1 Chapter 4 Money in Equilibrium

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1 1 Chapter 4 Money in Euilibrium 1.1 A Model of Divisible Money The environment is similar to chapter 3.2. The main difference is that now they assume the fiat money is divisible. In addtition, in this chapter, they do not restrict attention to stationary euilibria. As before, the model is solved in four steps 1. Characterize key properties of value functions in CM 2. Determine the terms of trade in match in DM 3. Characterize the Bellman euations in DM 4. Determine agents choice of money holdings in CM Firstly consider the value function for a buyer at CM W b t m) = max x y + βv b m m )} >0,x,y s.t x + m m ) = y or substituting the budget constraint, we have Wt b m) = m + max m + βv b m m )} >0 Similarly for the seller, we have Wt s m) = m + max t m + βv s m m )} >0 Secondly the terms of trade in DM are determined in a take-or-leave-it offer by buyers, d) that maximizes buyer s expected utility subject to seller s participation constraint. Buyer with money m s problem is max u) + W t b m d),d) s.t c) + W s t m s + d) W s t m s ) m s d m using the linearity in money holding of value function, we have max u) d,d m s.t c) + d 0 Note that the seller s participation constraint must hold, therefore, the solution is 1

2 = d = c) if m t c ) c 1 m t ) if m t < c ) The buyer can obtain socially effi cient uantitiy, if his real balances, m t are large enough to compensate seller for the disutility to produce. Thirdly The buyer s DM value function is V b t m) = σ u) + W b t m d) Now using the linearity, and bargaining solution, we have ] ) + 1 σ)wt b m Vt b m) = σ max u c 1 d) ] d } + m + W b 0) d 0,m] Thus, use this into CM value function, we have Wt b m) = m + max m >0 = m + max m >0 m + βv b m + β = m + βw b 0) + max m >0 m )} σ max d 0,m ] u c 1 +1 d )] +1 d } ]} + +1 m + W0) b β+1 ) m + β σ max u c 1 +1 d )] +1 d }]} d 0,m ] Now the buyer s problem can be reduced tp ) t max 1 m>0 β+1 m + σ max u c 1 +1 d ) +1 d ]} d 0,m] If If < β, then there is no solution since agent would demand infinite money balances. = β, then money is costless to hold. Thus buyer carries suffi cient balances in order to purchase.thus d = c ) and any m t c ) +1 is a solution If > β, then money is costly to hold. Thus buyer do not carry extra money balances into CM, i.e. d = m. Thus the F.O.C is u c 1 +1 m ) β c c 1 +1 m ) = σ Similarly, seller s DM value function is, using P.C. V s m s ) = m s + W s 0) Thus seller s choice of money balances problem in the CM is ) } t max 1 m>0 β+1 m β 1) 2

3 If If If +1 < β, then there is no solution +1 = β, seller is indifferent between holding money or not, m 0 +1 > β, then money is costly to hold, thus m = 0 Now the aggregate money demand correspondence is the sum of the individual money demands across buyers and sellers. ) c ) M d, ) = if t = β m}, where m solves 1) if > β +1 Thus the market clear reuires that the money supply M M d ) If M c ) +1, +1 = β If M < c ) +1, solve the differential euation 1) with m = M. Note the difference euation 1) can be written as u = β+1 1 c 1 + σ m ) ] + } c c 1 +1 m ) 1 2) The price of money is eual ] to its discounted price in next period plus a liuiedity factor + u β+1 σ c 1 +1 m) c c 1 +1 m) 1, which captures the marginal benefit of holding real balances in DM. If money is costly to hold, then buyers don t bring enough real balances in the DM to purchase, since M <. c ). Therefore, the liuidity factor is positive since buyer would value having an additional unit of money to spend in DM. ) If money is costless to hold, buyers accumulate suffi cient balances in CM M. c ) to purchase, thus buyer would not value such addtion unit of money to spend in DM. Definition 1 The euilibrium of an economy with divisible money is a seuence } sloves the First-Order diff erence euation in 2). Note that the value of money is bounded from above. Note that this implies Vt b m) = σ max u c 1 d) ] d } + m + W b 0) d 0,m] V s m) = m + W s 0) V b t m) + V s m) M On the other hand, given that the surplus of a match is bounded above by u ) c ), thus we have 3

4 Thus the feasibility reuires that Steady-State Euilibria V b t m) + V s m) σ u ) c )) 1 β M σ u ) c )) 1 β Consider the S.S in which = +1 = SS. 1 Now consider stationary euilibria in which t = = SS > 0. In addition, by the bargaining problem, we have Thus the 2) becomes LHS is decreasing in SS and u 0) c 0) = SS = min c 1 SS M ), ] Thus there is a uniue SS satisfies 3) and SS = css ) M The output is ineffi ciently low when r > 0, i.e. SS < SS r u SS ) c SS ) = r σ + 1 3) < 0 and SS σ > 0. Note that r σ is a measure of cost of holding real balances: product of rate at which agents depreciate future utility r), and average number of period it takes to get matched 1 σ ). Thus cost of holding increase, the DM output falls One-time change in M does not affect SS, the aggregate real balances, SS M is also constant since eual to c SS )). But the change in price level, 1, is proportional to the change in M Nonstationary Euilibria Now consider the following functional forms: c) = u) = 1 a 1 a with a < 1 For this specification, we have = 1. The difference euation is = β 1 σ) +1 + σ +1 ) 1 a M ) a } if +1 M < 1 β+1 if +1 M 1 The following figure shows the phase diagram for difference euation 1 There always exists the non-monetary euilibruim since money has no intrinsic value. 4

5 Figure 1: Phase diagram for u) = 1 a 1 a, c) = If 0 < SS, then 0 as t. Thus we have a positive inflation though the money supply is constant 1 t increase ), even If 0 = SS, the nthe euilibrium is stationary If 0 > SS, } diverges and thus the feasibility is violated, thus the euilibrium does not exist. In the book, they also consider another example for different specifications as well as sunspot euilibruim. 1.2 Alternative Trading Protocols In this section, they propose a number of different trading protocols for the DM, including alternative bargaining solutions, a Walrasian protocol, and a price-posting protocal Alternative Bargaining Solutions Bargaining Set Consider a match between buyer with m and seller holding zero money at beginning of DM. The agreement is, d) transferred by buyer to seller If an agreement is reached, then buyer s and seller s utility are If no agreement, then the utilies are u b = u) + W b m d) u s = c) + W S d) u b 0 = W b m) u s 0 = W S 0) Now since value functions are linear in money, we have 5

6 u b = u) d + u b 0 u s = c) + d + u s 0 therefore, the total surplus is ) u b u b 0 + u s u s 0) = u) c) To illustrate the role that money plays in exchange, suppose buyer cannot spend more than τ m units of money. Denote S τ) represent the set of feasible utility levels for the buyer and seller when buyer cannot spend more than τ S τ) = ) u) d + u b 0, c) + d + u s 0 The euation for the Pareto frontier of S is derived from } : d 0, τ] and 0 u b = max,d u) d] + ub 0 s.t c) + d u s u s 0 d τ the solution can be summarized as ), d) =, c )+u s u s 0 if τ c ) + u s u s 0 c 1 τ u s + u s 0 ), τ) if τ < c ) + u s u s 0 Thus the euation for the Pareto frontier is u s u s u 0 = ) c ) u b u b ) 0 if τ c ) + u s u s 0 τ c u 1 u b u b 0 + τ)) if τ < c ) + u s u s 0 If τ is suffi cient to purchase, the total surplus is split according to u b u b 0) and u s u s 0 ) If τ is insuffi cient to purchase, then buyer will spend all τ and < Figure 2: Bargaining Set Figure 2 displays that for τ 3 > τ 2 > τ 1, the bargaining set is larger, thus fiat money alows trader to achieve utility and output levels that otherwise would not be attainable. 6

7 Nash Solution The Nsh solution to the bargaining problem is based on three axioms: Pareto effi ciency, invariance to rescaling of agents payoffs, and independentce to irrelevant alternatives. These three axioms imply that the solution maximizes the weighted geometric average of buyer s and seller s surpluses from trade. if d < m, the solution is m), d m)] = arg max,d m u) d]θ c) + d] 1 θ = d = 1 θ) u ) + θc ) m If m < m then d = m contrait binds), we have The solution is m) = arg max θ ln u) m] + 1 θ) ln c) + m]} m = z θ ) = 1 θ) c )u) + θu )c) θu ) + 1 θ) c ) 4) The outcome m), d m)] is independent of seller s money balances due to seller s linear value function. Since money is costly to hol, and seller s money holding does not affect terms of trade, the seller will not accumulate money in CM to bring into DM. Now restrict to the S.S. The buyer s value function is V b t m) = σ using the linearity we have Now the CM value function is u) + W b t m d) ] + ) 1 σ)wt b m V b t m) = σ u) d] + m + W b t 0) Wt b m) = m + max m + βv b m m )} >0 )} = m + max m + β σ u) d] + m + W b m t 0) >0 = m + βwt b 0) + max 1 β)m + β σ u) d]) } m >0 Therefore the money holding problem for buyer is using 4) we have max rm + σ u) d]} m>0 max rz θ) + σ u) z θ )]} 5) 0, ] 7

8 Now assuming interior solution, the first order condition we have u ) z θ ) = 1 + r σ if θ = 1, z θ ) = c), the solutions is as before However, as r 0, < if θ < 1 Rewrite z θ ) = 1 Θ)) u) + Θ)c), where Θ) = Θ ) = θ and Θ ) < 0. Thus we have θu ) θu )+1 θ)c ) z θ ) = u ) Θ ) u ) c )) > u ) Thus as, buyer s surplus u) z θ ) decreases. There are twoo effects associated with an increase in < Total match surplus u ) c ) increases Buyer s share of surplus decreases For close to, the second effect dominates the first. So in addition to the monetary ineffi ciency created by discounting, there is an ineffi ciency from Nash bargaining. Due to the conseuence of the fact the buyer s surplus is not always increasing in his real balances If θ = 0, z θ ) = u), then the solution to 5) is = 0. Since money is costly to hold and buyer receives no surplus from purchasing DM good, as a result, the trade does not occur. Necessary condition for trade to take place is θ > 0 The Proportional Solution Kalai) In contrast to the Nash, the proportional bargaining solution reuires that agents surpluses increases as bargaining set expands, which implies that solution is monotonic. Each player receives a constant share of match surplus or we have u) d = θ u) c)] c) + d = 1 θ) u) c)] In our setting the problem is u b u b 0 = θ 1 θ us u s 0) 8

9 From constraint, we have, d) = arg max u) d],d m s.t u) d = θ c) + d) 1 θ thus the problem can be reduced to s.t d = 1 θ) u) + θc) = arg max θ u) c)] 1 θ) u) + θc) m If constraint binds, then m = z θ ) = 1 θ) u) + θc) 6) note this expression is similar to that in Nash except that the buyer s share in Nash is Θ) which is a function of, but for kalai, it is constant. The constraint always binds as long as r > 0 Similarly, the buyer s money holding problem is now using 6) we have max rm + σ u) d]} m>0 max rz θ) + σθ u) c)]} 0, ] = max σθ r 1 θ)] u) r + σ) θc)} 0, ] A necessary condition for the buyer s CM problem to admit a positive solution is that σθ r 1 θ) > 0, thus the buyers must have enought bargaining power is money is to be valued. The F.O.C is u ) c ) z θ ) = r θσ LHS is marginal increase of match surplus generated by an increase of buyer s real balances RHS is a monetary wedge introduced by discounting, search frictions and bargaining power. An increase in seller s bargaining power θ, raises the wedge through a holdup problem. Buyer underinvests in real balances since holding money incurs cost r σ but only receive a fraction θ of total surplus. The resulting is increasing with θ As r 0,. This contracts the result in Nash. With proportional bargaining, the buyer s surplus is strictly increasing in his real balances until the is reached. Thus if the money holding cost is zero, then buyer will accumulate suffi cient money to purchase the effi cient level of search good 7) 9

10 1.2.2 Walrasian Price Taking Since the notion of competitive markets is pervasive in economics, they accomodate such a trading protocol in the DM by assuming that buyers and sellers meet in large groups during day in a competitive market and they are anonymous. Since agents are anonymous at day, they are unable to use credit arrangements. They reinterpret the idiosyncratic matching shocks, σ, as preference and productivity shocks. In particular, a fraction σ of buyers seller) want to consume produce) at day, and the rest do not. Denote p as the price of day good in terms of night good, i.e., if ˆp is the dollar price for a unit of DM output and is amount of CM goods can be purchased with a dollar, then p = ˆp. The problem for an active seller who wants to sell in the DM is F.O.C reads as s = arg max c) + p) p = c s ) The problem that buyer faces in CM is how much money to bring into DM and makes choice before he learns if he is active in CM F.O.C reads as b = arg max rp + σ u) p]) u b) = 1 + r ) p σ r σ is the monetary wedge between buyer s marginal utility of consumption and price of good. This wedge arises because buyer must accumulate real balances the period before. In addition, there is always a risk that buyer will not needthe real balances at all if he receives a negative preference shock Since the measure of active buyers and sellers are eual, the market clear condition reuires that s = b =, thus we have u ) c ) = 1 + r σ whish delievers the identical result under bargaining protocol with buyer has all bargaining power. If r 0, The value of money is or using F.O.C from seller we have p = M = c ) M Remember that when buyer offers a take-or-leave-it offer, = c) M. Thus if c) is strictly convex, the value of money is larger e.g. c ) > c) when c) = 2 ) 10

11 When buyer makes offer, DM goods are priced according to average cost In Walrasian pricing, DM goods are priced according to marginal cost Competitive Price Posting In this section, they use the concept of competitive search. In particular, they assume that the economy is composed of different submarkets in DM, where submarket is identified by its terms of trade, d). Terms of trade for DM good in period t are posted by sellers at the beginning of the previous night t 1. Seller can commit to their posted prices Buyers in CM observe all of the terms of trade in all submarkets, and decide which particular submarket they will visit in the subseuent DM, and amount of real balances bring into DM. Submarkets are not frictionless. In each submarket, buyers and sellers face the risk of being unmatched. Matching process: suppose there is a measure of B buyers and S sellers in a submarket, d). The measure of matches is σ min B, S} Denote n = B S as the length of the uene. Conseuently, the matching rate of a buyer is min B, S} σ = σ min 1, 1 } B n the matching rate of a seller is σ min n, 1} When sellers post his terms of trade at beginning of the night, he takes the utility that buyers expect tot receive as given. In particular, the expected surplus of a buyer in the DM net of cost of holding money is U b = rd + σ min 1, 1 } u) d) 8) n A seller s choice of his terms of trade,, d), determines the length of the ueue n in the submarket. The length of the ueue is such that buyers are indifferent between going to each submarket with U b. The seller s posting problem is max,d,n σ min n, 1} c) + d] 9) 1, 1 } u) d) n s.t. U b = rd + σ min Now to determine U b. Let Ū b represent the upper bound of buyer s expected utility in the euilibrium. This upper bound is attained if buyer receives the entire match surplus u) c)) and if his matching probability is at its maximum value σ. 11

12 In this case, buyer will only bring enough cash to compensate seller s production cost, thus Ū b = max rd + σ u) d]} s.t c) + d = 0 or we have Ū b = max rd + σ u) c)]} U b > Ū b, the seller have no incentive to make markets, thus this is inconsistent with an euilibrium U b = Ū b, then buyer s surplus is at its maximum value. Any solution to 9) implies that buyer will have entire surplus from trade, thus d = c), and the buyers are on the short side of the market, n 1. U b 0, Ū b), then u ) d > 0. n > 1 is not an euilibruim. Since if it were, seller could slightly increase d such that n decreases but still greater than 1 by 8), hence seller s utility increase, which is a contradiction. Thus n 1. Then 8) becomes and using this 9) becomes The F.O.C w.r.t is max,n = max,n U b = rd + σ u) d) d = σu) U b r + σ c) σn + σu) U b ] r + σ ] rc) + σ u) c)] U b σn r + σ u ) c ) = 1 + r σ This coincides with F.O.C for Walrasian and Buyer s offer cases Finally, since U b 0, Ū b), rc)+σu) c)] U b r+σ > 0, thus n = 1 U b = 0. In this case, a buyer is indifferent between participating or not in DM. If buyer participates, then solution to 9) delivers n = 1 u ) c ) = 1 + r σ and the value of transfer d is adjusted so that 8) is holding with U b = 0 12

13 For buyers who do not participate, and thus enter an inactive submarket in which d = = 0 and thus n = The euilibrium value of U b is then determined such that the ratio of buyers per seller across submarkets is consistent with measures of buyers and sellers in the economy. Suppose that there is a unit measure of sellers and N measure of buyers where N > 0 Denote aggregate demand for active buyers as N d and aggregate supply of active buyers N S by N d = nj)dj = N S N n 0 nj) is measure of buyers per seller in submarket of seller j n 0 is measure of buyers who do not participate. Now for each cases discussed above If U b = Ū b, nj) 0, 1] and nj)dj 0, 1] If U b 0, Ū b), nj) = 1 j, and nj)dj = 1 If U b = 0, n 0 0, N] Therefore we have three cases to consider in the Euilibruim If N > 1, then U b = 0 In any euilibrium, nj) = 1 for all sellers, and a measure N 1 of buyers go to inactive market and gaining zero utility, and a unit measure will allocate themselves one-for-one with sellers. The unit measure buyers who are active receive zero utility Seller s posted price is solves If N < 1, then U b = Ū b In any euilibrium, nj) 1 for all sellers. 0 = rd + σ u) d) The seller s posted contract, d) is the one maximizes the expected surplus of the buyer and thus seller gains zero utility, thus d = c). This outcome is identical to the outcome under bargaining protocol with θ = 1. Since buyer is on the short side of the market and have all market power. If N = 1, then U b 0, Ū b]. In any euilibrium, nj) = 1 for all sellers and is determined by and d c), σu) r+σ ] u ) c ) = 1 + r σ The steady state value of money is indeterminate since market value of buyer U b is indeterminate 13