Tourist Trap Model with Downward-Sloping Demand Curve

2010 97 0203

Introduction

This paper will attempt to relax the unitary demand assumption of the tourist trap model that we saw in class. The others assumptions are conserved. We will now have a linear downward-sloping demand-curve:

p=G-gq

I will first discuss what could be the equilibrium price and how we can deduce it. Then, I will explain the conditions that must be fulfill to sustain this equilibrium. Finally, I’ll discuss the economic interpretation of these conditions.

Equilibrium price

Since we now consider a downward sloping demand curve, the quantity the consumer will buy could be more than 1 and will depends on the price. Therefore, a relevant measure to compare consumer choices would be the consumer surplus. To simplify the following demonstrations, I will use the following convention: Consumer surplus=fp=12(G-p)q(p) `

I assume in the following analysis that the consumer will buy in the shop, whatever the price is, a quantity depending on his demand function and that as long as his surplus is maximized. Equilibrium at p = MC?

Let’s suppose first that the firms all charge p=MC. If one firm charges p’ such as: f(p')<fp-c

Then the unfortunate consumer will still buy in this shop since it would results in a lower surplus if he tries to search for another shop. Therefore, as in the more simplified model, p = MC is not a feasible equilibrium. Monopoly price as an equilibrium price

We will now consider the single price equilibrium at the monopoly price and try to find if it would be profitable for a single firm to deviate from the equilibrium. Consider n-1 shops practicing the monopolist price pm and a single shop practicing a lower price p'. If the consumer is lucky enough to find the cheaper shop right away, he will buy according to his demand curve and his surplus will be fp' On the other hand, the unlucky consumer will still buy in the monopolist pricing shop if his surplus to do so is superior to his surplus to search for the cheaper shop, or in equation: fpm>1n-1(fp'-c)

From the last equation, we can solve for the limit quantity at which the consumer will start to search. fpm=1n-1(fp'-c)

n-1fpm=(fp'-c)

n-1fpm=12G-p'-cq'p'

n-1fpm=12G-p'-cG-p'g

Expanding and rearranging, we eventually find:

p'22g-p'*Gg+G22g-c-fpmn-1=0

Which lead to the following formula for p’:

p'= Gg2--2g-c-fpmn-1 g

The limit quantity is easily found by putting q’ in the inverse demand function. The price p’ is the price a deviating firm should adopt in order to attract all the market to its shop. Indeed, with such a price, the surplus of the consumer to buy at p’ minus the search cost is just equal to his surplus should he buy at price pm.

Conditions for a single-price equilibrium

Profit condition

Let’s now compare the profits for the firm. If the firm does not deviate, its profit will be: π=mnpm-MCqm

But if the firm deviates and charge p’ instead, its profit becomes: π'=mp'-MCq'

Therefore, by the solving the equality π'>π we will find the deviation condition: mp'-MCq'>mnpm-MCqm

From the previous analysis, we see that the pertinent variables are the number of firm n and the search cost c. Unfortunately, at this point I was unable to simplify the formulas enough to isolate n and c. Therefore I don’t have an equation for the conditions. So I proceed to a graphical analysis. Number of firm

Let’s consider the number of firm n first:

As we can see in figure 1, a larger number of firms will decrease the profit from both the monopolistic pricing firm and the deviating firm. However, the deviating firm sees its profit decline faster as n increases. Therefore, there exists a number of firm n’ high enough that, ceteris paribus, the deviating strategy won’t be profitable anymore and the single price equilibrium at the monopolistic price can be sustains. So the existence of single price equilibrium at the monopolistic price is...