Monetary Economics. Chapter 5: Properties of Money. Prof. Aleksander Berentsen. University of Basel

Monetary Economics Chapter 5: Properties of Money Prof. Aleksander Berentsen University of Basel Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 5 1 / 40

Structure of this chapter 1 Introduction 2 The model 3 Divisibility of money Currency shortage Divisible money 4 Portability of money 5 Recognizability of money 6 Conclusion Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 5 2 / 40

Introduction In the previous chapters, we introduced at money. Here, we will study how the physical properties of money, such as divisibility, portability and recognizability impact its value and its ability to perform its role as a medium of exchange. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 5 3 / 40

The model DAY (DM) σ bilateral matches are formed. Buyers receive a preference shock ε. Buyers make a take it or leave it offer. NIGHT (CM) Money is traded competitively against the general good at the price φ. Figure 5.1: Timing of a representative period Only buyers hold some units of money Again, we solve the model backward: we rst solve for CM and then for DM. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 5 5 / 40

Divisibility of money The utility of a buyer in a biliateral match is εu(q) : ε is the realization of an idiosyncratic preference shock over the output produced by the seller in the DM, drawn from some cumulative distribution function F (ε). These preference shocks are independent across time and across matches. A high ε means that a buyer in a match has a very high marginal utility for the good produced by the seller in the same match. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 5 7 / 40

Divisibility of money Currency shortage Currency shortage: M < 1. Not all buyers are able to hold a unit of indivisible money. In the money market CM, a fraction M of buyers will end up with one unit of money. The remaining 1 M will hold no money. Buyers will have to be indi erent between holding one unit of money and holding zero unit. Hence, the following must hold for a buyer in the CM: φ + βv 1 = βv 0 (5.1) φ : V 1 : V 0 : price of one unit of money in terms of the general good value of a buyer holding one unit of money in the DM value of a buyer holding no money Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 5 8 / 40

Divisibility of money Currency shortage The value of a buyer without money in the DM solves V 0 = βv 0 = 0 (5.2) For a buyer with one unit of money in the DM: Participation constraint of the seller: c(q) + φ 0, Value function at the beginning of the DM Z V 1 = σ max [εu(q) φ + βv 1, βv 1 ] df (ε) + (1 σ)βv 1 (5.3) Z = σ max [εu(q) φ, 0] df (ε) + βv 1 ) If the trading surplus is positive, i.e. εu(q) φ 0, then the buyer makes an o er. Otherwise, he chooses not to trade. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 5 9 / 40

Divisibility of money Currency shortage φ Let ε R (φ) = denote the threshold for ε below which the uc 1 (φ) buyer chooses not to trade u c 1 (φ) is strictly concave and increasing u c 1 (0) = 0. ε R (φ) φ > 0 : As money becomes more valuable, buyers become more choosy in terms of the trade they are willing to make. Using (5.1), (5.3) can be rewritten as rφ = σ Z ε R (φ) εu c 1 (φ) φ df (ε) (5.4) According to (5.4), the value of money in equilibrium is such that the opportunity cost of holding one unit of money is equal to the expected surplus from a trade in the DM. A steady-state equilibrium of the economy corresponds to a φ solution to (5.4). Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 5 10 / 40

Divisibility of money Currency shortage It is straightforward to prove that there exists a unique φ > 0 that satis es (5.4), with the following properties: φ M φ σ = 0: φ is independant of the quantity of money M. > 0: as σ increases, money becomes more valuable, the quantities increase. φ r < 0. as r increases, the value of money falls, and agents trade less in the DM. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 5 11 / 40

Divisibility of money Currency shortage Social welfare W is the discouted sum of utilites of buyers and sellers: W = σ(1 β) 1 M Z ε R [εu(q ε ) c(q ε )] df (ε) q ε : output traded in a match with idiosyncratic shock ε Here, a change in M has no e ect on the quatity produced - the intensive margin- but it does a ect the number of trade matches - extensive margin.! Money is not neutral This extensive margine result would disappear when money is perfectly divisible. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 5 12 / 40

Divisibility of money Currency shortage In term of the allocations, the planner would choose ε R and q ε such that In our equilibrium: ε R = 0 εu 0 (q ε ) = c 0 (q ε ). ε R > εr = 0. Buyers do not trade in matches where they have a low valuation for the seller s output. When ε = ε R, ε R u(q) c(q) = 0 whereas ε R u(qε R ) c(qε R ) > 0. In this case, agents trade too much, q > qε R. For values of ε su ciently large, agents trade too little, q < q ε. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 5 13 / 40

Divisibility of money Currency shortage q ε * q ε ε R No trade Too much trade Too little trade Figure 5.2: Trade ine ciencies with indivisible money Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 5 14 / 40

Divisibility of money Currency shortage To summarize, in the presence of indivisible money, we have the following ine ciencies: 1 The number of trade matches can be too low if there is a shortage of currency, M < 1, and not all buyers are endowed with money. 2 For low values of ε, buyers do not trade even though it would be socially optimal to do so. 3 For intermediate values of ε, agents trade too much. 4 For high values of ε, agents trade too little. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 5 15 / 40

Divisibility of money Divisible money A take-it-or-leave-it o er by the buyer in the DM is now characterized by the pair (q ε, d ε ) This pair solves max [εu(q) dφ] s.t. c(q) + dφ = 0, and 0 d m. (5.5) q,d The solution is q ε = q ε, d ε = c(q ε ) φ, if c(q ε ) mφ, and q ε = q = c 1 (mφ) and d = m otherwise. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 5 17 / 40

Divisibility of money Divisible money The divisibility of money, just like the use of lotteries when money is indivisible, removes the no-trade and too-much-trade ine ciencies: ε R = 0 and q ε q ε. The divisibility of money does not remove the too-little trade ine ciency. This ine ciency arises because of the cost of holding real balances due to discounting. Moreover, currency shortages cannot occur and the number of trade matches is maximum since it is feasible to endow all buyers with M units of money at the beginning of a period, even if M < 1. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 5 18 / 40

Portability of money Portability describes the ease at which an object can be carried to where it is needed, i.e., into bilateral meetings. Here, portability is the cost of bringing money into the DM. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 5 20 / 40

Portability of money Buyer s choice of money holdings in the CM of period t: n max m0 o φ t m + βvt+1(m) b (5.6) Value of being a buyer in the DM: Vt+1(m) b = κm +σ max u c 1 (φ t+1 d) d 2[0,m] φ t+1 d +φ t+1 m + W b t+1(0), (5.7) q t+1 = c 1 (φ t+1 d) Wt+1 b (m) = φ t+1 m + W t+1 b (0) κm represents the proportional cost from holding m units of money. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 5 21 / 40

Portability of money Solving problem (5.7) at a steady state gives us the following : u 0 (q ss ) c 0 (q ss ) = 1 + r σ + κm σc(q ss ) (5.8) Assuming that c(q) = q, u(q) = q 1 a /(1 a), a < 1, and σ = 1. Then, equation (5.8) can be rewritten as (q ss ) 1 a = (1 + r) q ss + κm. (5.9) The left side is a strictly concave function of q ss while the right side is linear with a positive intercept. Consequently, if κ is below a threshold, then there are two solutions q ss > 0 to the equation (5.9) : qh ss and qss Otherwise, there is no monetary equilibrium. L. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 5 22 / 40

Portability of money q t +1 q t +1 = q t < < < < < < 1 (κm ) 1 a ss q L ss q H q t Figure 5.3: Dynamic equilibria under imperfect portability Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 5 23 / 40

Portability of money When at money is more costly to carry, the output of the search qh good traded in the DM falls: ss κ < 0. Money is no longer neutral. As M increases, qh ss decreases since carrying money involves additional real resources. The comparative statics at the low steady-state monetary equilibrium are opposite to those at the high steady-state monetary equilibrium. There are a continuum of trajectories leading to the low steady-state monetary equilibrium, while there is a unique trajectory the stationary one that leads to the high steady-state monetary equilibrium. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 5 24 / 40

Portability of money If κ < 0 : The medium of exchange can be interpreted as a commodity money, or a real asset since it provides its holder with a real dividend There is a unique monetary equilibrium, and it is the stationary monetary equilibrium, q t = q t+1 = q ss. Hence, there is a bene t to society associated with a commodity money system. In a at monetary system, there is a continuum of equilibria that lead to the autarkic outcome, and in all these equilibria the value of money at any date is lower than what would prevail in a stationary (monetary) equilibrium. As a result, the stationary monetary equilibrium dominates, from a social welfare perspective, any of the in ationary equilibria. Since the presence of a commodity component eliminates any equilibria where money loses value overtime, there is a welfare gain associated with having a commodity money system. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 5 25 / 40

Structure of this chapter 1 Introduction 2 The model 3 Divisibility of money Currency shortage Indivisible money and lotteries Divisible money 4 Portability of money 5 Recognizability of money 6 Conclusion Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 5 26 / 40

Recognizability of money Buyers can produce counterfeit bills at night, after the CM has closed. Seller in the DM are unable to distinguish genuine from counterfeit money. Fixed cost k to engage counterfeiting activities. Marginal cost of producing a counterfeit note is zero. Technology to produce counterfeits becomes obsolete after one period. Counterfeits produced in t of period t. Buyers make take-it-or-leave-it o ers in the DM. 1 are con scated as agents enter the CM Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 5 27 / 40

Recognizability of money In the case where σ = 1, we have the following "reverse-ordered"game: Buyer Buyer Offer (q,d) [η ] [ 1 η ] Honest buyer Seller Counterfeiter Yes No Yes No ] [ 1 π ] [π [π ] [ 1 π ] Figure 5.4: The reverse-ordered game π 2 [0, 1] is the probability that a seller accepts the o er (q, d) and η 2 [0, 1] the probability that a buyer chooses to accumulate genuine money instead of producing counterfeits. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 5 28 / 40

Recognizability of money Given η, the decision of a seller to accept or reject an o er satis es 8 < π : = 1 2 [0, 1] = 0 if c(q) + ηφd 8 < : > 0 = 0 < 0 (5.10) Given π, a buyer is willing to accumulate genuine money if φd + β fσπ [u(q) φd] + φdg k + βσπu(q). (5.11) The left side of (5.11) is the expected discounted utility of a genuine buyer, net of the cost of accumulating d units of genuine money. The right side is the expected discounted utility of a counterfeit buyer, net of the cost of producing counterfeits. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 5 29 / 40

Recognizability of money The decision rule for the buyer to accumulate genuine real balances, η, is given by 8 < η : = 1 2 [0, 1] = 0 if [1 β(1 σπ)] φd 8 < : < = > k. (5.12) There are two costs associated with accumulating genuine money instead of counterfeits: the cost from holding money, (1 β)φd, the expected value of transferring genuine real balances if a trade match occurs in the DM, βσπφd. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 5 30 / 40

Recognizability of money A Nash equilibrium is a pair (π, η) 2 [0, 1] 2 that satis es (5.10) and (5.12). If (π, η) = (1, 1), then: c(q) φd k 1 β(1 σ). (5.13) The transfer of real balances must be su ciently large to compensate the seller for his disutility of work, but not so high as to give buyers an incentives to produce counterfeits. If (π, η) 2 (0, 1) 2, then both the buyer and seller are indi erent between their respective choices, and, from (5.10) and (5.12), we have π = η = c(q) φd (5.14) k (1 β)φd. (5.15) βσφd Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 5 31 / 40

Recognizability of money There are also other equilibria: where π = 1, and some buyers produce counterfeits. In this case: [1 β(1 σ)] φd = k and c(q) + φd 0. where, π 2 (0, 1), even though η = 1. This is the case if: c(q) = φd and φd! The di erent equilibria are reprensented in (5.5). k 1 β(1 σ). Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 5 32 / 40

Recognizability of money k 1 β d d c(q) (0,1) η (0,1) k 1 β (1 σ ) π = 1 (, ) {1} (0,1) η = 1 (, ) (0,1) {1} Figure 5.5: Nash equilibria of the subgame following an o er (q, d) Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 5 33 / 40

Recognizability of money By construction, the equilibrium o er will be such that (i) π = 1 and η = 1, and (ii) (q, d) satis es (q, d) = arg max f (1 β) φd + βσ [u(q) φd]g (5.16) subject to c(q) + φd 0 (5.17) k and φd 1 β(1 σ). (5.18) The solution to (5.16)-(5.18) has a unique equilibrium o er (q, d) characterized by π(q, d) = 1 and η(q, d) = 1. Constraint (5.17) ensures that the seller will accept the o er with probability one, Constraint (5.18) ensures that the buyer has no incentive to produce counterfeit notes. A property of this equilibrium is that no counterfeitung ever takes place. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 5 34 / 40

Recognizability of money The determination of the equilibrium level of search good production, q, is illustrated by: r 1 + σ u'( q) c'( q) 1 k c 1 β (1 σ ) Figure 5.6: Determination of the equilibrium Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 5 35 / 40

Recognizability of money When constraint (5.18) is not binding or equivalently if k > k the equilibrium q is given by the intersection of the horizontal line representing the cost of holding money, 1 + r σ, and a downward sloping curve representing the function u0 (q). In this case, we have a c 0 (q) monetary equilibrium. When constraint (5.18) is binding i.e., k < k the equilibrium level q is given by the intersection of the horizontal line representing the cost of holding money, 1 + r σ, and the vertical line emanating from q = c 1 (k/ (1 β (1 σ))). When the threat of counterfeiting is binding, a reduction in the trading frictions will actually reduce the level of output and the value of money. This result is in contrast to the one we had for an economy where there is no recognizability problem. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 5 36 / 40

Recognizability of money An increase in k shifts the vertical line to the right, resulting in a higher production level of the search good; as a result, money becomes more valuable. policies designed to make it harder to counterfeit at money e.g., the use of special paper and ink, the frequent redesign of the currency and so on can have real e ects even when counterfeiting does not take place. Although an increase in σ shifts the horizontal line 1 + r σ down, it also shifts the vertical line at c 1 (k/ (1 β (1 σ))) to the left, the net result being a decrease in q. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 5 37 / 40

Recognizability of money When there is a recognizability problem, as the trading frictions are reduced: a potential counterfeiter has a higher chance to passing a counterfeit; the surplus of a counterfeiter, u(q), is greater than the surplus of a genuine buyer, u(q) c(q); this reduction in trading frictions increases the buyer s incentive to produce counterfeits and, hence, the no-counterfeiting constraint becomes tighter. On the other hand, if the economy does not have a recognizability problem, as trading frictions are reduced: there is a higher chance that a buyer and seller will be matched, which reduces the cost of holding money. As a result, both the output of the search good and the value of money will increase. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 5 38 / 40

Conclusion When money is indivisible and scarce, the number of trades is too low. Moreover, it distorts the quantities traded. Lotteries eliminate the no-trade and too-much trade ine ciencies but do not overcome the technological impossibility to endow all buyers with money. Indivisible money is not neutral. In terms of portability, if the cost of carrying an asset is higner than a threshold, this asset will not be used as money. Under our assumptions, we showed that the lack of recognizability manifests itself in an upper bound on the quantity of real balances that a buyer can transfer in a match. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 5 40 / 40