Product Cycles and Prices: Search Foundation Mei Dong 1 Yuki Teranishi 2 1 University of Melbourne 2 Keio University and CAMA, ANU April 2018 1 / 59
In this paper, we Show a fact for product cycles and price dynamics. Make a simple price model with search friction in a goods market. Introduce a simple price model into general equilibrium model. Multiple price model (product chain and price chain). 2 / 59
Motivation: Product Turnover and Price Change Broda and Weinstain (2010) and Nakamura and Steinsson (2012) show that a price change in product replacement holds a nontrivial effect on a price index. Nakamura and Steinsson (2012) show that 40 percent of products are replaced without price change and 70 percent are replaced with two or less price. Ueda, Watanabe and Watanabe (2016) show that price adjustment occurs in timing of product turnover and more than half of products do not experience price changes until their exit from the market. Observations tell us that firms match with a new product, then set a new price with negotiation and fix the price until the product exits from a market. 3 / 59
Search Theory for Goods Market Michaillat and Saez (2015) assume a search and matching in a goods market. They show that productive capacity is idle in U.S. and such no full operating rate implies that sellers have a search friction to find buyers. They match a model to data and show that fixed-price model describes the data better than flexible price model does. Drozd and Nosal (2012) introduce search and matching into goods trade between countries in a model to solve puzzles regarding the correlation of real export and import prices and the volatility of the real exchange rate. Eaton, Jinkins, Tybout, and Xu (2016) assume a search and matching process for international buyer-seller connections for goods to explain various empirical issues. These papers imply the validity of search and match for a goods market. 4 / 59
Search Theory for Goods Market: Evidence Barrot and Sauvagnat (2016) show that there exist search and match frictions in production networks using firm level data. They find that the occurrence of natural disasters on suppliers reduces output on their customers when these suppliers produce specific input goods and justify that specific input goods are not traded in a centralized market that does not need search frictions. Carvalho, Nirei, Saito, and Tahbaz-Salehi (2016) also show that individual firms can not quickly find suitable alternatives under a decentralized goods market with search friction when firms are faced with a supply-chain disruption by a natural disaster in Japan. They also show that a disruption of the micro supply chain is a key driver of macro aggregated fluctuations. 5 / 59
Other Approach for Price Dynamics Calvo (1983) - Yun (1996) s price adjustment. Rotemberg (1982) s quadratic adjustment cost. New Keynesian model. Mankiw and Reis (2002) build up a sticky-information model. Information diffusion is slow and information updating is costly to reset goods price. Gertler and Leahy (2008) develop a tractable state dependent Phillips curve in contrast to a time dependent Phillips curve. Firms being in a state to get benefit over cost optimally reset new price. 6 / 59
Agents and Matching Market Two types of firms: Firms A (measure one) and Firms B (infinite). Firms A supply goods A. Firms B demands goods A. Two types of firms search for and match with a partner in a market for goods A. Only in a timing of match, they set price. This is a idea of Shimer (2004) and Hall (2005). Matching friction in a goods market. 7 / 59
Firm B (1) Firm B can either producer for goods B or (only) a seeker for goods A. At the beginning of period t, with probability s t, a seeker firm B is matched with a firm A. The flow cost of searching for a vacancy is κ > 0. Firm B then receives Z A units of goods A with a price P t A, produces Zt B (now random variable including a price) of goods B, give it to the next agent (could be Firms C), and pay back P t A Z A to firm A. At the end of period t, a product chain is dissolved with probability ρ (0, 1) (separation rate), then firm A and B separate and search for new matches in the next period t + 1. With probability 1 ρ, a contract survives. A survived firm keeps the same price. 8 / 59
Firm B (2) Free entry into a market of goods A and the value of a seeker firm is zero. κ = βs t E t W t+1 ( P A t+1). (1) W t ( P A t ) is the value of a productive firm B as and so W t ( P A t ) = Z B t Z A PA t + β(1 ρ)e t W t+1 ( P A t ), (2) W t ( P A t ) =Z B t Z A 1 β(1 ρ) P A t (3) + Z A β(1 ρ) 1 β(1 ρ) E t P A t+1 + β(1 ρ)e t W t+1 ( P A t+1). 9 / 59
Firm A (1) Firm A produces goods A using an exogenous resource with a cost. This resource can include a labor input through a production function even though we do not specify it in this stage. To search for seeker of firm B, firm A must post offers, which we call vacancies. Posting vacancies is costless, but total goods production by firm A is capped at Z A L. The number of vacancies v t is expressed as where L t is the number of match. v t = L (1 ρ)l t, (4) 10 / 59
Firm A (2) The value of a match for a firm A is J 1 t ( P A t ) = Z A PA t X t + βe t [(1 ρ)j 1 t+1( P A t ) + ρj 0 t+1 The value of a vacancy for a firm A is J 0 t = βe t [q t J 1 t+1( P A t+1) + (1 q t )J 0 t+1 The surplus of a firm A from a match is Z A ]. (5) ]. (6) Jt 1 ( P t A ) Jt 0 = 1 β(1 ρ) P t A β(1 ρ) 1 β(1 ρ) E t P t+1 A Z A + βe t [(1 ρ q t )(Jt+1( 1 P ] t+1) A Jt+1) 0 X t. (7) 11 / 59
Frictional Goods Market The number of new matches in a period is given by a Cobb-Douglas matching function m (u t, v t ) = χu 1 α t v α t, χ, α (0, 1). (8) Defining supply and demand for goods A in a market as θ t = u t v t, (9) we obtain s t = χθ α t, (10) q t = χθ 1 α t, (11) L t+1 = (1 ρ)l t + χθ 1 α t v t. (12) 12 / 59
Price Setting: Sharing Condition The price of goods A is determined according to Nash bargaining between the newly matched firm A and firm B. max W t ( P t A ) 1 b (Jt 1 ( P t A ) Jt 0 ) b, (13) P t A where b (0, 1) is the bargaining power for firm A. The first-order condition with respect to P A t yields bw t ( P A t ) = (1 b)(j 1 t ( P A t ) J 0 t ). (14) The aggregate price P A t of goods A is given by L t P A t = (1 ρ)l t 1 P A t 1 + χθ 1 α t 1 v t 1 P A t. (15) 13 / 59
Price Equation under Search Friction: Linearization Search-based Phillips Curve is given by π A t = β (1 ρ) E t π A t+1 + (1 α)γ q θt + x t, (16) π A t p A t (1 ρ) p A t 1, (17) γ q b Z B Z A ρβ q 1 α Z A, (18) x t ρb [1 β(1 ρ)] Z B Z A Ẑ X t B + ρ(1 b) [1 β(1 ρ)] Z A X t. (19) 14 / 59
Supplement: Price Level Expression p t A β(1 ρ) = 1 + β(1 ρ) 2 E t p t+1 A 1 ρ + 1 + β(1 ρ) 2 pa t 1 (20) + (1 α)γq 1 + β(1 ρ) 2 E t θ 1 t+1 + 1 + β(1 ρ) 2 x t. 15 / 59
Supplement: Real Profit Maximization When a firm maximizes a real profit in price setting, a value of a productive firm B is given by W t ( P A t ) = Z B t Z A P A t P A t where P A t is an aggregate price of goods A. A value of a match for a firm A is J 1 t ( P A t ) = Z A P A t P A t + β(1 ρ)e t W t+1 ( P A t ), (21) X t + βe t [(1 ρ)j 1 t+1( P A t ) + ρj 0 t+1 Other equations do not change and we finally derive a Search-based Phillips curve. ]. (22) π t = βe t π t+1 + (1 α)γ q θt + x t. (23) 16 / 59
Agents Now we extend our model to a general equilibrium model. We basically introduce a Search-based Phillips curve into an economy of Trigari (2009) and Monacelli, Perotti, and Trigari (2010). There are three types of agents, firms, a household, and a central bank. Household is a large family as given by Merz (1995). There are shopper-workers in a family. They match with firms and then provide labor forces through wage negotiation with firms and buy consumption goods through price negotiation with firms. Firms produce consumption goods by a production function using labor force. Firm sets price only when firms match with shopper-workers. 17 / 59
Household Problem (1) An infinitely lived representative household derives utility u from real aggregated consumption C t and disutility from labor supply l t (i), and discounts the future with discount factor β (0, 1). In period t, the household enjoys total real aggregate consumption and receives Π t as a real lump-sum profit from firms, and T t as a net real lump-sum transfer from the government. In addition, the household deposits D t into a bank account, to be repaid at the end of period t with a nominal interest rate R D t 1, where R D t is a policy variable of monetary policy. 18 / 59
Household Problem (2) For an aggregated consumption, we assume that the consumer s utility from consumption is increasing and concave in the consumption index of bundles of differentiated goods as in Benigno and Benigno (2003). C t [ ( 1 N t ) 1 ε N t 0 ] ε c t (i) ε 1 ε 1 ε di, (24) where ε (1, ) is the elasticity of substitution among intermediate goods. This consumption index has a property that prefers product variety out of symmetric equilibrium in which firms are homogeneous in any aspect. 19 / 59
Household Problem (3) Household chooses each c t (i) to minimize cost Nt 0 p t (i)c t (i)di, given the level of C t and the price of each intermediate good, p t (i). This minimization yields [ ] pt (i) ε C t c t (i) =, (25) N t P t where the price index is given by [ 1 P t N t Nt 0 ] 1 p t (i) 1 ε 1 ε di. (26) In a demand function, when the number of a product increases, a demand for each good decreases due to preference of product variety through a cost minimization problem, i.e., the same amount of consumption across goods is better. 20 / 59
Household Problem (4) Then, the household s intertemporal problem is max E t {C t+i,d t+i } i=0 i=0 subject to the budget constraint β i u(c t+i, G t+i ), (27) P t C t + D t = R D t 1D t 1 + W t L t + Π t + T t, (28) where P t denotes the price of C t, W t denotes the wage of L t, and L t = Nt The household s period utility function is 0 l t (i)di. (29) u(c t, G t ) C 1 σ t 1 σ G t, (30) where σ > 0 is relative risk aversion. The variable G t denotes the family s disutility from supplying labor. 21 / 59
Household Problem (5) Worker-shoppers in a large family go to a goods market and match with firms under a search friction and then provide labor forces with price and wage negotiations with firms. To decide a goods price and wages, each worker-shopper maximizes an additional utility contribution to a large family by a new match in a goods market Q t. Q t (p t (i), W t ) = W t l t,t (i) g (l t,t (i)) P t λ t + (1 ρ) β t,t+1 E t Q t+1 (p t (i), W t+1 ), (31) where λ t is a marginal utility of consumption, β t,t+1 β λ t+1 λ t is a stochastic discount factor, and g (l t,t (i)) is disutility from the labor supply of each member in a large family as follows. g (l t,t (i)) l 1+φ t,t (i) 1 + φ. (32) 22 / 59
Household Problem (6) φ 1 is Frisch elasticity of the labor supply to wages. Note that wage is common for all workers and is set by newly matched worker-shoppers and firms. Thus, wage is flexible in a sense that wage is set every period. For worker-shoppers, we assume free entry into a goods market. Thus, in equilibrium, the value of a seeker is zero, and hence the cost of searching must equal the expected revenue, or κ = s t Q t (p t (i), W t ), (33) where κ is a constant search cost. To simplify a model, a search cost does not consume consumption goods and is a tax to enter a market. This cost is finally returned to a household through transfer. 23 / 59
Firm Problem (1) Firms exist in an infinite number of a measure one. They can be productive when they match with worker-shoppers in a goods market. Firm i employs worker l t,t (i) from a household to produce an consumption goods through a linear production function. y t,t (i) = A t l t,t (i), (34) where A t is a technology shock. To search for a worker-shopper, a firm must post offers, which we call vacancies. Posting vacancies is costless, but total number of matches is capped at N = 1. The number of vacancies v t is expressed as v t = N B t (1 ρ)n t 1, (35) where N t is the number of matches between firms and a household and B t is a shock to the goods market. In period t, a vacancy is filled with probability q t. 24 / 59
Firm Problem (2) N t evolves according to N t = (1 ρ)n t 1 + q t v t. (36) The value of a new match for an intermediate goods producer i is Jt 1 (p t (i), W t ) = p t(i) y t,t (i) W t l t,t (i) (37) P t P [ t +β t,t+1 E t (1 ρ)j 1 t+1 (p t (i), W t+1 ) + ρjt+1] 0. The value of a vacancy for a firm is J 0 t = β t,t+1 E t [ qt+1 J 1 t+1(p t+1 (i)) + (1 q t+1 )J 0 t+1]. (38) 25 / 59
Firm Problem (3) These two equations imply that the surplus of a firm from a new match is Jt 1 (p t (i), W t ) Jt 0 = p t(i) y t,t (i) W t l t,t (i) (39) P t P { [ t + β t,t+1 E t (1 ρ) J 1 t+1 (p t (i), W t+1 ) Jt+1 0 ] q t+1 [ J 1 t+1 (p t+1 (i), W t+1 ) J 0 t+1]. 26 / 59
Goods Market and Resource Constraint The number of new matches in a period is given by a Cobb-Douglas matching function m (u t, v t ) = χu 1 α t v α t, χ, α (0, 1). (40) Defining supply and demand for consumption goods in a market as θ t = u t v t, (41) we obtain s t = χθ α t, (42) q t = χθ 1 α t, (43) N t = (1 ρ)n t 1 + χθ 1 α t v t. (44) 27 / 59
Nash Bargaining for Price and Wage A consumption goods price and wage for labor supply are determined according to Nash bargaining. Given a demand function for consumption goods, p t and W t solve max p t, W t [ Q t ( p t, W t ) ] 1 b [ Jt 1 ( p t, W ] b t ) Jt 0, (45) where p t is a relative price pt P t and b (0, 1) is the bargaining power for a firm. The first-order conditions with respect to p t and W t yields bq t ( p t, W t ) [ Jt 1 ( p t, p W ] t ) Jt 0 (46) t [ = (1 b) Jt 1 ( p t, W ] t ) Jt 0 Q t ( p t, p W t ), t bq t ( p t, W [ t ) = (1 b) Jt 1 ( p t, W ] t ) Jt 0. (47) 28 / 59
Aggregate Price and Resource Constraint The aggregate price P t of consumption goods is given by P 1 ε t = q tv t N t p 1 ε t A resource constraint is given by + (1 ρ) N t 1 Pt 1 1 ε N. (48) t C t = A t L t. (49) 29 / 59
Closed and Linearized Economy (1) We linearize a model around a constant steady state. From consumer s optimization problem, we have a standard IS equation. Ĉ t = E t Ĉ t+1 σ 1 ( Rt E t π t+1 ), (50) where π t p t p t 1. From a price setting problem, we have a Search-based Phillips curve. 1 (1 ρ) β ρ [ π t = βe t π t+1 + (σ + φ) 1 + εφ 1 ρ Ĉt φ N ] t φât. (51) This Search-based Phillips curve defines a relationship between demand, market conditions, and price. 30 / 59
Closed and Linearized Economy (2) From equations of a goods market, we have an equation for the number of match. N t = ψ N 1 E t Nt+1 +ψ N 2 N t 1 +ψ N 3 Ĉt+ψ N 4 E t βt,t+1 +ψ N 5 Ât+ψ N 6 B t, (52) where ψ N are parameters. This equation shows the number of matches depends on consumption, discount factor, and shocks. To close an economy, we assume a following Taylor type rule. R t = (1 ψ R )ψ π π t + (1 ψ R )ψ C Ĉ t + ψ R Rt 1, (53) where ψ R, ψ π, and ψ C are positive parameters. 31 / 59
Eliminating Goods Market Friction When we restrict the role of a market, a Search-based Phillips curve is independent from the number of matches in a market and is similar to a standard New Keynesian model. For example, we assume a constant match in a market, i.e., v t q t = N and an entry cost is time-varying. Then variation of the number of matches is zero, i.e., ˆN t = 0, and an entry cost changes to keep a constant number of matches. Then, a Search-based Phillips curve induces a New Keynesian Phillips curve with the Calvo s price adjustment mechanism as a special case. π t = βe t π t+1 + 1 (1 ρ) β 1 + εφ ρ [ ] (σ + φ) 1 ρ Ĉt φât. (54) 32 / 59
Parameter Values Param Value Explanation β 0.99 Discount Factor ρ 0.08 Product Entry Rate b 0.5 Bargaining Power for Firms α 0.5 Elasticity of Match of Supplier φ 1 0.1 Frisch Elasticity of Labor Supply to Wages ε 3.8 Elasticity of Substitution across Goods σ 2 Relative Risk Aversion ψ R 0.85 Coef of Lagged Interest Rate in Taylor Rule ψ π 2 Coef of Inflation Rate in Taylor Rule ψ C 0.5 Coef of Consumption in Taylor Rule N 0.83 Number of Match in a Steady State q 0.28 Probability of Match in a Steady State 33 / 59
Effect of Separation Rate Nakamura and Steinsson (2012) show that 70 percent of goods are replaced with two or less price. We assume a more frequent separation and set ρ = 0.16. 34 / 59
Response of Inflation Rate, Consumption, Number of Match, and Interest Rate to Productivity Shock for Different Separation Rates 0 Inflation Rate 0.15 Number of Match -0.05-0.1-0.15-0.2 rho=0.08 rho=0.16 0.1 0.05-0.25 0 5 10 15 0 0 5 10 15 0.4 Consumption 0 Interest Rate 0.3-0.05 0.2-0.1 0.1-0.15 0 0 5 10 15-0.2 0 5 10 15 35 / 59
Response of Inflation Rate, Consumption, Number of Match, and Interest Rate to Goods Market Shock for Different Separation Rates 0 Inflation Rate 2.5 Number of Match -0.05 rho=0.08 rho=0.16 2-0.1-0.15-0.2 1.5 1-0.25 0.5-0.3 0 5 10 15 0 0 5 10 15 0.5 Consumption 0 Interest Rate 0.4-0.05 0.3-0.1 0.2-0.15 0.1-0.2 0 0 5 10 15-0.25 0 5 10 15 36 / 59
Interpretation For a positive productivity shock, the inflation rate decreases. For a larger separation rate, a price is more flexible, i.e., a steeper slope in the curve, and a negative price response is larger. The number of matches increase in response to the shock. A larger separation rate gives a smaller increase in the number of matches since coefficients for the future and past number of matches are smaller in equation (52). An increase of the number of products causes a long term deflation under an economic expansion. For a larger separation rate, price response is larger. Note that a separation rate appears in a Search-based Phillips curve and an equation for the number of matches. Thus, these outcomes are given by combined effect from two equations. An effect from a Search-based Phillips curve, however, is dominant for an inflation rate, consumption, and interest rate since these variables show larger responses for a larger separation rate. 37 / 59
Effect of Matching Probability We change the probability of match in a goods market in a steady state, i.e., q, and assume q to be double at 0.56. 38 / 59
Response of Inflation Rate, Consumption, Number of Match, and Interest Rate to Productivity Shock for Different Matching Probability 0 Inflation Rate 0.1 Number of Match -0.02-0.04-0.06 q=0.28 q=0.56 0.08 0.06 0.04 0.02-0.08 0 5 10 15 0 0 5 10 15 0.12 Consumption 0 Interest Rate 0.1-0.01 0.08-0.02 0.06-0.03 0.04-0.04 0.02-0.05 0 0 5 10 15-0.06 0 5 10 15 39 / 59
Response of Inflation Rate, Consumption, Number of Match, and Interest Rate to Goods Market Shock for Different Matching Probability 0 Inflation Rate 2.5 Number of Match -0.05 q=0.28 q=0.56 2 1.5-0.1 1 0.5-0.15 0 5 10 15 0 0 5 10 15 0.25 Consumption 0 Interest Rate 0.2-0.02 0.15 0.1-0.04-0.06-0.08 0.05-0.1 0 0 5 10 15-0.12 0 5 10 15 40 / 59
Interpretation (1) For the larger probability of a match in a steady state, the probability of a match gives a weaker response to a shock as shown in equation (52) in which a coefficient for the expected probability of a match becomes smaller. This is because a negative response of vacancy to the number of matches in the last period become stronger as the probability of a match in a steady state increases. This reduces fluctuations in the number of matches as shown in equation (48). Eventually, the number of matches increase less since coefficients for the future and past number of matches in equation (52) are smaller. Then, an inflation rate decreases less in response to the shock. Note that the probability of a match gives an effect only on an equation for the number of matches (52). Thus, these outcomes are purely given by the equation. 41 / 59
Interpretation (2) Figure 4 shows responses of the inflation rate, consumption, the number of matches, and the interest rate to a 1 percent goods market shock. For a larger probability of a match, all variables show smaller responses. 42 / 59
A Price Chain New observations for product cycles and prices reveal that we can find a basic mechanism of Search-based Phillips curve in several stages in a product chain. Broda and Weinstein (2010) for whole sales and retail sales, Nakamura and Steinsson (2012) for international trades, Ueda, Watanabe, and Watanabe (2016) for retail sales, and Cavallo, Neiman, and Rigobon (2014) for online markets. These multiple Search-based Phillips curves make aggregate price dynamics. 43 / 59
Assumption for Two Goods Markets (1) A new type of firms C is introduced into a model. There are two markets for goods, a first market between firms A and firms B for goods A and a second market between firms B and firms C for goods B. Structures of two markets are as follows in a time sequence. First, dissolved firms in the last period search for matches for goods transactions in the first market. Second, newly matched firms B in the first market can search for matches for goods transactions in the second market. Only when a match is successful for firms B in the second market, firms B can make goods transactions in two markets. 44 / 59
Assumption for Two Goods Markets (2) Third, a first market opens and firms A and firms B newly matched in the market set price for goods A. When firm B sets a price in the first market, firm B knows the result of price setting in the second market as in Wasmer and Weil (2004) that assume two sequential bargaining in a loan market and a labor market. Forth, a second market opens and firms B and firms C newly matched in the market set price for goods B. When firms set a price in the second market, a price of goods A is as given since price negotiation finished in the first market before the second market opens. 45 / 59
Assumption for Two Goods Markets (3) Fifth, firms A and firms B are dissolved with a separation rate ρ (0, 1). When a firm B is dissolved in the first market, a match between the firm B and the firm C is also dissolved since an interruption of a product chain as shown in Barrot and Sauvagnat (2016) and Carvalho, Nirei, Saito, and Tahbaz-Salehi (2016). They confirm that a disruption of a part of a product chain destroys a whole product chain. Sixth, firms B and firms C are dissolved with a separation rate ρ (0, 1). When a firm B is dissolved in the second market, a match between the firm A and the firm B is also dissolved similarly as in the first market. 46 / 59
Homogeneous Matching/Separation in Two Markets In two markets, transitions of firms matching are given by v t = L (1 ρ)(1 ρ)l t 1, (55) v t = q t v t, (56) L t = (1 ρ)(1 ρ)l t 1 + q t v t. (57) 47 / 59
Firms C There is free entry into a market of goods B. The value of a seeker firm C is zero, and hence the cost of searching must equal the expected revenue, or κ = s t ṡ t W t ( P t B ). (58) The value of firm C W t ( P t B ) is the value of a productive firm C as W t ( P B t ) = Z C t Z B 1 β(1 ρ)(1 ρ) P B t (59) + Z B β(1 ρ)(1 ρ) 1 β(1 ρ)(1 ρ) E t P B t+1 + β(1 ρ)(1 ρ)e t W t+1 ( P B t+1). 48 / 59
Firms B (1) Firms B join in two markets. First, firms B join in the first market of goods A. There is free entry into a market of goods A. A cost of searching must equal the expected revenue is κ = s t s t W t ( P t A, P t B ), (60) where P t A is a newly set price in the first market for goods A at time t. A value function of firms B in the market of goods A is given as W t ( P A t, P B t ) = Z B PB t Z A PA t +β(1 ρ)(1 ρ)e t W t+1 ( P A t, P B t ). (61) When firms B set price of P A t, firm B guesses price setting for P B t. 49 / 59
Firms B (2) After price setting in the first market of goods A, firms B join in a second market of goods B and set a price of P t B given price of P t A. Firms B must post offers, which we call vacancies in the second market of goods B, to search for seeker of firm C. Posting vacancies is costless. A surplus of a firm B from a new match is J t 1 ( P t A, P t B ) J t 0 = Z B PB t Z A PA t Xt B (62) + βe t (1 ρ)(1 ρ) [ J 1 t+1( P A t, P B t ) J 0 t+1 βe t q t+1 q t+1 [ J 1 t+1( P A t+1, P B t+1) J 0 t+1 ]. ] 50 / 59
Firms A A surplus of a firm A from a new match is Jt 1 ( P t A ) Jt 0 = Z A PA t Xt A (63) [ + E t (1 ρ)(1 ρ) Jt+1( 1 P ] t A ) Jt+1 0 E t q t+1 q t+1 [Jt+1( 1 P ] t+1) A Jt+1 0. 51 / 59
Two Goods Markets with Search Frictions The number of new matches in a period is given by a Cobb-Douglas matching functions in two markets m (u t, v t ) = χu 1 α t v α t, χ (0, 1), α (0, 1), (64) 1 α m ( u t, v t ) = χ u t v t α, χ (0, 1), α (0, 1), (65) where χ, χ, α, and α are parameters. Defining supply and demand for goods A and goods B, respectively, in markets as θ t = u t, s t = χθt α, q t = χθt 1 α, (66) v t θ t = u t v t, ṡ t = χ α 1 α θ t, q t = χ θ t. (67) 52 / 59
Price Setting and Aggregate Price (1) A price of goods A is determined according to Nash bargainings between the newly matched firms. Thus, P t A solves max W t ( P t A, P t B ) 1 b (Jt 1 ( P t A ) Jt 0 ) b, (68) P t A where b (0, 1) is the bargaining power for firm A. The first-order condition with respect to P t A yields bw t ( P A t, P B t ) = τ(1 b)(j 1 t ( P A t ) J 0 t ), (69) where τ b. Note that τ is from an assumption that firm B guess price setting in the second market when firm B set a price in the first market. The aggregate price Pt A of goods A is given by L t P A t = (1 ρ)(1 ρ)l t 1 P A t 1 + q t v t PA t. (70) 53 / 59
Price Setting and Aggregate Price (2) For a new price P B t, we have max P B t W t ( P t B ) 1 ḃ( J t 1 ( P t A, P t B ) J t 0 )ḃ, (71) where ḃ (0, 1) is the bargaining power for firm B in the second market. The first-order condition with respect to P B t yields ḃ W t ( P B t ) = (1 ḃ)( J 1 t ( P A t, P B t ) J 0 t ). (72) The aggregate price P B t of goods B is given by L t P B t = (1 ρ)(1 ρ)l t 1 P B t 1 + q t v t PB t. (73) 54 / 59
Two Price Equations: Linearization (1) Two Search-based Phillips Curves are given by πt a = β(1 ρ)(1 ρ)e t πt+1 a (74) b [ ] + πt b β(1 ρ)(1 ρ)e t πt+1 b b + τ(1 b) + γ fq E t q t+1 + x f t, πt b = β(1 ρ)(1 ρ)e t πt+1 b (75) + (1 ḃ) [ πt a β(1 ρ)(1 ρ)e t πt+1 a ] + γ s q E t q t+1 + xt s, where π b t p B t (1 ρ)(1 ρ) p B t 1, (76) π a t p A t (1 ρ)(1 ρ) p A t 1, (77) 55 / 59
Two Price Equations: Linearization (2) γ fq b Z B β q q [1 (1 ρ)(1 ρ)] PB Z A PA 1 α Z A P A [b + τ(1 b)], (78) γ s q ḃ Z C Z B P B β q [1 (1 ρ)(1 ρ)] 1 α Z B P B, (79) xt f (1 b) [1 (1 ρ)(1 ρ)] [1 β(1 ρ)(1 ρ)] X A Z A P X A t A, [b + τ(1 b)] (80) xt s [1 (1 ρ)(1 ρ)] [1 β(1 ρ)(1 ρ)] (81) [ḃ Z C Z B P B Ẑ X t C B ] + (1 ḃ) Z B P X B B t. 56 / 59
Transformation To more closely look at interaction effects between markets, we can combine two Phillips curves as π a t =β(1 ρ)(1 ρ)e t π a t+1 + + 2 b 1 b + ḃ γfq E t q t+1 (82) 1 1 b + ḃ γs q E t q t+1 + 2 b 1 b + ḃ x f t + 1 1 b + ḃ x s t, πt b =β(1 ρ)(1 ρ)e t πt+1 b + 2 b 1 b + ḃ γs q E t q t+1 (83) (1 ḃ)(2 b) + γ fq E t q t+1 + 2 b 1 b + ḃ 1 b + ḃ x t s (1 ḃ)(2 b) + xt f. 1 b + ḃ 57 / 59
Market Interactions Market interactions give new features for price dynamics. First, a market tightness in one market makes a disturbance to a price in another market. A price response to own and another market tightness strengthens and weakens according to bargaining powers of b and ḃ in two markets. Second, a shock in another market also plays an important role to price dynamics since a shock in another market is included in each curve. A price response to a shock in own and another market can increase and decrease according to bargaining powers of b and ḃ. 58 / 59
Points Match a model with micro (macro) data. Estimate parameters with micro (macro) data. Evaluate a performance of Search-based Phillips curve in a rich general equilibrium model. 59 / 59