Detecting multiple equilibria for continuous dependent variables

Abstract

This article considers structural equations where continuous dependent variables are related to independent variables and unobservables through a nonparametric function. Multiple equilibria may arise when the structural equations admit multiple solutions. This article proposes a detecting criterion for the existence of multiple equilibria. The main finding is that multiple equilibria would reveal itself in the form of jump(s) in the density function of the dependent variables. When there is a unique equilibrium, the density function of dependent variables will be continuous, whereas when there are multiple equilibria, the density will have jump(s) under reasonable conditions.

Keywords:

• Multiple equilibria
• nonparametric simultaneous equations
• density discontinuity

JEL Classification::

• C14
• C30
• C57

Notes

1 We use the terms dependent variables, endogenous variables, and outcome variables interchangeably.

2 For example, Berry et al. (1999) assume that “the equilibrium is unique (or at least that we solve for the relevant one.)” (p. 418)

3 An alernative identification approach relies on the completeness condition of the joint distribution of the endogenous and exogenous variables [see Assumption 4 in the work by Berry and Haile (2014, p. 1961)], which resembles the one used in nonparametric IV literature (Newey and Powell, 2003). However, the completeness condition is high level and difficult to interpret in economic models. Berry and Haile (2018) commented that “a high-level assumption like completeness implicitly places further restrictions on the model, although the nature of these restrictions is typically unclear.” (p. 290)

4 Throughout this article, continuity at a point means that the limits from all directions coincide. It does not place any condition on the value of the function at the limit point. A discontinuity or a jump refers to a jump discontinuity where the directional limits are not all equal.

5 For example, equilibrium $(y_1,y_2)$ is characterized by $r_1(y_1,y_2)=u_1$ and $r_2(y_{1,}{y}_2)=u_2.$ One can derive the best response function $y_2={\varphi }_2(y_{1,}{u}_2)$ from the second equation and then substitute it into the first equation. This yields $r_1(y_1,{\varphi }_2(y_{1,}{u}_2))=u_1,$ where the unobservable u₂ is not separated from (y₁, y₂).

6 For the aforementioned example, Echenique and Komunjer (2009) began with $r_1(y_1,y_2)=0$ and $r_2(y_1,y_2)=0,$ reduce it to $r_1(y_1,{\varphi }_2(y_1))=0$ by substitution, and then add a disturbance term $u$ to yield $r_1(y_1,{\varphi }_2(y_1))=u.$

7 Tamer (2003, p. 150) defines an incomplete econometric model as the one “where the relationship from (x, u) to y is a correspondence and not a function.”

8 See Assumption 1 of Berry and Haile (2014), which leads to $s_{jt}={\sigma }_j(p_t,{\delta }_t,x_t^{(2)}),$ where ${\delta }_{jt}=x_{jt}^{(1)}+{\zeta }_{jt},x_{jt}^{(1)}$ is one element of x_jt and $x_{jt}^{(2)}$ are the remaining elements. As in their article, $x_{jt}^{(2)}$ is suppressed because we can conditional on an arbitrary value of $x_{jt}^{(2)}.$ Let ${\delta }_{jt}=x_{jt}+{\zeta }_{jt}.$

9 See Assumption 2 of Berry and Haile (2014), which leads to ${\delta }_{jt}={\sigma }_j^{-1}(p_t,s_t)$ for all $j=1,\dots ,J$ and for any $(p_t,s_t).$

10 See Berry and Haile (2014, p. 1767–1770) for details.

11 By Eqs. (3) and (4), function $r=({\sigma }_1^{-1},\dots ,{\sigma }_J^{-1},{\pi }_1^{-1},\dots ,{\pi }_J^{-1})\prime .$ The component ${\sigma }_j^{-1}$ comes from the market share equations $s_{jt}={\sigma }_j(x_{jt}+{\zeta }_{jt},p_t)$ for $j=1,2,\dots ,J,$ where $p_t=(p_{1t},\dots ,p_{Jt})\prime .$ The functional form of σ_j is determined by individual utility (2) and the joint distribution of the idiosyncratic terms. Therefore, the smoothness of ${\sigma }_j^{-1}$ follows from smoothness of the demand function σ_j for product j, which is guaranteed by commonly used joint distributions of ${\epsilon }_{\mathit{jti}},j=1,2,\dots ,J.$ ${\pi }_j^{-1}$ comes from the condition that marginal revenue equals marginal cost: $c_j(s_t,w_{jt}+{\omega }_{jt})={\psi }_j(s_t,p_t)$ and thus $w_{jt}+{\omega }_{jt}=c_j^{-1}({\psi }_j(s_t,p_t),s_t),$ where c_j and ψ_j are marginal cost and marginal revenue functions, respectively. ${\pi }_j^{-1}$ is a composite of ψ_j and $c_j^{-1},$ thus its smoothness naturally follows from smoothness of the marginal revenue and marginal cost functions, see Berry and Haile (2014, p. 1767–1770) for derivation of the structural equations.

12 Suppose there are $v_1\ne v_2$ in B_m satisfying $y=q_m(v_1)=q_m(v_2).$ Then we have $r(y)=v_1\ne v_2=r(y),$ which cannot hold.

13 We require it to be positive in order to guarantee that the jump occurs at the interior of the support.

14 For example, let $y_d={\overline{A}}_{11}\cap {\overline{A}}_{12}.$ If the probability of picking up equilibrium 1 is ${\lambda }_1(r(y))=-\frac{1}{3}(y-y_d)+1$ for $y\in A_{12},$ then $\underset{y\in A_{12},y\to y_d}{\lim }{\lambda }_1(r(y))=1=\underset{y\in A_{11},y\to y_d}{\lim }{\lambda }_1(r(y))$ and it will prevent a jump of $f_Y(y)$ at y_d.

15 In (p. 1288) they write:“For a given x, different realizations of u can affect the support of ${\mathcal{P}}_{xu},$ but not the probabilities assigned to different outcomes in the support.”

16 Since ${\varphi }_1(\mathcal{M}(r(A_{12})))+{\varphi }_3(\mathcal{M}(r(A_{31})))\le 1,$ the two components cannot both equal to 1.

17 For asymptotic validity of our test, the rates must satisfy ${\gamma }_p+{\gamma }_h<1,3{\gamma }_p+{\gamma }_h>1,$ and ${\gamma }_p+3{\gamma }_h>1.$

Additional information

Funding

This work is supported by JSPS KAKANHI Grant numbers 17K13713 and 19K13666.