions and they should expect to see these prices in the future. Additionally, they are asked to ignore changes in prices due to inflation. Further details about the variation in prices and number of titles available across different choice tasks is present in Section 4.