Preferences and income effects in monopolistic competition models

Abstract

This paper develops a novel approach to modeling preferences in monopolistic competition models with a continuum of goods. In contrast to the commonly used constant elasticity of substitution preferences, which do not capture the effects of consumer income and the intensity of competition on equilibrium prices, the present preferences can capture both effects. The relationship between consumers’ incomes and product prices is then analyzed for two cases: with and without income heterogeneity.

Appendices

Appendix A

In this part of Appendix, I analyze the properties of the price function $p(\varphi ,\,Q)$ determined by the profit maximization problem. Recall that if the solution of the maximization problem in (8) is interior, then the optimal price solves

$$\begin{aligned} \frac{1}{\varphi p}=1-\frac{1-F(pQ)}{pQf(pQ)}. \end{aligned}$$

(22)

If $F(\varepsilon )$ satisfies the IPFR property, then $\frac{\varepsilon f(\varepsilon )}{1-F(\varepsilon )}$ is strictly increasing on $\left[ \varepsilon _{L},\,\varepsilon _{H}\right] .$ This implies that the right-hand side of the above equation is strictly increasing as a function of $p.$ Remember that for the interior solution, $p$ must belong to $\left[ \frac{\varepsilon _{L}}{Q},\,\frac{ \varepsilon _{H}}{Q}\right] .$ Hence, taking into account that the left-hand side is strictly decreasing in $p,$ Eq. (22) has a unique solution on $\left[ \frac{\varepsilon _{L}}{Q},\,\frac{\varepsilon _{H}}{Q}\right]$ if and only if

$$\begin{aligned} \frac{1}{\varphi \frac{\varepsilon _{L}}{Q}}&\ge 1-\frac{1-F\left( \frac{ \varepsilon _{L}}{Q}Q\right) }{\frac{\varepsilon _{L}}{Q}Qf\left( \frac{\varepsilon _{L}}{ Q}Q\right) }\quad \text{ and } \\ \frac{1}{\varphi \frac{\varepsilon _{H}}{Q}}&\ge 1-\frac{1-F(\frac{ \varepsilon _{H}}{Q}Q)}{\frac{\varepsilon _{H}}{Q}Qf(\frac{\varepsilon _{H}}{ Q}Q)}. \end{aligned}$$

The latter is equivalent to

$$\begin{aligned} \varphi \in \left[ \frac{Q}{\varepsilon _{H}},\,\frac{f(\varepsilon _{L})Q}{ f(\varepsilon _{L})\varepsilon _{L}-1}\right] . \end{aligned}$$

Next, I provide the proofs of Lemmas 2 and 3.

1.1 The Proof of Lemma 2

From Lemma 1, if $\varphi >\frac{f(\varepsilon _{L})Q}{ f(\varepsilon _{L})\varepsilon _{L}-1},$ then $p(\varphi ,\,Q)$ is equal to ${\varepsilon _{L}}/{Q}$ and, therefore, is decreasing in $Q.$ If $\varphi \in \left[ \frac{Q}{\varepsilon _{H}},\,\frac{f(\varepsilon _{L})Q}{ f(\varepsilon _{L})\varepsilon _{L}-1}\right] ,$ then $p(\varphi ,\,Q)$ is the solution of

$$\begin{aligned} \frac{1}{\varphi p}=1-\frac{1-F(pQ)}{pQf(pQ)}. \end{aligned}$$

As $F(\varepsilon )$ satisfies the IPFR property, $\frac{1-F(pQ)}{pQf(pQ)}$ is decreasing in $Q$ for any $p.$ This implies that for any $p,$ the right-hand side of the above equation is increasing in $Q\!:$ higher $Q$ shifts the function $1-\frac{1-F(pQ)}{pQf(pQ)}$ upward. As a result, the value of $p(\varphi ,\,Q)$ decreases.

1.2 The Proof of Lemma 3

Consider $\varphi \in \left[ \frac{Q}{\varepsilon _{H}},\,\frac{f(\varepsilon _{L})Q}{f(\varepsilon _{L})\varepsilon _{L}-1}\right] .$ It is straightforward to see that in this case, $p(\varphi ,\,Q)Q$ solves

$$\begin{aligned} \frac{Q}{\varphi x}=1-\frac{1-F(x)}{xf(x)}, \end{aligned}$$

with respect to $x.$ Higher $Q$ shifts the left-hand side of the equation upward. This means that the value of $p(\varphi ,\,Q)Q$ increases, as the right-hand side is increasing in $x.$

Appendix B

In this Appendix, I explore the effects of a rise in population size $L$ on the equilibrium outcomes in the long run. In the long run, the equilibrium is determined by

$$\begin{aligned}&\displaystyle \int \limits _{0}^{\infty }\pi (\varphi ,\,Q)dG(\varphi )=f_{e}, \\&\displaystyle M_{e}\int \limits _{\varphi ^{*}}^{\infty }p(\varphi ,\,Q)(1-F( p(\varphi ,\,Q)Q))dG(\varphi ) =y, \end{aligned}$$

where $\pi (\varphi ,\,Q)$ is given by (11). As can be seen from the equations, population size affects the equilibrium only through $\pi (\varphi ,\,Q).$ Specifically, a rise in $L$ increases $\pi (\varphi ,\,Q)$ for any $\varphi ,\,Q.$ In other words, higher $L$ increases the firm’s expected profits, implying that the left-hand side of the free entry equation rises. From Lemmas 2 and 3, it is straightforward to show that $\pi (\varphi ,\,Q)$ is a decreasing function of $Q$ (for any $\varphi$). Thus, in order the free entry condition is satisfied, a rise in $L$ has to be compensated by a rise in $Q.$ A rise in $Q$ in turn results in lower prices charged by firms (as it is shown in Lemma 2).

Finally, the mass of entrants can be found from

$$\begin{aligned} M_{e}=\frac{y}{\int _{\varphi ^{*}}^{\infty }p(\varphi ,\,Q)(1-F(p(\varphi ,\,Q)Q)) dG(\varphi )}. \end{aligned}$$

As the denominator is decreasing in $Q$ and consumer income $y$ is fixed, a rise in $L$ results in more entry into the market (higher $M_{e}$).