Network structure, equilibrium and dynamics in a monopolistically competitive economy

Abstract

Although network theory has been busy to emphasize the role of connection structures in shaping aggregate level phenomena of complex systems, there are only few attempts in economic modeling which try to build this dimension into the analysis. Macroeconomic models typically build on complete connectedness among economic actors (frictionless flow of information, perfect information on prices), thus these models typically oversee the possible effects of complex, incomplete network structures among economic agents on emergent macroeconomic phenomena. In this paper we try to fill this gap by incorporating possibly incomplete relationship structures between economic actors in a standard model of monopolistic competition and then analyze the effect of different network structures on the equilibrium and dynamic properties of the model. Analytical and simulation results of the model show that incomplete connectedness give rise to deadweight loss, shrinking output below the level observed in standard models with complete networks. Also, the dynamics of link formation has an effect on the steady state of the economy as well as on its response to shocks.

Appendix

Derivation of the demand function

In the model of monopolistic competition household i faces the following optimization problem:

$$\underset{x_{ij}}{\max} \quad X_{i}=\left( \sum\limits_{j = 1}^{F} s_{ij}{x_{ij}}^{\frac{\varepsilon-1}{\varepsilon}}\right)^{\frac{\varepsilon}{\varepsilon-1}} $$

$$s.t. \quad I_{i}=\sum\limits_{j = 1}^{F}p_{j}x_{ij} $$

For the sake of simplicity and because the problem is static in time the time indices can be omitted. The objective of the household is to reach as much composite consumption as possible with the given budget.

The Lagrange-function and the first-order conditions of the problem are:

$$\begin{array}{@{}rcl@{}} L_{i}(x_{ij},\lambda_{i})&=&\left( \sum\limits_{j = 1}^{F} s_{ij}{x_{ij}}^{\frac{\varepsilon-1}{\varepsilon}}\right)^{\frac{\varepsilon}{\varepsilon-1}}-\lambda_{i}\left( I_{i}-\sum\limits_{j = 1}^{F}p_{j}x_{ij}\right)\\ \frac{\partial L_{i}}{\partial x_{ij}}&=&\frac{\varepsilon}{\varepsilon-1}\left( \sum\limits_{j = 1}^{F} s_{ij}{x_{ij}}^{\frac{\varepsilon-1}{\varepsilon}}\right)^{\frac{\varepsilon}{\varepsilon-1}-1}\frac{\varepsilon-1}{\varepsilon}s_{ij}x_{ij}^{\frac{\varepsilon-1}{\varepsilon}-1}-\lambda_{i}p_{j}= 0 \quad j = 1,\ldots,F\\ \frac{\partial L_{i}}{\partial \lambda_{i}}&=&I_{i}-\sum\limits_{j = 1}^{F}p_{j}x_{ij}= 0 \end{array} $$

From the first-order conditions we get:

$$\begin{array}{@{}rcl@{}} s_{ij}x_{ij}^{-\frac{1}{\varepsilon}}X_{i}^{\frac{1}{\varepsilon}}&=&\lambda_{i}p_{j}\\ x_{ij}&=&s_{ij}^{\varepsilon}X_{i}\left( \lambda_{i}p_{j}\right)^{-\varepsilon} \end{array} $$

Take a firm k for which s_ik≠ 0 and take the ratio of the demands for the two products:

$$\begin{array}{@{}rcl@{}} \frac{x_{ij}}{x_{ik}}&=&\frac{s_{ij}^{\varepsilon}X_{i}\left( \lambda_{i}p_{j}\right)^{-\varepsilon}}{s_{ik}^{\varepsilon}X_{i}\left( \lambda_{i}p_{k}\right)^{-\varepsilon}}=\left( \frac{s_{ij}}{s_{ik}}\right)^{\varepsilon}\left( \frac{p_{j}}{p_{k}}\right)^{-\varepsilon}\\ x_{ij}&=&\left( \frac{s_{ij}}{s_{ik}}\right)^{\varepsilon}\left( \frac{p_{j}}{p_{k}}\right)^{-\varepsilon}x_{ik} \end{array} $$

Substitute this term into our initial condition:

$$I_{i}=\sum\limits_{j = 1}^{F}p_{j}x_{ij}=\sum\limits_{j = 1}^{F}p_{j}\left( \frac{s_{ij}}{s_{ik}}\right)^{\varepsilon}\left( \frac{p_{j}}{p_{k}}\right)^{-\varepsilon}x_{ik}=s_{ik}^{-\varepsilon}p_{k}^{\varepsilon}x_{ik}\sum\limits_{j = 1}^{F}s_{ij}^{\varepsilon}p_{j}^{1-\varepsilon} $$

Define the price index perceived by the i th household as follows:

$$P_{i}=\left( \sum\limits_{j = 1}^{F}s_{ij}^{\varepsilon}p_{j}^{1-\varepsilon}\right)^{\frac{1}{1-\varepsilon}} $$

Then the previous term for the income can be rewritten as follows:

$$I_{i}=s_{ik}^{-\varepsilon}p_{k}^{\varepsilon}x_{ik}P_{i}^{1-\varepsilon} $$

Expressing this in terms of x_ik we arrive to the demand functions for each product variety:

$$x_{ik}=s_{ik}^{\varepsilon}p_{k}^{-\varepsilon}I_{i}P_{i}^{\varepsilon-1} $$

the value of the objective function in optimum is:

$$\begin{array}{@{}rcl@{}} X_{i}&=&\left[\sum\limits_{j = 1}^{F} s_{ij}\left( s_{ij}^{\varepsilon}p_{j}^{-\varepsilon}I_{i}P_{i}^{\varepsilon-1}\right)^{\frac{\varepsilon-1}{\varepsilon}}\right]^{\frac{\varepsilon}{\varepsilon-1}}=I_{i}P_{i}^{\varepsilon-1}\left( \sum\limits_{j = 1}^{F}s_{ij}^{\varepsilon}p_{j}^{1-\varepsilon}\right)^{\frac{\varepsilon}{\varepsilon-1}}\\ &=& I_{i}P_{i}^{\varepsilon-1}\left( P_{i}^{1-\varepsilon}\right)^{\frac{\varepsilon}{\varepsilon-1}}=\frac{I_{i}}{P_{i}} \end{array} $$

That is, if the consumer chooses optimally, then her disposable income can be expressed by means of her perceived price index and composite consumption:

$$I_{i}=P_{i}X_{i} $$

Using the latter expression, consumption of each product can be expressed with the price of the product, the composite consumption and the perceived price index:

$$x_{ik}=s_{ik}^{\varepsilon}X_{i}\left( \frac{p_{k}}{P_{i}}\right)^{-\varepsilon} $$

where

$$P_{i}=\left( \sum\limits_{j = 1}^{F}s_{ij}^{\varepsilon}p_{j}^{1-\varepsilon}\right)^{\frac{1}{1-\varepsilon}} $$

If connections are binary (that is, the intensity of the relations does not matter, only their existence), then the s_ij parameters can have the value 0 or 1 and the equations can be simplified as follows:

$$x_{ik}=s_{ik}X_{i}\left( \frac{p_{k}}{P_{i}}\right)^{-\varepsilon} $$

where

$$P_{i}=\left( \sum\limits_{j = 1}^{F}s_{ij}p_{j}^{1-\varepsilon}\right)^{\frac{1}{1-\varepsilon}} $$

The firm’s optimal price decision

The firm aims to reach the highest possible profit for a given demand (??) and production technology (??); that is, it maximizes the following object function subject to the constraints (??) and (??):

$$\underset{p_{j}}{\max} \quad{\Pi}_{j}=p_{j}y_{j}-TC_{j}(y_{j})=p_{j}^{1-\varepsilon}\sum\limits_{i = 1}^{H}s_{ij}^{\varepsilon}X_{i}{P_{i}}^{\varepsilon}-TC_{j}\left( p_{j}^{-\varepsilon}\sum\limits_{i = 1}^{H}s_{ij}^{\varepsilon}X_{i}{P_{i}}^{\varepsilon}\right) $$

where TC_j(⋅) is the total cost function of firm j and we used the fact that in product market equilibrium the demand for the firm’s production equals its supply.

The first order condition of the optimization problem is the following:

$$\begin{array}{@{}rcl@{}} \frac{\partial {\Pi}_{j}}{\partial p_{j}}&=&(1-\varepsilon)p_{j}^{-\varepsilon}\sum\limits_{i = 1}^{H}s_{ij}^{\varepsilon}X_{i}{P_{i}}^{\varepsilon}-\frac{\partial TC_{j}}{\partial y_{j}}(-\varepsilon)p_{j}^{-\varepsilon-1}\sum\limits_{i = 1}^{H}s_{ij}^{\varepsilon}X_{i}{P_{i}}^{\varepsilon}\\ &=&p_{j}-\frac{\varepsilon}{\varepsilon-1}MC_{j}= 0 \end{array} $$

where we used that ∂TC_j/∂y_j = MC_j. According to this the optimal price of firm j is:

$$p_{j}=\frac{\varepsilon}{\varepsilon-1}MC_{j} $$

The firm’s total cost function is the following:

$$TC_{j}(y_{j})=WL_{j}=W\left( \frac{y_{j}}{A_{j}}\right)^{\frac{1}{1-\alpha}} $$

Deriving this with respect to y_j leads to the marginal cost function:

$$MC_{j}=\frac{\partial TC_{j}}{\partial y_{j}}=\frac{W}{(1-\alpha)A_{j}^{\frac{1}{1-\alpha}}y_{j}^{\frac{-\alpha}{1-\alpha}}}=\frac{W}{MP_{L_{j}}} $$

Accordingly, the optimal price can be written as follows:

$$p_{j}=\frac{\varepsilon}{\varepsilon-1}MC_{j}=\frac{\varepsilon}{\varepsilon-1}\frac{W}{MP_{L_{j}}} $$

This equation holds for all firms in every period t.