A model of the effect of affect on economic decision making

Abstract

The standard economic model of decision making assumes a decision maker’s current emotional state has no impact on his or her decisions. Yet there is a large psychological literature that shows that current emotional state, in particular mild positive affect, has a significant effect on decision making, problem solving, and behavior. This paper offers a way to incorporate this insight from psychology into economic modelling. Moreover, this paper shows that this simple insight can parsimoniously explain a wide variety of behaviors.

Appendices

Appendix 1: Comparison of affect with rational addiction

The rational addiction literature (see, e.g., Pollak 1970; Becker and Murphy 1988; and, for a survey, von Auer 1998) would seem related to the model of affect presented here. This appendix provides a brief comparison.

In a rational addiction model, utility at time t is given by the formula:

$$ u_t = V\left(x_t,S_{t-1}\right)\,, $$

(23)

where x _t is time-t consumption of the addictive good and S _t − 1 is some “stock” of the addictive good consumed to the beginning of period t. The stock follows the process:

$$ S_t = x_t + \rho S_{t-1}\,, $$

(24)

where 0 < ρ < 1. The key behavioral assumption is that the cross-partial derivative of V is positive: A larger stock of the addictive good raises the marginal utility of consuming it today.

Observe that instead of working in terms of per-period consumption, one can work in terms of the stock: Using Eq. 24, Eq. 23 can be rewritten as

$$ u_t = V\left(S_t - \rho S_{t-1},S_{t-1}\right)\,. $$

(25)

Given an initial stock, S ₀, the rational addict chooses $\{S_t\}_{t=1}^{\infty}$ to maximize the discounted utility flow of Eq. 25 subject to the constraint that S _t  ≥ ρS _t − 1 (i.e., negative consumption is not permitted). For a constant discount factor (i.e., $\omega_t = \delta^t$), the optimal S _t solves

$$ V_1\left(S_t-\rho S_{t-1},S_{t-1}\right) - \delta\rho V_1\left(S_{t+1}-\rho S_t,S_t\right) + \delta V_2\left(S_{t+1}-\rho S_t,S_t\right) = 0 $$

(26)

if the constraint does not bind and equals ρS _t − 1 otherwise.

The affect model also has a stock variable. But in the affect model it is utility itself. That is, loosely, in the affect model, expression (23), with u _t − 1 substituted for S _t − 1, defines both the flow of utility and the transformation of the stock variable (i.e., it is as if it combines Eqs. 23 and 24 in one expression). It is this difference—combined with the fact the affect model requires no particular sign on the cross-partial derivative of V—that distinguishes these models. It also explains why the dynamic programming problem in the affect model is so much more straightforward than in the rational addiction literature.

Appendix 2: Dynamics without monotonicity

As discussed in the text, the never-regret-a-good-mood assumption, expression (4), served to vastly simplify the dynamics in our model. While we believe it to be a reasonable assumption, we briefly consider, to be complete, what would happen were it not to hold. Let ${\mathbf{x}^{\ast\ast}_t(u_{t-1})}$ denote the optimal decision rule. Observe, by the envelope theorem, that a change in x _τ , τ < t will not have an impact on future utility through ${\mathbf{x}^{\ast\ast}_t(u_{t-1})}$. Hence, the ${\mathbf{x}^{\ast\ast}_t(u_{t-1})}$ satisfy

$$ \left(\omega_t + \sum_{s=t+1}^T \omega_s \prod_{n=t+1}^s \frac{\partial U_n\big(\mathbf{x}^{\ast\ast}_n(u_{n-1}),u_{n-1}\big)}{\partial u}\right)\frac{\partial U_t\big({\mathbf{x}^{\ast\ast}_t(u_{t-1})},u_{t-1}\big)}{\partial x} = 0\,. $$

(27)

We can, thus, conclude

Proposition 1

Assume that a solution, $\mathbf{x}_t^{\ast}\left( u\right)$, exists for the program (3) for all possible u. Assume further that U _t (·,u) is strictly concave for all uand all t. Finally, assume that a solution${\mathbf{x}^{\ast\ast}_t(u_{t-1})}$exists to the decision maker’s dynamic programming and that solution satisfies

$$ \left(\omega_t + \sum_{s=t+1}^T \omega_s \prod_{n=t+1}^s \frac{\partial U_n\big(\mathbf{x}^{\ast\ast}_n(u_{n-1}),u_{n-1}\big)}{\partial u}\right) > 0 $$

(28)

along the equilibrium path. Then${\mathbf{x}^{\ast\ast}_t(u_{t-1})} = \mathbf{x}^\ast_t(u_{t-1})$; that is, the optimal path involves maximizing per period utility.

Proof

If Eq. 28 holds, then the first-order condition (27) can hold if and only if

$$ \frac{\partial U_t\big({\mathbf{x}^{\ast\ast}_t(u_{t-1})},u_{t-1}\big)}{\partial x} = 0\,. $$

But given that U _t (·,u) is strictly concave, this implies ${\mathbf{x}^{\ast\ast}_t(u_{t-1})} = \mathbf{x}^\ast_t(u_{t-1})$. Given that t was arbitrary, this completes the proof.□

For example, suppose T = ∞ and $\omega_t=\delta^{t-1}$, where δ is a constant discount factor. Further assume $U_t(x,u) = x-ux^2/2$ (observe this utility function does not satisfy expression (4)). It is readily shown that $x^\ast_t = 1/u_0$ if t is odd and $x^\ast_t = 2u_0$ if t is even. Substituting these values into Eq. 28 we have that the sign of Eq. 28 is the same as the sign of

$$ 1+\delta \frac{\delta - 2u_0^2}{1-\delta^2} $$

if t is even and

$$ 1+\delta \frac{2u_0^2\delta - 1}{2u_0^2(1-\delta^2)} $$

if t is odd. Both expressions are positive if $\max\{\frac{1}{2u_0^2},2u_0^2\}<1/\delta$. This condition would hold, for instance, if u ₀ = 1 and δ < 1/2. Observe that if this condition holds, then utility oscillates period to period between u ₀ and $\frac{1}{2u_0}$. Similarly, x oscillates between 1/u ₀ and 2u ₀. Such a “yoyo”-path with regard to the choice of x seems at odds with observed behavior with regard to many decisions and is one reason for questioning models that do not satisfy condition (4).