Does Restricting Outsiders Always Lower Price and Benefit Insiders?

Abstract

Policies that restrict outsiders are common. Some justifications include protecting insiders from high price and leaving more of the concerned products to insiders. Sometimes these policies fail to work because outsiders can get around the restrictions. In a model in which a policy of restricting outsiders is anticipated, we find that if the policy works, it only sometimes lowers the price. When the price does decrease, the product quality decreases too. Not every insider would benefit equally; those insiders who likely suffer are identified. While restricting outsiders may or may not reduce insiders’ consumer surplus, outsiders and the producer are always worse off. They therefore would find ways to get around the restrictions. Evaluating these policies must (a) take into account the possibility that they might not work at all, (b) check their effects beyond just price if they do work.

Appendices

Appendix A: Proofs

Proof of Lemma 1.

The first order conditions for the profit maximization problem in Eq. 7 are:

$$ \begin{array}{@{}rcl@{}} &&\frac{\partial \pi}{\partial p} = \frac{m}{s+mt} \left\{ (s-p)-[p-c(s)]\right\} = 0, \end{array} $$

(A.1)

$$ \begin{array}{@{}rcl@{}} &&\frac{\partial \pi}{\partial s} = \frac{m}{s+mt}\left\{ -c^{\prime}(s)(s-p) + [p-c(s)]\frac{p+mt}{s+mt} \right\} = 0. \end{array} $$

(A.2)

Simplifying these two equations gives $p^{*}=\frac {s^{*}+c(s^{*})}{2},$ and $c^{\prime }(s^{*}) = \frac {p^{*}+mt}{s^{*}+mt}$. □

Proof of Proposition 1.

To find the sign of $\frac {\partial p^{*}}{\partial m}$ and $\frac {\partial s^{*}}{\partial m}$, we totally differentiate (A.1) and (A.2) with respect to m to obtain the following system of equations:

$$ \begin{array}{@{}rcl@{}} \left[\begin{array}{cc} \frac{\partial^{2} \pi}{\partial p^{2}} & \frac{\partial^{2} \pi}{\partial p\partial s} \\ \frac{\partial^{2} \pi}{\partial p\partial s} & \frac{\partial^{2} \pi}{\partial s^{2}} \end{array}\right] \left[\begin{array}{c} \frac{\partial p^{*}}{\partial m} \\ \frac{\partial s^{*}}{\partial m} \end{array}\right] = \left[\begin{array}{c} -\frac{\partial^{2} \pi}{\partial p\partial m} \\ -\frac{\partial^{2} \pi}{\partial s\partial m} \end{array}\right]. \end{array} $$

(A.3)

Let H denote the Hessian matrix. By Cramer’s rule:

$$ \begin{array}{@{}rcl@{}} & \frac{\partial p^{*}}{\partial m} = \frac{\left|\begin{array}{cc} -\frac{\partial^2 \pi}{\partial p\partial m} & \frac{\partial^2 \pi}{\partial p\partial s} \\ -\frac{\partial^2 \pi}{\partial s\partial m} & \frac{\partial^2 \pi}{\partial s^2} \end{array}\right|}{\left|\begin{array}{cc} \frac{\partial^2 \pi}{\partial p^2} & \frac{\partial^2 \pi}{\partial p\partial s} \\ \frac{\partial^2 \pi}{\partial p\partial s} & \frac{\partial^2 \pi}{\partial s^2} \end{array}\right|} =\frac{ -\frac{\partial^{2} \pi}{\partial p\partial m}\frac{\partial^{2} \pi}{\partial s^{2}} +\frac{\partial^{2} \pi}{\partial s\partial m}\frac{\partial^{2} \pi}{\partial p\partial s} }{|H|}, \end{array} $$

(A.4)

$$ \begin{array}{@{}rcl@{}} & \frac{\partial s^{*}}{\partial m} = \frac{\left|\begin{array}{cc} \frac{\partial^2 \pi}{\partial p^2} & -\frac{\partial^2 \pi}{\partial p\partial m} \\ \frac{\partial^2 \pi}{\partial p\partial s} & -\frac{\partial^2 \pi}{\partial s\partial m} \end{array}\right|}{\left|\begin{array}{cc} \frac{\partial^2 \pi}{\partial p^2} & \frac{\partial^2 \pi}{\partial p\partial s} \\ \frac{\partial^2 \pi}{\partial p\partial s} & \frac{\partial^2 \pi}{\partial s^2} \end{array}\right|} =\frac{ -\frac{\partial^{2} \pi}{\partial p^{2}}\frac{\partial^{2} \pi}{\partial s\partial m} +\frac{\partial^{2} \pi}{\partial p\partial s}\frac{\partial^{2} \pi}{\partial p\partial m} }{|H|}. \end{array} $$

(A.5)

Observe that at (p^∗,s^∗), $\frac {\partial ^{2} \pi }{\partial p\partial m}=0$, $\frac {\partial ^{2} \pi }{\partial s\partial m}=\frac {mt[p^{*}-c(s^{*})](s^{*}-p^{*})}{(s^{*}+mt)^{3}}\geq 0$, and $\frac {\partial ^{2} \pi }{\partial p\partial s}=\frac {m[1+c^{\prime }(s^{*})]}{s^{*}+mt}>0$. By the second order condition, $\frac {\partial ^{2} \pi }{\partial p^{2}}<0$ and |H| > 0. Therefore:

$$ \begin{array}{@{}rcl@{}} & \frac{\partial p^{*}}{\partial m} =\frac{ \frac{\partial^{2} \pi}{\partial s\partial m}\frac{\partial^{2} \pi}{\partial p\partial s} }{|H|} = \frac{\frac{mt[p^{*}-c(s^{*})](s^{*}-p^{*})}{(s^{*}+mt)^{3}} \times \frac{m[1+c^{\prime}(s^{*})]}{s^{*}+mt}}{|H|}, \end{array} $$

(A.6)

$$ \begin{array}{@{}rcl@{}} & \frac{\partial s^{*}}{\partial m} =\frac{ -\frac{\partial^{2} \pi}{\partial p^{2}}\frac{\partial^{2} \pi}{\partial s\partial m} }{|H|} = \frac{-\frac{\partial^{2} \pi}{\partial p^{2}} \times \frac{mt[p^{*}-c(s^{*})](s^{*}-p^{*})}{(s^{*}+mt)^{3}}}{|H|}. \end{array} $$

(A.7)

With crowding out (i.e., t = τ > 0), $\frac {\partial p^{*}}{\partial m}>0$ and $\frac {\partial s^{*}}{\partial m}>0$. That is, $p^{*}_{\alpha } < p^{*}_{1}$ and $s^{*}_{\alpha } < s^{*}_{1}$, indicating that restricting outsiders lowers the producer’s equilibrium price and quality.

Without crowding out (i.e., t = 0), $\frac {\partial p^{*}}{\partial m}=0$ and $\frac {\partial s^{*}}{\partial m}=0$. That is, $p^{*}_{\alpha } = p^{*}_{1}$ and $s^{*}_{\alpha } = s^{*}_{1}$, suggesting that restricting outsiders does not affect the producer’s equilibrium price and quality. □

Proof of Proposition 2.

Let r^∗ = s^∗/p^∗ be the quality-per-dollar ratio in equilibrium. When t = 0: Proposition 1 suggests that $p^{*}_{\alpha } = p^{*}_{1}$ and $s^{*}_{\alpha } = s^{*}_{1}$ so that $r^{*}_{\alpha }=r^{*}_{1}$.

When t = τ > 0: Differentiating r^∗ with respect to m, we have $\frac {\partial r^{*}}{\partial m} = \frac {p^{*}\frac {\partial s^{*}}{\partial m} - s^{*} \frac {\partial p^{*}}{\partial m} }{(p^{*})^{2}}$. By Lemma 1, $p^{*}=\frac {s^{*}+c(s^{*})}{2}$ so that $\frac {\partial p^{*}}{\partial m}=\frac {1}{2}\left [\frac {\partial s^{*}}{\partial m} + c^{\prime }(s^{*}) \frac {\partial s^{*}}{\partial m}\right ]$. Therefore, we can rewrite $\frac {\partial r^{*}}{\partial m}$ as:

$$ \frac{\partial r^{*}}{\partial m} = \frac{\frac{s^{*}+c(s^{*})}{2} - s^{*} \left\{\frac{1}{2}\left[1 + c^{\prime}(s^{*})\right]\right\}}{(p^{*})^{2}} \frac{\partial s^{*}}{\partial m} = \frac{c(s^{*}) - s^{*}c^{\prime}(s^{*})}{2(p^{*})^{2}} \frac{\partial s^{*}}{\partial m}. $$

(A.8)

By Lemma 1 again, we have $c^{\prime }(s^{*}) = \frac {p^{*}+mt}{s^{*}+mt} = \frac {\frac {s^{*}+c(s^{*})}{2}+mt}{s^{*}+mt}$. Therefore:

$$ c(s^{*}) - s^{*}c^{\prime}(s^{*}) = c(s^{*}) - s^{*}\left[\frac{\frac{s^{*}+c(s^{*})}{2}+mt}{s^{*}+mt}\right] = \frac{[c(s^{*})-s^{*}](s+2mt)}{2(s^{*}+mt)}. $$

(A.9)

Non-negative quantity demanded implies p^∗ < s^∗. By Lemma 1, we have $\frac {s^{*}+c(s^{*})}{2}<s^{*}$ or s^∗ > c(s^∗). Therefore, $c(s^{*}) - s^{*}c^{\prime }(s^{*})<0$ and thus $\frac {\partial r^{*}}{\partial m}<0$. That is, $r^{*}_{\alpha }>r^{*}_{1}$, i.e., restricting outsiders raises the quality-per-dollar ratio. □

Proof of Proposition 3.

Fixing w and given (p,s), the difference between the producer’s profit when outsiders are welcomed (denoted as π₁) and that when outsiders are restricted (denoted as π_α) is:

$$ \begin{array}{@{}rcl@{}} \pi_{1}(p,s)-\pi_{\alpha}(p,s) &=\left\{[p-c(s)] \frac{s-p}{s+t} - w\right\} - \left\{[p-c(s)] \frac{\alpha(s-p)}{s+\alpha t} - w\right\} \end{array} $$

$$ \begin{array}{@{}rcl@{}} &=[p-c(s)]\frac{(s-p)s(1-\alpha)}{(s+t)(s+\alpha t)}, \end{array} $$

(A.10)

which is positive for all t ≥ 0. Therefore, π₁(p,s) > π_α(p,s) for all (p,s), including $(p^{*}_{\alpha },s^{*}_{\alpha })$. Moreover, by revealed preference, the optimal profit when m = 1 at $(p^{*}_{1},s^{*}_{1})$ must be larger than that at $(p^{*}_{\alpha },s^{*}_{\alpha })$, i.e., $\pi _{1}(p^{*}_{1},s^{*}_{1})>\pi _{1}(p^{*}_{\alpha },s^{*}_{\alpha })$. Taken together:

$$ \pi_{1}(p^{*}_{1},s^{*}_{1}) >\pi_{1}(p^{*}_{\alpha},s^{*}_{\alpha}) > \pi_{\alpha}(p^{*}_{\alpha},s^{*}_{\alpha}). $$

(A.11)

That is, the producer’s equilibrium profit is always higher when outsiders are unrestricted; this observation is true regardless of whether there is a crowding out effect. □

Proof of Proposition 4.

Note that given c(s) and t, the shape of the utility function when m = 1 does not change. In the case of Fig. 2a, it must be true that the consumer with 𝜃_i = 1 has a higher utility when m = 1 than when m = α, i.e.:

$$ s^{*}_{\alpha} [1 -c^{\prime}(s^{*}_{\alpha})] < s^{*}_{1} [1 -c^{\prime}(s^{*}_{1})]. $$

(A.12)

If the values of α (mass of insiders) and τ (extent of crowding out), and the shape of the cost function c(s) is such that the above condition holds, then at least some insiders are better off when outsiders are restricted. □

Proof of Proposition 5.

Here we show that in the presence of crowding out, restricting outsiders can increase or decrease the consumer surplus of insiders. Let $A_{m} = {\int \limits }_{\bar {\theta }_{m}}^{1} u_{i} \mathrm {d} \theta _{i}$ denote the area under the utility function when m = {α, 1}. It is the area of a triangle with base $1-\bar {\theta }_{m}=1-c^{\prime }(s^{*}_{m})$ and height $s^{*}_{m} [1 -c^{\prime }(s^{*}_{m})]$. By simple geometry, we have:

$$ A_{m} = \frac{s^{*}_{m} [1 -c^{\prime}(s^{*}_{m})]^{2}}{2}. $$

(A.13)

Considering that $s^{*}_{\alpha }<s^{*}_{1}$ and $1-c^{\prime }(s^{*}_{\alpha })>1-c^{\prime }(s^{*}_{1})$, we do not know whether $A_{\alpha }\gtrless A_{1}$, that is, we cannot confirm whether the consumer surplus of insiders is higher when outsiders are welcomed. □

Appendix B: Further Details for Footnote 26

We use the data of Lee and Ooi (2018) to do the following exercise.³⁹ We first run the following regression using their matched sample of Executive condominiums (ECs) and Private condominiums (PCs):

$$ CONQUAS_{ikt} = \alpha + \upbeta Controls_{ik} + \tau_{k} + \varphi_{t} + \tau_{k}\times \varphi_{t} + \varepsilon_{ikt}, $$

(B.1)

where i, k, and t are respectively indexes of a condominium unit, a neighborhood, and year, CONQUAS_ikt represents the CONQUAS score of the unit,⁴⁰Controls_ik is a vector of physical attributes of the unit and locational characteristics of the neighborhood (including Development size, Distance to the nearest subway station, Distance to central business district, presence of different amenities (Swimming pool, Barbecue pit, Gym, Mini-mart, Pavillion, Playground, Sauna, Clubhouse, Exercise area, Basketball court, Tennis court, Library, Lounge, and Game room)), τ_k and φ_t are respectively the neighborhood and year fixed-effects, and ε_ikt is the residual.

We then obtain the residuals from the regression and perform a t-test to compare the residuals of ECs and those of PCs. Table 1 summarizes the results. These results seem to suggest that the CONQUAS scores for ECs are lower than those for PCs.

Table 1 Comparing the CONQUAS scores of the Executive and Private condominiums