Present bias and health

Abstract

This study uses a dynamic discrete choice model to examine the degree of present bias and naivete about present bias in individuals’ health care decisions. Clinical guidelines exist for several common chronic diseases. Although the empirical evidence for some guidelines is strong, many individuals with these diseases do not follow the guidelines. Using persons with diabetes as a case study, we find evidence of substantial present bias and naivete. Counterfactual simulations indicate the importance of present bias and naivete in explaining low adherence rates to health care guidelines.

Appendix

This appendix follows Section 2 and provides more details on the identification and estimation of the dynamic discrete choice model. Specifically, given the Extreme Value Distribution assumption, the probability of action i being chosen given x, P_i,t(x_t), is:

$$ P_{i, t}(x_{t}) =\Pr \left[W_{i, t}\left( x_{t}\right) +\varepsilon_{i, t}\geq W_{j, t}\left( x\right) +\varepsilon_{j, t}, \forall j\neq i \right] =\frac{ \exp \left[W_{i, t}\left( x_{t}\right) \right]} {{\sum}_{j = 0}^{1}\exp \left[ W_{j, t}\left( x_{t}\right) \right]}. $$

(9)

While W, defined in (4), is not observable, actual choice probabilities P_i,t (x_t) are observable in the data and can be used to infer W.

With the choice-specific value function of the next-period self perceived by the current self Z_i,t+ 1 (x_t+ 1), defined in (5), the current self’s perception of her future self’s choice, σ, can be defined as

$$\begin{array}{@{}rcl@{}} \sigma\left( x_{t + 1},\varepsilon_{i, t + 1}\right) &=&{ arg\max_{i\in \mathcal{I}}}\left[ u_{i, t + 1}\left( x_{t + 1}\right) +\varepsilon_{i, t + 1}+{\tilde{\beta}\delta \sum\limits_{x_{t + 2}\in \mathcal{X}}{V_{t + 2}(}}x_{t + 2} {)\pi (}x_{t + 2}{{|x_{t + 1}, i)}}\right] \\ &=&{arg\max_{i\in \mathcal{I}}}\left[ Z_{i, t + 1}\left( x_{t + 1}\right) +\varepsilon_{i, t + 1}\right] . \end{array} $$

Then the probability perceived by the current period self of choosing alternative i by the next period’s self when the next period’s state, again assuming an Extreme Value Distribution, is x_t+ 1, $\tilde {P}_{i, t + 1}\left (x_{t + 1}\right )$, is:

$$\begin{array}{@{}rcl@{}} \tilde{P}_{i, t + 1}\left( x_{t + 1}\right) &=&\Pr \left[ \sigma\left( x_{t + 1},\mathbf{\varepsilon}_{t + 1}\right) =i\right] \\ &=&\Pr \left[Z_{i, t + 1}\left( x_{t + 1}\right) +\varepsilon_{i, t + 1}\geq Z_{j, t + 1}\left( x_{t + 1}\right) +\varepsilon_{j, t + 1},\forall j\neq i\right] \\ &=&\frac{\exp \left[ Z_{i, t + 1}\left( x_{t + 1}\right) \right]} {{\sum}_{j = 0}^{1}\exp \left[ Z_{j, t + 1}\left( x_{t + 1}\right) \right]} . \end{array} $$

(10)

The distinction between $\tilde {P}$ and P is that the sophisticated present-biased decision-maker knows the extent of her actual future present bias; by contrast, the naive person underestimates the extent of her present bias. That is, she thinks her β is larger than it actually will be. For sophisticated persons, $\tilde {P}= P$; for naive ones, $\tilde {P} \neq P$.

With non-stationarity and a finite horizon, at t = T, when the continuation value is zero, there is no distinction among W, Z, V, and u:

$$W_{i, T}=Z_{i, T}=V_{i, T}=u_{i, T}, $$

which, according to the Extreme Value Distribution assumption, leads to:

$$ V_{T}=\ln {\sum\limits_{i\in \mathcal{I}}\exp [Z_{i, T}]}=\ln {\sum\limits_{i\in \mathcal{ I}}\exp [u_{i, T}]}. $$

(11)

Combining (11) and (5) yields Z_i,T− 1. Given the link between Z_i,T− 1 and V_T− 1, using backward induction, V_t+ 1 can be determined, which in turn relates to W_i,t (4), and then to P_i,t (x_t) (9). By this reasoning, we link instantaneous utility u to P, which is observable in the data. Once this relationship between u and P is established empirically from the choice probabilities (P_i,t (x_t)) and transition probabilities (π (x_t+ 1|x_t,i)) for all $x\in \mathcal {X}$ and for i = (0, 1), we can estimate the utility parameters for a given $\left \langle \beta ,\tilde {\beta },\delta \right \rangle$.

The relationship between Z_i,t and V_t can be described in three steps. First, combining (5) and (6) yields

$$ V_{i, t + 1}\left( x_{t + 1}\right) =Z_{i, t + 1}\left( x_{t + 1}\right) +\left( 1- \tilde{\beta}\right) \delta {\sum\limits_{x_{t + 2}\in \mathcal{X}}{V_{t + 2}(}}x_{t + 2}{ )\pi (}x_{t + 2}{{|x_{t + 1}, i)}.} $$

(12)

Given (12), (7) can be rewritten as:

$$\begin{array}{@{}rcl@{}} V_{t + 1}\left( x_{t + 1}\right) &=&\mathrm{E}_{\varepsilon_{t + 1}}\left[ V_{\sigma\left( x_{t + 1},\mathbf{\varepsilon}_{t + 1}\right), t + 1} \left( x_{t + 1}\right) +\varepsilon_{\sigma\left( x_{t + 1}, \mathbf{\varepsilon}_{t + 1}\right), t + 1}\right] \\ &=&\mathrm{E}_{\mathbf{\varepsilon}_{t + 1}}\left[ \begin{array}{c} Z_{\sigma\left( x_{t + 1},\mathbf{\varepsilon}_{t + 1}\right), t + 1} \left( x_{t + 1}\right) +\varepsilon_{\sigma\left( x_{t + 1}, \mathbf{\varepsilon}_{t + 1}\right), t + 1} \\ +\left( 1-\tilde{\beta}\right) \delta {{\sum}_{x_{t + 2}\in \mathcal{X}}{V_{t + 2}(} }x_{t + 2}{)\pi (}x_{t + 2}{{|x_{t + 1}, \sigma\left( x_{t + 1},\mathbf{ \varepsilon}_{t + 1}\right) )}} \end{array} \right] \\ &=&\mathrm{E}_{\mathbf{\varepsilon}_{t + 1}}{\max_{i\in \mathcal{I}}}\left[ Z_{i, t + 1}\left( x_{t + 1}\right) +{\varepsilon}_{i,t + 1}\right] \\ &&+\left( 1-\tilde{\beta}\right) \delta \mathrm{E}_{\mathbf{\varepsilon} _{t + 1}}{\sum\limits_{x_{t + 2}\in \mathcal{X}}{V_{t + 2}(}}x_{t + 2}{)\pi (}x_{t + 2}{{ |x_{t + 1}, \sigma\left( x_{t + 1},\mathbf{\varepsilon}_{t + 1}\right) )}} \\ &=&\mathrm{E}_{\mathbf{\varepsilon}_{t + 1}}{\max_{i\in \mathcal{I}}}\left[ Z_{i, t + 1}\left( x_{t + 1}\right) +{\varepsilon}_{i,t + 1}\right] \\ &+&\left( 1-\tilde{\beta}\right) \delta \sum\limits_{i\in \mathcal{I}}\tilde{P} _{i, t + 1}\left( x_{t + 1}\right) {\sum\limits_{x_{t + 2}\in \mathcal{X}}{V_{t + 2}(}} x_{t + 2}{)\pi (}x_{t + 2}{{|x_{t + 1}, i)}}. \end{array} $$

(13)

Given the Extreme Value Distribution assumption,

$$ \mathrm{E}_{\mathbf{\varepsilon}_{t + 1}}{\max_{i\in \mathcal{I}}} \{Z_{i, t + 1}(x_{t + 1})+\varepsilon_{i,t + 1}\}=\ln \left\{ \sum\limits_{i\in \mathcal{I}}\exp \left[ Z_{i, t + 1}\left( x_{t + 1}\right) \right] \right\}. $$

(14)

Combined with (14) and (10), (13) can be rewritten as

$$\begin{array}{@{}rcl@{}} &&V_{t + 1}\left( x_{t + 1}\right) =\ln \left\{ \sum\limits_{i\in \mathcal{I}}\exp \left[ Z_{i, t + 1}\left( x_{t + 1}\right) \right] \right\} \\ &&+\left( 1-\tilde{\beta}\right) \delta {\sum}_{i\in \mathcal{I}}\frac{\exp \left[Z_{i, t + 1}\left( x_{t + 1}\right) \right]} {{\sum}_{j = 0}^{1}\exp \left[ Z_{j, t + 1}\left( x_{t + 2}\right) \right]} {\sum\limits_{x_{t + 2}\in \mathcal{X}}{V_{(t + 2)}(}} x_{t + 2}{)\pi (}x_{t + 2}{{|x_{t + 1}, i)}}, \end{array} $$

(15)

which relates Z_i,t to V_t, a relationship that makes backward induction possible.