REGLAMENTO NUEVO DE ALIMENTOS Y BEBIDAS EN ECUADOR

Two-period models of switching costs affecting the market in terms of consumer’s behavior and competition were widely used in the 1980s, in which the representative works are Klemperer (1987a, b). These models usually set a Bertrand competition in a market with switching costs. However, allowing for switching costs in equilibrium leads to the non- existence of Nash-Bertrand equilibrium because of the assumption of homogeneous switching costs (Shy, 2001). In order to solve this problem, Shy (2002) built a model that was based on a new equilibrium concept, which is called the undercut-proof equilibrium. The model of consumers changing suppliers when facing switching costs is described as follows.

There are two firms A and B competing Bertrand style with brand A and brand B products, respectively. The marginal costs of the two firms are assumed to be 0. The consumers are distributed between the firms so that initially 𝑁_𝐴 consumers have already purchased brand A and 𝑁_𝐵 consumers have already purchased brand B. All of the consumers face switching

costs, SC>0, if they wish to change supplier. The utility function of each consumer type derived from the next purchase is given by:

𝑈_𝐴 ≝ {_−𝑃−𝑃𝐴 𝑖𝑓 𝑠𝑡𝑎𝑦𝑖𝑛𝑔 𝑤𝑖𝑡ℎ 𝐴

𝐵− 𝑆𝐶 𝑖𝑓 𝑏𝑢𝑦𝑖𝑛𝑔 𝑓𝑟𝑜𝑚 𝐵 (8)

𝑈𝐵 ≝ {−𝑃_−𝑃𝐴− 𝑆𝐶 𝑖𝑓 𝑏𝑢𝑦𝑖𝑛𝑔 𝑓𝑟𝑜𝑚 𝐴

𝐵 𝑖𝑓 𝑠𝑡𝑎𝑦𝑖𝑛𝑔 𝑤𝑖𝑡ℎ 𝐵 (9)

If firm A wishes to poach customers from firm B, it has to offer a lower price than firm B. Furthermore, the price difference has to be larger than the switching cost S to make it worthwhile for consumers to switch. Let 𝑁𝐴 denote the (endogenously determined) number of

brand A buyers (the next period purchase), and 𝑁𝐵 denote the number of brand B buyers (the

next period purchase). Then, (8) and (9) imply that:

𝑁𝐴= { 0 𝑖𝑓 𝑃_𝐴> 𝑃_𝐵+ 𝑆𝐶 𝑁_𝐴 𝑖𝑓 𝑃_𝐵− 𝑆𝐶 ≤ 𝑃_𝐴≤ 𝑃_𝐵+ 𝑆𝐶 𝑁_𝐴+ 𝑁_𝐵 𝑖𝑓 𝑃_𝐴< 𝑃_𝐵− 𝑆𝐶 (10) 𝑁𝐵 = { 0 𝑖𝑓 𝑃_𝐵> 𝑃_𝐴+ 𝑆𝐶 𝑁_𝐵 𝑖𝑓 𝑃_𝐴− 𝑆𝐶 ≤ 𝑃_𝐵 ≤ 𝑃_𝐴+ 𝑆𝐶 𝑁_𝐴+ 𝑁_𝐵 𝑖𝑓 𝑃_𝐵 < 𝑃_𝐴− 𝑆𝐶 (11)

Assume that the firms’ production costs are zero. Denote 𝜋_𝐴 and 𝜋_𝐵 as the profit of firm A and B. Thus, due to the equilibrium in Bertrand model, the profits of each firm are given as:

𝜋_𝐴(𝑃_𝐴, 𝑃_𝐵) = 𝑃_𝐴𝑁_𝐵 and 𝜋_𝐵(𝑃_𝐴, 𝑃_𝐵) = 𝑃_𝐵𝑁_𝐵 (12)

A Nash-Bertrand equilibrium would be a pair of non-negative prices {𝑃_𝐴, 𝑃_𝐵}. For a given 𝑃_𝐵, firm A chooses 𝑃𝐴 to maximize 𝜋𝐴, and the symmetry method to maximize 𝜋𝐵 of firm B.

Although the Nash-Bertrand equilibrium does not exist in pure strategies, an undercut-proof equilibrium does. According to Shy (2002) definition 1: firm i is said to undercut firm j, if it sets its price to 𝑃𝑖 < 𝑃𝑗 − 𝑆𝐶, i=A, B and 𝑖 ≠ 𝑗, That is, if firm i ‘subsidizes’ the switching

Prices represent an undercut-proof equilibrium if it is impossible for any firm to increase profits by undercutting the competitor while it is impossible for any firm to raise its price without being profitably undercut by the competitor. The under-proof property is formally designed (definition) as in Shy (2002):

A pair of prices {𝑃_𝐴𝑈_{, 𝑃}

𝐵𝑈} satisfies the undercut-proof property (UPP) if:

(a) For a given 𝑃_𝐵𝑈 _{and 𝑁}

𝐵𝑈, firm A chooses the highest price 𝑃𝐴𝑈 subject to the constraint

𝜋_𝐵𝑈_{= 𝑃}

𝐵𝑈𝑁𝐵𝑈≥ (𝑃𝐴− 𝑆𝐶)(𝑁𝐴+ 𝑁𝐵) (13)

(b) For a given 𝑃_𝐴𝑈 _{and 𝑁}

𝐴𝑈, firm B chooses the highest price 𝑃𝐵𝑈 subject to the constraint

𝜋_𝐴𝑈_{= 𝑃}

𝐴𝑈𝑁𝐴𝑈≥ (𝑃𝐵− 𝑆𝐶)(𝑁𝐴+ 𝑁𝐵) (14)

Although firm A sets the highest price possible in order to maximize profits, the price is still sufficiently low to prevent firm B from undercutting and taking the whole market. Firm A’s price is set low enough to make firm B’s profit from not undercutting, 𝑃_𝐵𝑈_𝑁

𝐵𝑈 larger than the

profit firm B would make when undercutting and capturing the whole market, (𝑃𝐴𝑈 −

𝑆𝐶)(𝑁𝐴+ 𝑁𝐵). But since both firms set prices as high as possible, the inequalities hold as

equalities. These equalities give the unique pair of prices {𝑃𝐴𝑈, 𝑃𝐵𝑈} where

𝑃_𝐴𝑈 ₌(𝑁𝐴+𝑁𝐵)(𝑁𝐴+2𝑁𝐵)∗𝑆𝐶

(𝑁𝐴)2+𝑁𝐴𝑁𝐵+(𝑁𝐵)2 (15)

and

𝑃_𝐵𝑈₌(𝑁𝐴+𝑁𝐵)(2𝑁𝐴+𝑁𝐵)∗𝑆𝐶

(𝑁𝐴)2+𝑁𝐴𝑁𝐵+(𝑁𝐵)2 . (16)

Then solve for the switching costs based on undercut-proof equilibrium. Inserting equations (13) and (14) in the equalities of definition gives that 𝑁_𝐴𝑈 _{= 𝑁}

𝐴 and 𝑁𝐵𝑈 = 𝑁𝐵. The solutions

𝑆𝐶_𝐴= 𝑃_𝐴− 𝑃𝐵𝑁𝐵

𝑁𝐴+𝑁𝐵 (17) 𝑆𝐶_𝐵 = 𝑃_𝐵− 𝑃𝐴𝑁𝐴

𝑁𝐴+𝑁𝐵 (18)

Shy (2002) extends the model described above to a multi-firm industry for estimating switching cost using merely information on market shares and prices, which is based on a solution to the non-existence of a Nash-Bertrand equilibrium. As to the estimation in the banking sector, price setting by banks is measured by the average lending interest rate. Define 𝑆_𝑖 to be the switching cost of a brand i consumer, and assume that 𝑆_𝑖 (i=1,2,……L) are known all firms and consumers. Then, each firm 𝑖 ≠ 𝐿 takes 𝑃𝐿 as given and sets maximal 𝑃𝑖

to satisfy:

𝜋_𝐿= 𝑃_𝐿𝑁_𝐿≥ (𝑃_𝑖− 𝑆𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔 𝑐𝑜𝑠𝑡𝑠_𝑖)(𝑁𝑖+ 𝑁𝐿) (19)

According to the method, the switching costs estimation model is designed as follows:

𝑆𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔 𝐶𝑜𝑠𝑡_𝑖𝑡 = 𝑃_𝑖𝑡− 𝑃𝐿𝑡𝑁𝐿𝑡

𝑁𝑖𝑡+𝑁𝐿𝑡 (20)

where the switching costs of bank 𝑖 are estimated as a function of the average interest P set by bank 𝑖 and 𝐿, and the market share of bank 𝑖 and 𝐿 at period t. 𝑃𝐿𝑡 𝑎𝑛𝑑 𝑁𝐿𝑡denote the average

interest rate and market share of bank L, which has the lowest market share in period t, respectively. Assume that the firm with the smallest market share, firm L, is a prey target of firm 1. Therefore, the price 𝑃𝐿 of firm L would make undercutting its price by firm 1

unprofitable; that is,

𝜋₁= 𝑃₁𝑁₁≥ (𝑃_𝐿− 𝑆𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔 𝑐𝑜𝑠𝑡𝑠_𝐿)(𝑁1+ 𝑁𝐿) (21)

Since 𝑃𝐿𝑡 is observed, the unobserved remaining switching cost 𝑆𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔 𝑐𝑜𝑠𝑡𝑠𝐿𝑡can be

solved as treating equation (12) equality. Thus, the switching costs of the bank which has lowest market share at period t can be estimated as:

𝑆𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔 𝐶𝑜𝑠𝑡𝐿𝑡 = 𝑃𝐿𝑡−_𝑁𝑃1𝑡𝑁1𝑡

1𝑡+𝑁𝐿𝑡 (22)