Appendix for: Price Setting in ForwardLooking Customer Markets


 Theodore Hutchinson
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1 Appendix for: Price Setting in ForwardLooking Customer Markets Emi Nakamura and Jón Steinsson Columbia University Appendix A. Second Order Approximations Appendix A.. A Derivation of a nd Order Approximation to the Firm s Value It is straightforward to show that the steady state price with full commitment is pz) = S, where variables without subscripts denote steady state values. Notice furthermore that equation ) in the paper implies that C = )cz) and equation ) in the paper implies that β)p = pz). A second order Taylor series approximation of the value of the firm around the steady state of the solution to the firm s problem with commitment is given by E 0 β t [cz)p t z) pz)) + pz)c tz) cz)) + p t z) pz))c t z) cz)) pz) S S t S)c t z) cz)) + cz) p β t t z) pz))m 0,t β t ) + ] pz) c β t t z) cz))m 0,t β t ) + ex. terms + O ξ 3 ), A.) where ex. terms stands for terms that are exogenous to the firm s decision problem, ξ stands for a vector of the exogenous variables and O ξ 3 ) denotes higher order terms. The exposition of our results is simplified if we make a change of variables. Let ĉ t z) = logc t z)/cz)) and define hatted versions of all other variables in the same way. Making use of the fact that c t z) = cz) + ĉ t z) + ) ĉtz) + O ξ 3 ),
2 we can rewrite equation A.) as E 0 [ pz)cz)β t ˆp t z) + ) ˆp tz) + ĉ t z) + ĉtz) ) + ˆp t z)ĉ t z) Ŝ t ĉ t z) +ˆp t z) ˆM 0,t + ĉtz) ˆM 0,t ] + ex. terms + O ξ 3 ).A.) Appendix A.. A Derivation of a nd Order Approximation to the Consumer Demand Curve Notice that consumer demand given by equation ) in the paper may be rewritten as P t C t t c t z) c t z)) t = p t z) + E t [M t,t+ P t+ Ct+ c t+ z) c t z)) t+ t+ ] A second order Taylor series approximation of this equation around the steady state of the solution to the firm s problem with commitment is given by. P t P ) + P C C t C) P cz) c t z) cz)) + P cz) c t z) cz)) P cz) P t P )c t z) cz)) + + P cz) c t z) cz)) P C cz) C t C)c t z) cz)) + P cz) c t z) cz)) t ) = p t z) pz)) β P cz) E t c t+ z) cz)) + s.o.ex.terms + O ξ, 3 ), A.3) where s.o.ex.terms denotes second order exogenous terms and O ξ, 3 ) denotes terms that are third order or higher) in ξ,. The norm ξ, is simply meant to denote the standard Euclidian distance norm in ξ, ) space. As in the case of the expression for the value of the firm, we find it convenient to rewrite equation A.3) in terms of the hatted variables and subsitute the E t ĉ t+ z) term our for a E tˆp t+ z) term. This yields ĉ t z) + ĉtz) ) ĉ t z) + )ĉtz) ˆP t Ĉt + ˆP t ĉ t z) + Ĉtĉ t z) ĉ t z)ˆ t = β)ˆp t z) ˆp tz) βe tˆp t+ z) + s.o.ex.terms + O ξ, A.4) 3 ),
3 Appendix B. Proofs of Propositions Appendix B.. Proposition Proof: We can rearrange equation 5) in the paper so that it says that β ĉtz) β ĉtz) β β E tĉ t+ z) = ˆp t z) β ĉtz) + + ) β)ĉtz) β ĉtz) ˆΥ t β ˆp tz) β ˆP t + Ĉt ˆP t ĉ t z) ) Ĉtĉ t z) + s.o.ex.terms + O ξ, 3 ). Now notice that equation 4) in the paper may be written [ ) E 0 pz)cz)β t β ĉtz) β ĉtz) β β ĉt+z) + ˆp t z) + ) ˆp tz) + ĉtz) + ˆp t z)ĉ t z) Ŝ t ĉ t z) +ˆp t z) ˆM 0,t + ] ĉtz) ˆM 0,t pz)cz) β ĉ0z) + ex. terms + O ξ, 3 ). Substituting consumer demand into this expression now yields [ E 0 pz)cz)β t ĉ t z) + β )ĉtz) + )ĉ t z) ˆΥ t + ˆp t z) ˆP t ĉ t z) )) Ĉtĉ t z) + ˆp tz) + ĉtz) + ˆp t z)ĉ t z) Ŝ t ĉ t z) +ˆp t z) ˆM 0,t + ] ĉtz) ˆM 0,t pz)cz) β ĉ0z) + ex. terms + O ξ, 3 ). If we now multiply this expression by ) β), use consumer demand to substitute for ĉ t z) and simplify, we get that [ ] E 0 pz)cz)β t ˆp t z) + ) ˆΥ tˆp t z) + )Ŝtˆp t z) + pz)cz) ˆp 0 z) +ex. terms + O ξ, 3 ). B.) Setting the derivative of this with respect to ˆp t z) for t equal to zero shows that the firm s optimal pricing policy under full commitment to a statecontingent rule for t is ˆp t z) = Ŝt + ˆΥ t up to an error of order O ξ, ). 3
4 Appendix B.. Proposition Proof: A derivation analogous to the derivation of expression B.) yields that the objective of the firm at time t can be written as E t j=0 pz)cz)β j [ ] ˆp t+j z) + ) ˆΥ t+j ˆp t+j z) + )Ŝt+j ˆp t+j z) + pz)cz) ˆp t z) +ex. terms + O ξ, 3 B.) ). We seek a Markov perfect equilibrium that is accurate up to a residual of order O ξ, ). The definition of a Markov perfect equilibrium implies that the strategies of the firms are functions of only the payoff relevant state of the economy. In our model, the state of the economy at time t is ĉ t z), Ŝt, ˆΥ t ). However, given our approximation, ĉ t z) contributes only terms of order O ξ, ) since it is multiplied by in consumer demand. This implies that, up to a residual of order O ξ, ), the stretegies of the firm are functions of only Ŝt, ˆΥ t ). Since these two variables are i.i.d., the firm can correctly assume that its decisions at time t have no effect on outcomes in any period T t +. The firm can therefore simply maximize expression B.) with respect to the current period price ˆp t z). This yields ˆp t z) = + Ŝt + ˆΥ t, B.3) up to an error of order O ξ, ). Appendix B.3. Proposition 3 Proof: We must show that the firm does not have a profitable deviation when it is setting p t z) = p c tz). The potential benefit from deviating at this point is a higher price in the current period. Denote this benefit by Π d t Π c t which could be negative). The loss is the change in future profits associated with playing the Markov perfect equilibrium in future periods rather than p t+j z) = p c t+jz). Denote the expected loss in the period after the deviation by E[Π c + Π m +] and the per period loss in subsequent periods by E[Π c Π m ]. The firm will refrain from deviating in the current period if Π d t Π c t < βe[π c + Π m +] + β E[Π c Π m ]/ β). This condition will hold for all β > β t where β t is implicitely defined by Π d t Π c t = β t E[Π c + Π m +] + β t E[Πc Π m ]/ β t ). The firm will never deviate if 4
5 β > β = max β t. If Π d t Π c t is negative, β = 0. It is relatively easy to show that β is independent of. Appendix B.4. Proposition 4 Proof: First, we show that the profit maximizing price rule assuming that prices can only be changed in odd numbered periods is given by equation 0) in the paper. The value of the firm s expected profits from period t on are given by expression B.). The firm maximizes this expression with respect to ˆp t subject to the constraint ˆp t z) = ˆp t+ z). We can use this constraint to eliminate ˆp t+ z) in expression B.) and rewrite it ignoring terms that are exogenous to the firm s time t problem. This yields [ ] ) + β) pz)cz) ˆp t z) + ) ˆΥ tˆp t z) + )Ŝtˆp t z) + ˆp t z) +ex. terms + O ξ, 3 ).B.4) Maximizing this with respect to ˆp t z) yields equation 0) in the paper. Next, we show that the value expected profits in future periods from adhering to the price path stated in the proposition is greater than from playing the Markov perfect equilibrium. The value of the expected future profits is equal to E t j= pz)cz)β j [ ] ˆp t+j z) + )Ŝt+j + ˆΥ t+j )ˆp t+j z) + ex. terms + O ξ, 3 ). Using equation 8) in the paper we can derive that, ignoring exogenous terms and terms of higher than second order, the value of the expected profits of a firm if it plays the Markov perfect equilibrium in all future periods is E t Π m β t+ = pz)cz) β [ ) + varŝt + ˆΥ t )]. Similarly, using equation 0) in the paper we can derive that the value of the expected profits of a firm that prices according the the rule in the proposition is larger than or equal 5
6 to [ [ min E t Π F Ŝ t+ ˆΥ t+ = min pz)cz)β ) ] + + β t Ŝ t+ ˆΥ t ) + β) + β) Ŝt + ˆΥ t ) [ ) ]] +pz)cz) β + + β β ) + β) + β) varŝt + ˆΥ t ) [ ) ] β β + β) = pz)cz) + β ) + β) + β) varŝt + ˆΥ t ) Comparing these expressions we get that minŝt+ ˆΥ t E t Π F t+ > E t Π m t+ if > ) + β β β + β varŝt + ˆΥ t ). Given this condition an argument analogous to the one given in the proof of Proposition 3 implies that the firm does not have a profitable deviation while it is setting its price according to equation 0) in the paper as long as β > β. Appendix B.5. Proposition 5 We must show that the portion of the price path described in the proposition that is played in equilibrium is the best feasible price path from the firm s perspective. Once we have shown this, an argument analogous to the proof of Proposition 3 implies that the firm does not have a profitable deviation from this price path. We begin by deriving the function in our model that correponds to Rx t, µ t, t ) in Athey et al. 004). Given the consumer s demand curve equation 5) one can view the decision the consumer makes at each point in time as a decision about what he expects prices to be in the next period. To see this, notice that we can rewrite equation 5) as ˆp t z) = β ĉtz) β ĉtz) β β E tĉ t+ z) + β ˆp tz) + β ĉtz) + ) β)ĉtz) ˆP t Ĉt + ˆP t + ) β) Ĉt + ) ˆΥ t ĉ t z) + s.o.ex.terms + O ξ, 3 ). Notice furthermore that ĉ t z) = ˆp t z) + ˆP t + Ĉt + O ξ, ). 6 B.5)
7 Using this fact, we can rewrite the consumer s demand curve as ˆp t z) = β ĉtz) β ĉtz) β + β E tˆp t+ z) + β ˆp tz) + β ˆp tz) ˆP tˆp t z) Ĉtˆp t z)) + ) ) β) ˆp tz) ˆP tˆp t z) Ĉtˆp t z)) ˆP t Ĉt ˆP t + ) β) Ĉt + ) ˆΥ t ˆp t z) + s.o.ex.terms + O ξ, 3 ). Notice that once the consumer has chosen what to expect about the firm s price in period t +, this equation determines his demand. One can therefore view the consumer s decision at each point given the form of the demand curve as a choice about what to expect about the firm s price in the next period. In equilibrium, the consumer will have rational expectations. We have used this fact by writing the consumer s expectation as E tˆp t+. However, more generally, we can denote the consumer s expectation about ˆp t at time t as x t. Using this notation, the consumer s demand curve becomes ˆp t z) = β ĉtz) β ĉtz) β + β x t+ + β ˆp tz) + β ˆp tz) ˆP tˆp t z) Ĉtˆp t z)) + ) ) β) ˆp tz) ˆP tˆp t z) Ĉtˆp t z)) ˆP t Ĉt ˆP t + ) β) Ĉt + ) ˆΥ t ˆp t z) + s.o.ex.terms + O ξ, 3 ). B.6) Next, notice that the second order approximation of the value of the firm equation 4) may be written E 0 pz)cz)β t [ ˆp t z) + β ĉtz) + ˆp tz) + ĉtz) + ˆp t z)ĉ t z) +ex. terms + O ξ 3 ) ) β ĉtz) Ŝ t ĉ t z) + ˆp t z) ˆM 0,t + ] ĉtz) ˆM 0,t β ĉtz) Using equation B.6) to eliminate the first two terms in this expression, equation B.5) to eliminate ĉ t z) and multiplying the resulting expression by β) ) gives [ E 0 pz)cz)β t ˆp t z) βx t+ + ] ˆp t + )Ŝt + ˆΥ t )ˆp t z) +ex. terms+o ξ, 3 ). B.7) 7
8 Collecting the terms in the sum that involve ˆp t z), x t and ˆΦ t = Ŝt + ˆΥ t we can define Rx t, ˆp t z), Ŝt) = ˆp t z) x t )ˆp t z) + )ˆΦ tˆp t z) B.8) for t. This is the function in our model that corresponds to Rx t, µ t, t ) in Athey et al. 004). Mapping our notation into the notation used by Athey et al. 004) we get that: x t = x t, µ t = ˆp t z) and t = ˆΦ t. In the notation used by Athey et al. 004), the firm s objective function is Rx t, µ t, t ) = µ t x t )µ t + ) t µ t. Notice that this function satisfies all the conditions required for the propositions in Athey et al. 004) to be valid. Specifically, R x x t, µ t, t ) = < 0, R µ x t, µ t, t ) = > 0 and R µµ x t, µ t, t ) = ) < 0. The main difference between our results and the results in Athey et al. 004) is that they consider a model in which Rx t, µ t, t ) is the social welfare function, i.e. it is the objective of all the agents in the model. The fact that Rx t, µ t, t ) in Athey et al. 004) is the social welfare function entails that the resulting policy is socially optimal. Here we use the objective of the firm as our Rx t, µ t, t ), which means that the resulting policy is not socially optimal but rather the best policy from the perspective of the firm. The proofs in Athey et al. 004) do not rely on Rx t, µ t, t ) being a social welfare function. Only their interpretation as solving for the socially optimal policy relies on this. Given equation B.8) and the following monotone hazard conditions: P ˆΦ t ))/pˆφ t ) is strictly decreasing in ˆΦ t and P ˆΦ t )/pˆφ t ) is strictly increasing in ˆΦ t, Proposition in Athey et al. 004) shows that the pricing policy that is optimal from the perspective of the firm is static. Here pˆφ t ) and P ˆΦ t ) denote the pdf and cdf of ˆΦ t, respectively. We assume that ˆΦ t [ˆΦ, ˆΦ]. Furthermore, Proposition in Athey et al. 004) shows that the firm s best pricing policy is either a constant price or it is a policy of bounded discretion, i.e., ˆp ˆΦ; z) if ˆΦ [ˆΦ, ˆΦ ] ˆpz) = ˆp ˆΦ ; z) if ˆΦ [ˆΦ, ˆΦ] 8 B.9)
9 where ˆp ˆΦ; z) denotes the static best response of a firm with a desired price equal to ˆΦ and ˆΦ ˆΦ ˆΦ. To complete the description of the policy most prefered by the firm, we must calculate four things: ) Under what conditions does the firm prefer a constant price? ) What is the optimal constant price from the firm s perspective? 34) When the firm prefers to set its price according to equation B.9), what is the optimal cutoff point ˆΦ and what is the firm s static best response ˆp ˆΦ; z)? The remainder of this section draws heavily on appendix D in Athey et al. 004). First, notice that the static best reponse of the firm solves Rˆpz) E ˆpz), ˆpz), ˆΦ) = 0. The solution is ˆp ˆΦ, z) = + ˆΦ. If the firm s pricing policy is of the form B.9), then B.0) E ˆpz) = ˆΦ ˆΦ ˆp ˆΦ, z)pˆφ)dˆφ + ˆΦ ˆp ˆΦ, z)pˆφ)dˆφ. Using equation B.0) to plug in for ˆp ˆΦ, z) in this equation we get that E ˆpz) = ˆΦ ˆΦ )pˆφ)dˆφ. ˆΦ Athey et al. 004) show that the objective of the firm, RE ˆpz), ˆpz), ˆΦ)pˆΦ)dˆΦ may be written RE ˆpz), ˆp ˆΦ, z), ˆΦ) + ˆΦ ˆΦ RˆΦE ˆpz), ˆp ˆΦ, z), ˆΦ)[ P ˆΦ)]dˆΦ + RˆΦE ˆpz), ˆp ˆΦ, z), ˆΦ)[ P ˆΦ)]dˆΦ. ˆΦ Since RˆΦE ˆpz), ˆpz), ˆΦ) = )ˆpz), this expression simplifies to ˆΦ ˆΦ ˆΦ )pˆφ)dˆφ + ) ˆΦ ˆΦ ˆΦ[ P ˆΦ)]dˆΦ + ) ˆΦ [ P ˆΦ)]dˆΦ + ex. terms. ˆΦ Differentiating this with respect to ˆΦ and setting the resulting expression equal to zero yields pˆφ)dˆφ + ) [ P ˆΦ)]dˆΦ = 0, ˆΦ ˆΦ 9
10 which is equivalent to [ P ˆΦ )] + ) [ P ˆΦ)]dˆΦ = 0. ˆΦ When ˆΦ < ˆΦ, P ˆΦ ) > 0, so this last equation is equivalent to P ˆΦ) pˆφ) + ) ˆΦ pˆφ) P ˆΦ ) dˆφ = 0. B.) Notice that the second term on the left hand side of this equaiton is the conditional mean of P ˆΦ))/pˆΦ) over the interval [ˆΦ, ˆΦ]. Since P ˆΦ))/pˆΦ) is strictly decreasing in ˆΦ monotone hazard assumption), its conditional mean is also strictly decreasing in ˆΦ. This implies that equation B.) has at most one interior solution. Since the expression on the left hand side of equation B.) is decreasing in both and ˆΦ, it is furthermore the case that ˆΦ is decreasing in. We have shown that equation B.) has at most one interior solutions. To show that such a solution in fact exists we must show that the left hand side of this equation is negative for ˆΦ close ˆΦ and positive for ˆΦ = ˆΦ. Notice that when ˆΦ ˆΦ, P ˆΦ))/pˆΦ) 0. This implies that for > 0 and ˆΦ close enough to ˆΦ, the left hand side of equation B.) is strictly less than zero. When ˆΦ = ˆΦ, equation B.) is not defined. However, ˆΦ = ˆΦ is a solution to the equation above equation B.). However, since the expression on the left hand side of that equation is strictly negative for ˆΦ < ˆΦ in the neighborhood of ˆΦ, this is not a local maximum. Athey et al. 004) show that at ˆΦ = ˆΦ the left hand side of equation B.) becomes ˆΦ. Since ˆΦ < 0, this is positive for 0, ˆΦ). So, there is an interior solution in this case. When > ˆΦ there is no interior solution to equation B.). This implies that for this range of the firm s best policy is a constant price. Finally, when > ˆΦ the firm chooses its constant price to maximize ˆΦ RE ˆpz), ˆpz), ˆΦ)pˆΦ)dˆΦ subject to E ˆpz) = ˆpz). The solution to this problem is ˆpz) = 0. 0
11 References Athey, S., Atkeson, A., Kehoe, P. J., 005. The optimal degree of discretion in monetary policy. Econometrica 73 5),