Quantitative Techniques for Competition and Antitrust Analysis by Peter Davis and Eliana Garcés_12 potx

408 8. Merger Simulation p 1 p 2 p 2 NE p 1 = R 1 ( p 2 ; c 1 ) * p 2 = R 2 ( p 1 ; c 2 ) * Price Post-merger (= Price Cartel ) Static ‘‘Nash equilibrium’’ prices where each firm is doing the best they can given the price charged by the other(s) p 1 = R 1 ( p 2 ; c 1 ) NE p 2 = R 2 ( p 1 ; c 2 ) NE p 1 NE NE NE Figure 8.6. A two-to-one merger in a differentiated product pricing game. the demand model. For example, the linear model could involve D 2 .p/ D a 2 C b 21 p 1 Cb 22 p 22 so that @D 2 .p/=@p 1 D b 21 and, analogously, @D 1 .p/=@p 2 D b 12 . Note that if the two products are substitutes and @D i .p/=@p j >0, then the equi- librium price for a firm maximizing joint profits will be higher, absent countervailing efficiencies. This is because the monopolist, unlike the single-product firm in the duopoly, gains the profits from the customers who switch to the competing product after a price increase. We illustrated this fact for the two-product game in chapter 2. The effect of a merger in a two-to-one merger in a market with two differentiated single-product firms is illustrated in figure 8.6. Because Bertrand price competition with differentiated products is a model where products are strategic complements, the reaction functions are increasing in the price of the other good. The intersection of the two pricing functions gives the optimal price for the Bertrand duopoly. After the merger, the firm will price differently since it internalizes the effect of changing the price of a product on the other product’s profits. This will result in higher prices for both products. In this case, the post-merger price is also that which would be associated with a perfect cartel’s prices. 8.3.2.2 Multiproduct Firms Let us now consider the case of a firm producing several products pre-merger. If a market is initially composed of firms producing several products, this means that firms’profit maximization already involves optimization across many products. The pricing equation of given goods will also depend on the demand and cost parameters of other goods which are produced by the same firm. A merger will result in a change in the pricing equation of certain goods as the parameters of the cost and demand of the products newly acquired by the firm will now enter the pricing equations of 8.3. General Model for Merger Simulation 409 all previously produced goods. This is because the number of products over which the post-merger firm is maximizing profits has changed relative to the pre-merger situation. Suppose firmf producesa setof productswhich wedenote = f Â=Df1;:::;Jg and which is unique to this firm. The set of products produced by the firm does not typically include all J products in the market but only a subset of those. The profit- maximization problem for this firm involves maximization of the profits on all the goods produced by the firm: max p f X j 2= f ˘ j .p f ;p f / D max p f X j 2= f .p j  mc j /D j .p/: Solving for the profit-maximizing prices will result in a set of first-order conditions. For firm f , the system of first-order conditions is represented as follows: D k .p/ C X j 2= f .p j  mc j / @D j .p/ @p k D 0 for all k 2= f : To these equations, we must add the first-order conditions of the remaining firms so that in the end we will, as before in the single-product-firms case, end up with a total of J first-order conditions, one for each product being sold. Solving these J equations for the J 1 vector of unknown prices p  will provide us with the Nash equilibrium in prices for the game. In comparison with the case where firms produced only a single product, the first-order conditions for multiproduct firms have extra terms. This reflects the fact that the firms internalize the effect of a change in prices on the revenues of the substitute goods that they also produce. Because of differences in ownership, first- order conditions may well not have the same number of terms across firms. To simplify analysis of this game, we follow the literature and introduce a J J ownership matrix  with the jkth element (i.e., j th row, kth column) defined by  jk D ( 1 if same firm produces j and k; 0 otherwise: We can rewrite the first-order conditions for each firm f D 1;:::;F as D k .p/ C J X j D1  jk .p j  mc j / @D j .p/ @p k D 0 for all k 2= f ; where the  jk terms allow the summation to be across all products in the market in all first-order conditions for all firms. The matrix  acts to select the terms that involve the products produced by firm f and changes with the ownership pattern of products in the market. At the end of the day, performing the actual merger simulations will only involve changing elements of this matrix from zero to one and tracing through the effects of this change on equilibrium prices. Once again, we will 410 8. Merger Simulation have a set of equations for every firm resulting in a total of J pricing equations, one first-order condition for each product being sold. In order to estimate demand parameters, we need to specify demand equations. For simplicity, let us assume a system of linear demands of the form, q k D D k .p 1 ;p 2 ;:::;p J / D a k C J X j D1 b kj p j for k D 1;:::;J: This specification conveniently produces @D k .p/ @p j D b kj : So that the first-order conditions become a k C J X j D1 b kj p j C J X kD1  jk .p j  mc j /b jk D 0 for all k 2= f and for all f D 1;:::;F: This will sometimes be written as q k C J X kD1  jk .p j  mc j /b jk D 0 for all j; k D 1;:::;J but one must then remember that the vector of quantities is endogenous and depen- dent on prices. Writing the system of equations this way and adding it together with the demand system provide the 2J equations which we could solve for the 2J endogenous variables: J prices and J quantities. Doing so provides the direct analogue to the standard supply-and-demand system estimation that is familiar for the homogeneous product case. Sometimes we will find it easier to work with only J equations and to do so we need only substitute the demand function for each product into the corresponding first-order condition. Doing so allows us to write a J -dimensional system of equations which can be solved for the J unknown prices. Large systems of equations are more tractable if expressed in matrix form. Fol- lowing the treatment in Davis (2006d) to express the demand system in matrix form, we need to define the matrix of demand parameters B 0 as B 0 D 2 6 6 6 6 6 6 6 4 b 11  b 1j  b 1J : : : : : : : : : b k1  b kj  b kJ : : : : : : : : : b J1  b Jj  b JJ 3 7 7 7 7 7 7 7 5 ; 8.3. General Model for Merger Simulation 411 where b kj D @D k .p/=@p j , and also define a D 2 6 6 6 6 6 6 6 4 a 1 : : : a k : : : a J 3 7 7 7 7 7 7 7 5 ; which is the vector of demand intercepts and where the prime on B indicates a transpose. The system of demand equations can then be written as 2 6 6 6 6 6 6 6 4 q 1 : : : q k : : : q J 3 7 7 7 7 7 7 7 5 D 2 6 6 6 6 6 6 6 4 a 1 : : : a k : : : a J 3 7 7 7 7 7 7 7 5 C 2 6 6 6 6 6 6 6 4 b 11  b 1j  b 1J : : : : : : : : : b k1  b kj  b kJ : : : : : : : : : b J1  b Jj  b JJ 3 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 4 p 1 : : : p j : : : p J 3 7 7 7 7 7 7 7 5 ; or, far more compactly in matrix form, as q D a CB 0 p. In order to express the system of pricing equations in matrix format, we need to specify the J J matrix  B, which is the element-by-element product of  and B, sometimes called the Hadamard product. 15 Note that B is the transpose of B 0 . Specifically, define   B D 2 6 6 6 6 6 6 6 4  11 b 11   j1 b j1   J1 b J1 : : : : : : : : :  1k b 1k  jk b jk  Jk b Jk : : : : : : : : :  1J b 1J   jJ b jJ   JJ b JJ 3 7 7 7 7 7 7 7 5 ; where b jk D @D j .p/=@p k . The rows will include the parameters of the pricing equation of a given product k. The term  jk will take the value of either 1 or 0 depending on whether the firm produces goods j and k or not and  jj D 1 for all j since the producer of good j produces good j . Recall the analytic expression for the pricing equations: D k .p/ C J X j D1  jk .p j  mc j / @D j .p/ @p k D 0 for all k 2= f and for all = f : The vector of all J first-order conditions can now be expressed in matrix terms as a C B 0 p C.  B/.p  c/ D 0; 15 Such matrix products are easily programmed in most computer programs. For example, in Gauss define A D B  C to define the Hadamard element-by-element product so that a jk D b jk c jk for j D 1;:::;J and k D 1;:::;J. 412 8. Merger Simulation where c D 2 6 4 mc 1 : : : mc J 3 7 5 and a D 2 6 4 a 1 : : : a J 3 7 5 : Alternatively, as we have already mentioned we may sometimes choose to work with the J pricing equations without substituting the demand equations: q C .  B/.p  c/ D 0. We will then need to work with a system of equations comprising these J equations and also the J demand equations. Written in matrix form, the equations that we need to solve simultaneously can then compactly be written as q C . B/.p  c/ D 0 and q D a C B 0 p: Using a structural form specification with all endogenous variables on the left side of the equations and the exogenous ones on the right side we have " .  B/ I B 0 I #" p q # D " .  B/ 0 .J J/ 0 .J J/ I .J J/ #" c a # ; which is equivalent to " p q # D " .  B/ I B 0 I # 1 " .  B/ 0 .J J/ 0 .J J/ I .J J/ #" c a # : This expression gives an analytic solution for all prices and all quantities for any ownership structure that can be represented in  since we may arbitrarily change the values of  jk from 0s to 1s to change the ownership structure provided only that we always respect the symmetry condition that  jk D  kj . With this system in place, once the parameters in B, c, and a are known, we can calculate equilibrium prices after a merger by setting the corresponding elements of  jk to 1. Indeed, we can calculate the equilibrium prices and quantities (and hence profits) for any ownership structure. 8.3.2.3 Example of Merger Simulation To illustrate the method, consider the example presented in Davis (2006f), a market consisting of six products that are initially produced by six different firms. Suppose the demand for product 1 is approximated by a linear demand and its parameters have been estimated as follows: q 1 D 10  2p 1 C 0:3p 2 C 0:3p 3 C 0:3p 4 C 0:3p 5 C 0:3p 6 : By a remarkably happy coincidence, the demands for other products have also been estimated and conveniently turned out to have a similar form so that we can write 8.3. General Model for Merger Simulation 413 the full system of demand equations in the form q j D 10  2p j C 0:3 X k¤j p k for j D 1;2;:::;6: Let us assume marginal costs of all products are equal to 1 and that the merger will generate no efficiencies so that c Pre j D c Post j D 1 for j D 1;2;:::;6. The pricing equation for the single-product firm is derived from the profit maximization first-order condition and takes the form @˘.p j / @p j D D j .p/ C .p j  c j / @D j .p/ @p j D 0: In our example this simplifies to q j D .p j  c j /.2/: The system of pricing and demand equations in the case of six firms producing one product each is then written as a total of twelve equations: " . Pre  B/ I B 0 I #" p q # D " . Pre  B/ 0 .J J/ 0 .J J/ I .J J/ #" c a # ; where  Pre takes the form of the identity matrix and B 0 D 2 6 6 6 6 6 6 6 4 2 0:3 0:3 0:3 0:3 0:3 0:3 2 0:3 0:3 0:3 0:3 0:3 0:3 2 0:3 0:3 0:3 0:3 0:3 0:3 2 0:3 0:3 0:3 0:3 0:3 0:3 2 0:3 0:3 0:3 0:3 0:3 0:3 2 3 7 7 7 7 7 7 7 5 ; . Pre  B/ D 2 6 6 6 6 6 6 6 4 200000 0 20000 002000 00020 0 000020 000002 3 7 7 7 7 7 7 7 5 ; c D 2 6 6 6 6 6 6 6 4 1 1 1 1 1 1 3 7 7 7 7 7 7 7 5 ;aD 2 6 6 6 6 6 6 6 4 10 10 10 10 10 10 3 7 7 7 7 7 7 7 5 : 414 8. Merger Simulation We can solve for prices and quantities: " p q # D " . Pre  B/ I B 0 I # 1 " . Pre  B/ 0 .J J/ 0 .J J/ I .J J/ #" c a # : If the firm that produced product 1 merges with the firm that produced product 5 the ownership matrix will change so that . Post-merger  B/ D 2 6 6 6 6 6 6 6 4 2 0 0 0 0:3 0 0 20000 002000 00020 0 0:3 0 0 0 20 000002 3 7 7 7 7 7 7 7 5 : This is because the new pricing equation for product 1 will be derived from the following first-order condition: @˘.p/ @p 1 D D 1 .p/ C .p 1  c 1 / @D 1 .p/ @p 1 C .p 5  c 5 / @D 5 .p/ @p 1 D 0; which in our example results in q 1 D .p 1  c 1 /.2/  .p 5  c 5 /.0:3/: New equilibrium prices and quantities can then be easily calculated using the new system of equations: " p q # D " . Post-merger  B/ I B 0 I # 1 " . Post-merger  B/ 0 .J J/ 0 .J J/ I .J J/ #" c a # : These kinds of matrix equations are trivial to compute in programs such as Mat- lab or Gauss. They may also be programmed easily into Microsoft Excel, making merger simulation using the linear model a readily available method. The predicted equilibrium prices for each product under different ownership structure are repre- sented in table 8.1. The market structure is represented by .n 1 ;:::;n F /, where the length of the vector F indicates the total number of active firms in the market and each of the values of n f represents the number of products produced by the f th firm in the market. The largest firm is represented by n 1 . Tables 8.1 and 8.2 show equilibrium prices and profits respectively for a variety of ownership structures. The results show, for example, that a merger between a firm that produces five products and one firm that produces one product, i.e., we move from market structure .5; 1/ to the market structure with one firm producing six products (6), increases the prices by more than 33%. Table 8.2 shows that the merger is profitable. 8.3. General Model for Merger Simulation 415 Table 8.1. Prices under different ownership structures. Market structure (n 1 ;:::;n F ) ‚ …„ ƒ Product .1; 1; 1; 1; 1; 1/ .2; 2; 2/ .3; 3/ .4; 2/ .5; 1/ 6 (Cartel) 1 4.8 5.3 5.9 6.62 7.87 10.5 2 4.8 5.3 5.9 6.62 7.87 10.5 3 4.8 5.3 5.9 6.62 7.87 10.5 4 4.8 5.3 5.9 6.62 7.87 10.5 5 4.8 5.3 5.9 5.77 7.87 10.5 6 4.8 5.3 5.9 5.77 5.95 10.5 Table 8.2. Profits under different ownership structures. Market structure .n 1 ;:::;n F / ‚ …„ ƒ Firms .1; 1; 1; 1; 1; 1/ .2; 2; 2/ .3; 3/ .4; 2/ .5; 1/ 6 (Cartel) 1 28.88 63.39 105 139 188.54 270.8 2 28.88 63.39 105 77.6 48.99 3 28.88 63.39 4 28.88 5 28.88 6 28.88 Industry profits 173 190 210 217 238 270.8 8.3.2.4 Inferring Marginal Costs In cases where estimates of marginal costs cannot be obtained from industry infor- mation, appropriate company documents, or management accounts, there is an alternative approach available. Specifically, it is possible to infer the whole vec- tor of marginal costs directly from the pricing equations provided we are willing to assume that observed prices are equilibrium prices. Recall the expression for the pricing equation in our linear demand example: a C B 0 p C.  B/.p  c/ D 0: In merger simulations, we usually use this equation to solve for the vector of prices p. However, the pricing equation can also be used to solve for the marginal costs c in the pre-merger market, where prices are known. Rearranging the pricing equation we have c D p C. B/ 1 .a C B 0 p/: More specifically, if we assumer pre-merger prices are equilibrium prices, then given the demand parameters in .a; B/ and the pre-merger ownership structure embodied 416 8. Merger Simulation in  Pre , we can infer pre-merger marginal cost products for every product using the equation: c Pre D p Pre C . Pre  B/ 1 .a C B 0 p Pre /: One needs to be very careful with this calculation since its accuracy greatly depends on having estimated the correct demand parameters and also having assumed the correct firm behavior. Remember that the assumptions made about the nature of competition determine the form of the pricing equation. What we will obtain when we solve for the marginal costs are the marginal costs implied by the existing prices, the demand parameters which have been estimated and also the assumption about the nature of competition taking place, in this case differentiated product Bertrand price competition. Given the strong reliance on the assumptions, it is necessary to be appropriately confident that the assumptions are at least a reasonable approximation to reality. To that end, it is vital to proceed to undertake appropriate reality checks of the results, including at least checking that estimated marginal costs are actually positive and ideally are within a reasonable distance of whatever accounting or approximate measures of marginal cost are available. This kind of inference involving marginal costs can be a useful method to check for the plausibility of the demand estimates and the pricing equation. If the demand parameters are wrong, you may well find that the inferred marginal costs come out either negative or implausibly large at the observed prices. If the marginal costs inferred using the estimated demand parameters are unrealistic, then this is a signal that there is often a problem with our estimates of the price elasticities. Alternatively, there could also be problems with the way we have assumed price setting works in that particular market. 8.3.3 General Linear Quantity Games In this section we suppose that the model that best fits the market involves com- petition in quantities. Further, suppose that firm f chooses the quantities of the products it produces to maximize profits and marginal costs are constant, then the firm’s problem can be written as max q f X j 2= f  j .q 1 ;q 2 ;:::;q J / D max q f X j 2= f .P j .q 1 ;q 2 ;:::;q J /  c j /q j ; where P j .q 1 ;q 2 ;:::;q J / is the inverse demand curve for product j . The represen- tative first-order condition (FOC) for product k is J X j D1  kj @P j .q/ @q k q j C .P k .q/ c k / D 0: We can estimate a linear demand function of the form q D a C B 0 p and obtain the inverse demand functions p D .B 0 / 1 q  .B 0 / 1 a: 8.3. General Model for Merger Simulation 417 In that case, the quantity setting equations become .  .B 0 / 1 /q C p  c D 0: And we can write the full structural form of the game in the following matrix expression: " I .B 0 / 1 B 0 I #" p q # D " I0 0I #" c a # : As usual, the expression that will allow us to calculate equilibrium quantities and prices for an arbitrary ownership structure will then be " p q # D " I .B 0 / 1 B 0 I # 1 " c a # : 8.3.4 Nonlinear Demand Functions In each of the examples discussed above, the demand system of equations had a convenient linear form. In some cases, more complex preferences may require the specification of nonlinear demand functions. The process for merger simulation in this case is essentially unaltered. One needs to calibrate or estimate the demand functions, solve for the pre-merger marginal costs if needed and then solve for the post-merger predicted equilibrium prices. That said, solving for the post-merger equilibrium prices is harder with nonlinear demands because it may involve solving a J  1 system of nonlinear equations. Generally, and fortunately, simple iterative methods such as the method of iterated best responses seem to converge fairly robustly to equilibrium prices (see, for example, Milgrom and Roberts 1990). Iterated best responses is a method whereby given a starting set of prices, the best responses of firms are calculated in sequence. One continues to recalculate best responses until they converge to a stable set of prices, the prices at which all first-order conditions are satisfied. At that point, provided second-order conditions are also satisfied, we will know we will have found a Nash equilibrium set of prices. The process is familiar to most students used to working with reaction curves as the method is often used to indicate convergence to Nash equilibrium in simple two-product pricing games that can be graphed. In practice, iterated best responses work as follows: 1. Define the best response for firm f given the rival’s prices as the price that maximizes its profits under those market conditions: R f .p f / D argmax p f X j 2= f .p j  mc j /D j .p/: 2. Create the following algorithm (following steps 3–5) in a mathematical or statistical package. [...]... !j // @qk D 0 for j 2 =f and also for each firm f: @pj These J equations, together with the J demand equations, provide us with the structural form for this model Note that the structural model involves a demand curve and a “supply” or pricing equation for each product available in the market, a total of 2J equations The only substantive difference between the linear and this nonlinear demand curve case... Suppose, by way of example, we aim to estimate the demand for sugar and for that purpose we have collected data on the quantity of sugar sold in millions of pounds and the price at which they were sold in cents per pounds The first step for analysis is to plot the available data Figure 9.1 shows the quantities and prices of sugar sold: (a) and (b) show respectively the quantity (deliveries) and (retail)... outside options, and imperfect monitoring In the theoretical literature, these issues have been tackled, for example, by Green and Porter (1984) and Rotemberg and Saloner (1986) and some empirical support for the patterns suggested by that literature are, for example, provided in Porter (1983) and Borenstein 434 8 Merger Simulation Table 8.7 Results from a coordinated effects merger simulation model... Source: Davis (2006f) Denotes IC constraint satisfied and Shephard (1986) While Davis and Huse (2009) provide a fully fledged merger simulation model for the complete information case, there are not yet any empirical coordinated effects merger simulation models under imperfect information In addition, an important role may be played by the presence of antitrust authorities, fines, leniency programs, and criminal... materially changed as a result of the merger 9 Demand System Estimation The previous chapters in this book have provided numerous illustrations of the importance of firm and market demand for understanding competition For example, we have seen that demand is important in determining firm behavior such as pricing decisions and we have also seen that demand is a central determinant of the effect of changes... than or close to 50% in Sweden, Norway, Finland, Denmark, and Ireland Ivaldi and Verboven (2005) details the simulation model developed for the case The focus of the analysis was on heavy trucks, which can be of two types known as “rigid” and “tractor,” the latter carrying a detachable container The demand for heavy trucks was modeled as a sequence of choices by the consumer, who in this case was a freight... values, EŒ j ˇ ; / j z1j  D 0 and EŒ!j ˇ ; / j z2j  D 0 (where z1j and z2j are sets of instrumental variables) in order to identify the demand and supply equations These moment conditions are exactly analogous to the moment conditions imposed on demand and supply shocks in the homogeneous product context.22 Ivaldi and Verboven undertake a simultaneous estimation of the demand and pricing equations using... of demand functions In general, though not exclusively, competition policy has a focus on price competition and as a result estimates of the own- and cross-price elasticities of demand are often important However, as in the rest of the book, much of the analysis can equally be applied should a nonprice variable such as advertising provide the main dimension of competition We begin the chapter by describing... Continuous Choice 9.1 437 Demand System Estimation: Models of Continuous Choice In this section, we introduce the simplest model of all, the demand for a single homogeneous good, and we progress to describe one of the most popular demand models for differentiated product markets, the almost ideal demand system (AIDS2 ) 9.1.1 Single-Product Demand Estimating the market demand in a market with a single... that collectively these forces (together with the actual process of collecting the data) are generating the data that we are 2 An unfortunate acronym, which has led some authors to describe the model as the nearly ideal demand system (NIDS) 438 9 Demand System Estimation observing and that our model is attempting to explain Only by a process of moving back and forth from data and regression results . for j 2= f and also for each firm f: These J equations, together with the J demand equations, provide us with the structural form for this model. Note that the structural model involves a demand curve. market consisting of six products that are initially produced by six different firms. Suppose the demand for product 1 is approximated by a linear demand and its parameters have been estimated as follows: q 1 D. 0:3p 6 : By a remarkably happy coincidence, the demands for other products have also been estimated and conveniently turned out to have a similar form so that we can write 8.3. General Model for Merger