Monetary Economics. Chapter 6: Monetary Policy, the Friedman rule, and the cost of in ation. Prof. Aleksander Berentsen. University of Basel

Monetary Economics Chapter 6: Monetary Policy, the Friedman rule, and the cost of in ation Prof. Aleksander Berentsen University of Basel Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 1 / 85

Structure of this chapter 1 Introduction 2 The Friedman rule Optimality Trading frictions 3 Distributional e ects of monetary policy 4 The hot-potato e ect of in ation 5 The welfare cost of in ation 6 Conclusion Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 2 / 85

Introduction Fiat money is useful in environments where credit arrangements are not feasible. The existence of at money allows the implementation of socially desirable allocations that otherwise could not be achieved. Thus far, the supply of money was constant. By changing the rate of growth of money supply, the monetary authority is able to a ect the rate of return of currency and, hence, agents incentives to hold real balances. This, in turn, has implications for equilibrium allocations and society s welfare. Monetary policy takes here the simple form of a constant money growth rate. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 3 / 85

The Friedman rule The Friedman rule Assuming that the marginal cost of creating money is zero, the Friedman rule advocates to set the nominal interest rate at zero. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 5 / 85

The Friedman rule Optimality M t is aggregate stock of money at the beginning of period t. γ M t+1 /M t is the gross growth rate of the money supply. Money is injected, or withdrawn, in a lump-sum fashion in the competitive market at night, CM. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 6 / 85

The Friedman rule Optimality Because agents have quasi-linear preferences in the CM preferences which eliminate wealth e ects we can assume without loss of generality that only buyers receive the monetary transfers. We focus on steady-state equilibria, where the real value of the money supply is constant over time, i.e., φ t M t = φ t+1 M t+1. Note that the gross rate of return on currency is φ t+1 /φ t = M t /M t+1 = γ 1. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 7 / 85

The Friedman rule Optimality Period t Period t+ 1 Transfers Transfers NIGHT (CM) DAY (DM) NIGHT (CM) Mt M M t+ 1 t+ 2 Agent s real balances: z m = φ γ 1 z = φ m t t + 1 Figure 6.1: Timing of a representative period Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 8 / 85

The Friedman rule Optimality Value function of the buyer at the beginning of the CM is n W b (z) = max x y + βv b z 0o (6.1) x,y,z 0 subject to z : d : T : x + φ t m 0 = y + z + T (6.2) z 0 = φ t+1 m 0 = φ t m0 γ buyer s real balances, i.e., z = φ t m transfer from the buyer to the seller in the DM real value of the lump-sum transfer (6.3) Or, alternatively: W b (z) = z + T + max z 0 0 n γz 0 + βv b z 0o. (6.4) Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 9 / 85

The Friedman rule Optimality Value function of the buyer at the beginning of the DM is n o V b (z) = σ u [q (z)] + W b [z d (z)] + (1 σ) W b (z) = σ fu [q (z)] d(z)g + W b (z), (6.5) The buyer s problem can be simpli ed by substituting V b (z) from (6.5) into (6.4), i.e., max f iz + σ fu [q(z)] d(z)gg, (6.6) z0 where 1 + i = (1 + r)γ and i is the nominal rate of interest. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 10 / 85

The Friedman rule Optimality Because it is costly to hold money: d = z. Since the quantity traded in a match satis es c(q) = z, the buyer s problem (6.6) can be rewritten as a choice of q, i.e., max f ic(q) + σ [u(q) c(q)]g. (6.7) q2[0,q ] The rst-order (necessary and su cient) condition to the buyer s problem (6.7) is simply u 0 (q) c 0 (q) = 1 + i σ. (6.8) Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 11 / 85

The Friedman rule Optimality The steady state solution q ss to (6.8) is depicted in Figure 6.2. u'(0) c'(0) u'( q) c'( q) i 1+ σ 1 ss q q* Figure 6.2: Stationary monetary equilibrium under a constant money growth rate Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 12 / 85

The Friedman rule Optimality The optimal monetary policy requires i = 0, or, equivalently, γ = 1 1+r < 1. This is the so-called Friedman rule. When buyers have all of the bargaining power, the allocation of the monetary equilibrium under the Friedman rule coincides with the socially e cient allocation of the search good, q = q. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 13 / 85

The Friedman rule Optimality The Friedman rule is the optimal monetary policy under many trading mechanisms. If terms of trade are determinied buy a Walrasian pricing mechanism or a competitive posting mechanism in the DM, the Friedman rule implements the e cient allocation, q. But it does not always achieve the e cient allocation. Under Nash bargaining, the buyer will choose an ine ciently low value for q. This ine ciency is due to a non-monotonicity property of the Nash solution Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 14 / 85

The Friedman rule Optimality If the buyer obtains the marginal social return of his real balances, as is the case under buyers-take-all or competitive price posting, then the Friedman rule implements the e cient allocation. And even if this condition does not hold, the Friedman rule can achieve the socially e cient allocation provided that the buyer s surplus from a trade increases with the total surplus of a match. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 15 / 85

The Friedman rule Optimality In many monetary environments, the Friedman rule ist the optimal policy. Why is it rarely implemented in practice? The government may lack the enforcement power required to implement the lump-sum tax needed to generate a de ation. The Friedman rule may not be needed if the mechanism that determines the terms of trade during the DM is appropriately designed. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 16 / 85

The Friedman rule Optimality In the following two models, we show that in environments with search externalities, or heterogenous agents, a deviation from the Friedman rule (i.e. some in ation) can be optimal. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 17 / 85

The Friedman rule Trading frictions We assume that there is a unit measure of ex ante identical agents that can choose to be either buyers or sellers in the DM. The decision to become a buyer or seller in period t is taken at the beginning of the previous CM, in period t 1. DAY (DM) NIGHT (CM) n buyers and 1 n sellers Choice of being are matched bilaterally buyers or sellers and at random in the next day Choice of real balances Figure 6.3: Timing of the representative period Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 18 / 85

The Friedman rule Trading frictions Let n denote the fraction of buyers in the DM and 1 of sellers. n the fraction A buyer meets a seller with probability 1 buyer with probability n. n, and a seller meets a Therefore, the number of matches in the DM is n(1 maximized when n = 1 2. n), and it is Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 19 / 85

The Friedman rule Trading frictions The value function at the beginning of the CM is analogous to (6.4), and satis es W j (z) = T + z + max γz 0 + βv j (z 0 ), (6.9) z 0 0 where j 2 fb, sg. The value of being a buyer in the DM satis es h i V b (z) = (1 n) fu [q(z)] zg + max W b (z), W s (z). (6.10) Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 20 / 85

The Friedman rule Trading frictions Substituting (6.10) into (6.9), and using the linearity of W b (z) and W s (z), the value of a buyer with z units of real balances at the beginning of the CM must satisfy W b (z) = T + z + max β f iz(q) + (1 n)[u(q) z(q)]g q2[0,q ] h i +β max W b (0), W s (0). (6.11) From (6.11), the buyer chooses the quantity to trade in the next DM, taking as given his matching probability, 1 n. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 21 / 85

The Friedman rule Trading frictions By similar reasoning, the value of being a seller with z units of real balances satis es h i W s (z) = T + z + βn[z(q) c(q)] + β max W b (0), W s (0). (6.12) Equation (6.12) embodies the result that sellers do not carry money balances into the DM since they do not need them and that the quantity traded q, or equivalently the buyers real balances, is taken as given. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 22 / 85

The Friedman rule Trading frictions Since both W b (z) and W s (z) are linear in z, the choice of being a buyer or a seller does not depend on z. In a monetary equilibrium, agents must be indi erent between being a seller or a buyer, otherwise there will be no trade and at money will not be valued. Consequently, we focus on active monetary equilibria where n 2 (0, 1) and W b (z) = W s (z). Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 23 / 85

The Friedman rule Trading frictions From (6.11) and (6.12), n must satisfy n[z(q) c(q)] = (1 n) [u(q) z(q)] iz(q). (6.13) The left side of (6.13) is the seller s expected surplus in the DM The right side is the buyer s expected surplus, minus the cost of holding real balances. In any monetary equilibrium: n = u(q) (1 + i)z(q). (6.14) u(q) c(q) Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 24 / 85

The Friedman rule Trading frictions From (6.11), q solves max f iz(q) + (1 n)[u(q) z(q)]g. (6.15) q2[0,q ] Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 25 / 85

The Friedman rule Trading frictions De nition A steady-state monetary equilibrium is a pair (q, n) such that q > 0 is a solution to (6.15) and n 2 (0, 1) satis es (6.14). Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 26 / 85

The Friedman rule Trading frictions In ation has a direct e ect on the equilibrium allocation by raising the cost of holding real balances and, therefore, by reducing q. If the pricing mechanism delivers q = q under the Friedman rule, then n decreases with in ation since dn z(q di = ) i=0 u(q ) c(q ) < 0. Since in ation is a direct tax on agents who hold money, as in ation increases agents have less incentives to be buyers. As a result the matching probability of buyers, 1 n, increases with in ation (close to the Friedman rule). So even though there are fewer buyers, they spend their money balances in the DM faster. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 27 / 85

The Friedman rule Trading frictions We measure social welfare by the sum of all trade surpluses in a period, i.e., W = n(1 n)[u(q) c(q)]. Welfare is maximized when the surplus of each match is maximized which requires q = q and when the number of matches in the DM is maximized which requires n = 1/2. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 28 / 85

The Friedman rule Trading frictions Suppose that the trading mechanism in the DM implements q at the Friedman rule. The rst condition for e ciency requires that the Friedman rule is implemented. From (6.14), the second condition for e ciency requires that u (q ) z (q ) u (q ) c (q ) = 1 2 (6.16) Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 29 / 85

The Friedman rule Trading frictions Equation (6.16) turns out to be a restatement of the so-called Hosios condition for e ciency in models with search externalities. Search externalities arise when agents decisions to participate in a market a ect the trading probabilities of other agents in the market. The buyer s contribution in the creation of matches in the DM must be rewarded by giving buyers a share in the match surplus that is equal to the fraction of matches that buyers are responsible for. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 30 / 85

The Friedman rule Trading frictions The welfare e ect of a change in i in the neighborhood of i = 0 is dw di = u0 (q) [u 0 (q) c 0 (q)] n 2 i=0 [u 00 (q) z 00 + (2n 1) z(q). (6.17) (q)] 1 n Assuming that q = q at the Friedman rule which is a valid assumption under proportional bargaining we can evaluate the welfare implications of a deviation from the Friedman rule by evaluating the second term in (6.17), (2n 1) z(q). If n 6= 1/2, then a deviation from the Friedman rule may actually be welfare improving. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 31 / 85

The Friedman rule Trading frictions A deviation will be optimal, i.e., d W di i=0 > 0, if and only if u(q ) z(q ) u(q ) c(q ) = n > 1/2. When the buyer s share of match surplus is greater than one-half, there are too many buyers from a social perspective. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 32 / 85

The Friedman rule Trading frictions A deviation from the Friedman rule will be optimal whenever θ 2 (0.5, 1): the policy maker is willing to trade o e ciency on the intensive margin (the quantities traded in each match) in order to improve the extensive margin (the number of trade matches in the DM) by raising in ation, which increases the number of sellers and reduces the number of buyers. If, on the other hand, θ 2 (0, 0.5), there will be too many sellers in equilibrium at the Friedman rule: the Friedman rule is the optimal policy since a deviation would only further increase the number of sellers in the economy. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 33 / 85

Distributional e ects of monetary policy In ation can have a positive e ect when the distribution of money balances across agents is not degenerate. Indeed, a positive in ation, which is engineered by lump-sum money injections, redistributes wealth from the richest to the poorest agents. If some agents are poor because of uninsurable idiosyncratic shocks, this redistribution can raise social welfare. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 35 / 85

Distributional e ects of monetary policy To capture a distributional e ect of monetary policy, we modify the benchmark model as follows: Buyers and sellers live for only three subperiods: they are born at the beginning of the night and die at the end of the following period. Agents can, therefore, potentially trade three times. For simplicity, assume that agents do not discount across periods, i.e., r = 0. Generation t Productivity shocks Transfers Competitive markets Bilateral trades Generation t+1 Figure 6.4: Overlapping generations Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 36 / 85

Distributional e ects of monetary policy The utility function of buyers is x y + u(q) + x o, x y 2 R is the utility of consumption net of the disutility of production in the CM when young, x o is the net utility of consumption in the CM when old, u(q) is the utility of consumption in the DM. Similarly, the utility function of sellers is x y c(q) + x o. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 37 / 85

Distributional e ects of monetary policy We assume that newly-born buyers di er in terms of their productivity in the rst period of their lives. a fraction ρ 2 (0, 1) of newly-born buyers are productive, while the remaining fraction is unproductive. as a result, the productive buyers can participate in the CM to accumulate money balances, while unproductive buyers cannot. The government is not able to make di erentiated transfers to productive and unproductive buyers. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 38 / 85

Distributional e ects of monetary policy The problem of a productive newly-born buyer, which is similar to (6.6), is max φt m + σ u[q(φ m0 t+1 m)] c[q(φ t+1 m)] + φ t+1 m. (6.18) The buyer who has access to the CM when he is born produces φ t m units of the general good in exchange for m units of money. If he doesn t meet a seller in the next DM, then he spends his money balances in the CM before he dies. If he does meet a seller, we assume that the buyer captures the entire surplus from the match. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 39 / 85

Distributional e ects of monetary policy Denote z = φ t+1 m as the choice of real balances for the next DM. The productive buyer s problem (6.18) can then be simpli ed to read max f (γ 1)z + σ fu[q(z)] c[q(z)]gg. (6.19) z0 Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 40 / 85

Distributional e ects of monetary policy Since dq dz = 1, the rst-order condition for problem (6.19) is c 0 (q) u 0 (q) c 0 (q) = 1 + γ 1 σ. (6.20) If the money supply is constant, i.e., γ = 1, newly-born productive buyers consume q units of the DM good. However, unproductive newly-born buyers cannot produce in exchange for money balances and, therefore, cannot consume in the DM. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 41 / 85

Distributional e ects of monetary policy Assume now that there is a constant (positive) in ation, γ > 1, and that money is injected into the economy through lump-sum transfers to all newly-born buyers in the CM. Let t denote a transfer at night in period t 1 which can be used in the DM of period t. We have t = M t M t 1 = γ 1 γ M t. (6.21) Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 42 / 85

Distributional e ects of monetary policy Let m t represent the money balances of buyers in the DM of period t who had access to the CM when they were young. Hence, equilibrium in the money market requires that ρm t + (1 ρ) t = M t. (6.22) The fraction ρ of productive buyers hold m t units of money while the 1 ρ unproductive buyers hold t. The sum of the individual money holdings must add up to the money supply, M t. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 43 / 85

Distributional e ects of monetary policy Substituting t from (6.21) into (6.22) and rearranging, we get m t = M t 1 + ρ(γ 1), (6.23) ρ γ and, from (6.21) and (6.23), we get t = ρ(γ 1) 1 + ρ(γ 1) m t. (6.24) Equation (6.24) implies that t < m t : unproductive buyers are poorer than productive ones. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 44 / 85

Distributional e ects of monetary policy Let q denote the DM consumption of unproductive buyers. Since productive buyers spend their m t units of money in the DM, the same is true for unproductive buyer who hold fewer money balances, t < m t. From the buyer-takes-all assumption, c(q t ) = φ t m t and c( q t ) = φ t t. Hence, (6.24) implies c( q t ) = ρ(γ 1) 1 + ρ(γ 1) c(q t). (6.25) Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 45 / 85

Distributional e ects of monetary policy From (6.25) q t < q t. As γ increases, q t decreases through a standard in ation-tax e ect, see (6.20). But in ation also a ects the distribution of real balances across buyers. Indeed, the dispersion of real balances, as measured by, decreases as γ increases. c(q t ) c( q t ) c(q t ) = 1 1+ρ(γ 1) So the policy-maker faces a trade-o between smoothing consumption across buyers and preserving the purchasing power of real balances. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 46 / 85

Distributional e ects of monetary policy In terms of social welfare: W = σρ[u(q) c(q)] + σ(1 ρ)[u( q) c( q)]. (6.26) In the neighborhood of price stability, an increase in in ation only has a second-order e ect on the match surpluses of productive buyers, d [u(q) c(q)] = 0. γ=1 + d γ However, it has a rst-order e ect on the match surpluses of unproductive buyers. Di erentiating (6.25) with respect to γ, we get d q t = ρc(q ) dγ c 0 (0). γ=1 + Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 47 / 85

Distributional e ects of monetary policy The welfare e ect of an increase in ation from price stability is, from (6.26), given by dw u dγ = σ(1 ρ) 0 (0) γ=1 + c 0 1 ρc(q ) > 0. (0) Hence, an increase in in ation from γ = 1 is welfare-improving because it allows unproductive buyers to consume, while the negative e ect on productive buyers welfare is of only a second-order consequence. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 48 / 85

The hot-potato e ect of in ation There is a commonly held view that higher in ation makes agents spend their money holdings faster (hot-potato e ect), implying that in ation can increase the velocity of money and the frequency of trades. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 50 / 85

The hot-potato e ect of in ation In this section we want to give an explicit choice to buyers to spend their money faster: we will let buyers increase or decrease the speed at which they spend their money balances by altering the rate at which they search for trading opportunities. We will investigate whether the model can generate an hot-potato e ect of in ation and we will explore the consequences of such an e ect for the optimal monetary policy. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 51 / 85

The hot-potato e ect of in ation Sellers are evenly distributed across several market places that work like Walrasian markets. Buyers are able to nd a market place with a good they like with probability e, where e 2 [0, 1] can be interpreted as a search e ort. The cost of search e ort, e, is ψ(e), where ψ(0) = 0, ψ 0 > 0, ψ 00 > 0, ψ 0 (0) = 0, and ψ 0 (1) =. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 52 / 85

The hot-potato e ect of in ation The price of a DM good in terms of the general good is p. The value function of a buyer in the DM is V b (z) = maxf e ψ(e) + e max [u(qb ) pq b ] q b +W b (z)g (6.27) subject to e 2 [0, 1] and pq b z. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 53 / 85

The hot-potato e ect of in ation The quantity of the search good demanded by the buyer is given by u q b 0 1 (p) if u 0 (z/p) p (z) = otherwise. z p (6.28) Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 54 / 85

The hot-potato e ect of in ation If we substitute V b (z) into the CM value function, then the buyer s problem can be expressed as n h io iz ψ(e) + e u[q b (z)] pq b (z) (6.29) max z,e where q b (z) is given by (6.28). subject to z 0 and e 2 [0, 1], Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 55 / 85

The hot-potato e ect of in ation The rst-order conditions for the choices of real balances and search e ort are: u 0 q b = p 1 + i (6.30) e ψ 0 (e) = u q b pq b. (6.31) If i = 0, then the solution to (6.30)-(6.31) is unique: q b = u 0 e = ψ 0 1 u q b pq b. 1 (p) and Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 56 / 85

The hot-potato e ect of in ation We will focus our attention on monetary policies close to i = 0. The problem of a seller in the DM is to supply an amount of the search good, q s, taking the price, p, as given, i.e., max fpq s c (q s )g. (6.32) q s The solution is c 0 (q s ) = p. (6.33) Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 57 / 85

The hot-potato e ect of in ation The market-clearing condition in the DM is simply eq b = q s. (6.34) From (6.30), (6.31) and (6.34), e and q b q will satisfy: u 0 (q) c 0 (eq) = 1 + i e (6.35) ψ 0 (e) = u (q) qc 0 (eq) (6.36) Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 58 / 85

The hot-potato e ect of in ation Society s welfare is measured by the sum of the utility ows of buyers and sellers in the DM net of the buyers search costs, W = eu(q) c(eq) ψ(e). Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 59 / 85

The hot-potato e ect of in ation The rst-order conditions that characterize the social optimum are: u 0 (q) = c 0 (eq), (6.37) ψ 0 (e) = u(q) qc 0 (eq). (6.38) The rst-best allocation under the Friedman rule. q fb, e fb is unique and can be implemented Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 60 / 85

The hot-potato e ect of in ation Q i fb e SI fb q Figure 6.5: E ects of an increase in i in the neighborhood of the Friedman rule Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 61 / 85

The hot-potato e ect of in ation In Figure 6.5, we represent the condition (6.35) in the space (q, e) by the curve labelled Q and the condition (6.36) by the curve labelled SI. Following an increase in i from i = 0, e increases but q decreases. From (6.31), for e to increase, p must decrease, which implies from (6.33) that aggregate production eq decreases. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 62 / 85

The hot-potato e ect of in ation Summary The model where buyers can choose their search e ort in the DM generates a hot-potato e ect: as in ation increases (away from the Friedman rule), agents spend their cash faster.! This is because the price of the DM good decreases, which raises buyers surplus.! Still, aggregate output and welfare fall. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 63 / 85

The hot-potato e ect of in ation Summary The optimal monetary policy is the Friedman rule, and it achieves the rst-best allocation. So the presence of an hot-potato e ect does not imply that the policy-maker should deviate from the Friedman rule. This result di ers from the before, where the Friedman rule was sometimes suboptimal, because here buyers search e orts do not generate a trading externality. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 64 / 85

The welfare cost of in ation Let us now turn on the welfare cost of in ation. In ation distorts the allocations by inducing agents to reduce their real balances, and hence the quantities they trade in the DM. If the costs associated with a moderate level of in ation are very small, then in ation will not be an important policy concern. Search-theoretic monetary models predict a welfare cost of in ation which can be signi cantly higher than traditional estimates. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 66 / 85

The welfare cost of in ation A typical calibration procedure adopts a representative-agent version of the model studied so far. The night-time utility function: B ln x h, where x is consumption, h is the hours of work, and h hours produces h units of the general good. Given these preferences, the production in the CM is B. One can interpret B as the quantity of goods that do not require money to be traded. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 67 / 85

The welfare cost of in ation The functional forms for utility during the day are u(q) = q 1 η /(1 η) and c(q) = q. The parameters (η, B) are chosen to t money demand as described in the model to the data. The cost of holding real balances, i, is measured by the commercial paper rate and M is measured by M1, which is cash plus liquid deposits. The typical measure of the cost of in ation is the fraction of total consumption that agents would be willing to give up to have zero in ation instead of 10 percent in ation. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 68 / 85

The welfare cost of in ation The results of existing studies are summarized in Table 1. Trading mechanism Cost of in ation (% of GDP) Buyers-take-all 1.2-1.4 Nash solution 3.2-3.3 Generalized Nash up to 5.2 Egalitarian 3.2 Price-posting (private info) 6.1-7.2 Price-taking 1-1.5 Gen. Nash w/ ext. margin 3.2-5.4 Proportional w/ ext. margin 0.2-5.5 Comp. search w/ ext. margin 1.1 Table 1: Summary of studies on the cost of in ation Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 69 / 85

The welfare cost of in ation Under the buyer-takes-all bargaining solution, the welfare cost of 10 per cent in ation is typically between 1 percent and 1.5 per cent of GDP per year. One nds a similar magnitude for the welfare cost of in ation under price-taking or competitive price posting. This is a sizeable number. Graphically, this number is approximately equal to the area underneath the money demand curve. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 70 / 85

The welfare cost of in ation To see this, integrate the inverse n (individual) o money demand function, which is given by i(z) = u σ 0 [q(z)] 1, see (6.8), to obtain c 0 [q(z)] Z z1 z 0 i(z)dz = σ fu [q(z 1 )] c [q(z 1 )]g σ fu [q(z 1.1 )] c [q(z 1.1 )]g, where z 1 represents real balances when γ = 1 and z 1.1 represents real balances when γ = 1.1. The left side of the above expression is the area underneath the individual money demand curve while the right side is the change in society s welfare. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 71 / 85

The welfare cost of in ation In Figure 6.6 we represent the individual money demand function, i(z). As the nominal interest rate approaches to 0, real balances approach their maximum level, z. Under buyers-take-all, z = c(q ). Consider two nominal interest rates, i > 0 and i 0 > i. The welfare cost from raising the nominal interest rate from i to i 0 corresponds to the area underneath money demand, ABDE. The welfare cost from raising the interest rate from its optimal level given by the Friedman rule to i 0 is the area ABC. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 72 / 85

The welfare cost of in ation i u' q( z) i( z) = σ 1 z '[ q( z) ] i' B i 0 A z' D E C z* Figure 6.6: Welfare cost of in ation and the area underneath money demand Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 73 / 85

The welfare cost of in ation If sellers have some bargaining power, then the welfare cost of in ation is bigger. Under the (symmetric) Nash solution or the egalitarian solution (i.e., proportional with θ = 0.5), the welfare cost of 10 per cent in ation is between 3 and 4 per cent of GDP. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 74 / 85

The welfare cost of in ation Whenever θ < 1 and i > 0 any bargaining solution generates a holdup problem on money holdings: buyers incur a cost from investing in real balances in the CM that they cannot fully recover once they are matched in the DM. The severity of this holdup problem depends on the seller s bargaining power, 1 θ, and the average cost of holding real balances, i/σ. As in ation increases, the holdup problem is more severe, which induces buyers to underinvest in real balances. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 75 / 85

The welfare cost of in ation Once again, this argument can be illustrated using the area underneath the money demand function. The inverse n (individual) o money demand function is i(z) = u σ 0 [q(z)] 1. z 0 [q(z)] The area underneath money demand is Z z1 z 0 i(z)dz = σ fu [q(z 1 )] z 1 g σ fu [q(z 1.1 )] z 1.1 g. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 76 / 85

The welfare cost of in ation Under proportional bargaining, u [q(z)] z = θ fu [q(z)] c [q(z)]g, the area underneath the money demand function is Z z1 z 0 i(z)dz = θσ fu [q(z 1 )] c [q(z 1 )]g θσ fu [q(z 1.1 )] c [q(z 1.1 )]g The private loss due to an increase in the in ation rate corresponds to left side of the above expression. It is equal to a fraction θ of the welfare loss for society, the right side of the above expression. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 77 / 85

The welfare cost of in ation In Figure 6.7 we represent the individual demand for real balances as well as the social return of those real balances (the dashed curve). The welfare cost from raising the nominal interest rate from 0 to i is given by the area ADC, while the welfare cost to the buyer is the area underneath money demand, ABC. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 78 / 85

The welfare cost of in ation So the individual money demand does not accurately capture the social value of holding money since it ignores the surplus that the seller enjoys when the buyer increases his real balances. If, for example, θ = 1/2 (the egalitarian solution), then the social welfare cost of in ation is approximately twice the private cost for money holders. This private cost has been estimated to be about 1.5 percent of GDP, so the total welfare cost of in ation for society is then about 3 percent of GDP. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 79 / 85

The welfare cost of in ation A B D C 0 i i z* [ ] = 1 ) ( ' ) ( ' ) ( z q z z q u z i σ [ ] θ σ ) ( ) ( ' ) ( ' ) ( ' z i z q z z q c z q u = Figure 6.7: Holdup problem and the cost of in ation Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 80 / 85

The welfare cost of in ation In the case where θ = 0, there is no monetary equilibrium since buyers get no surplus from holding money. If buyers are heterogenous and the buyer s willingness to trade is private information, then a monetary equilibrium can be restored. In this case, the welfare cost of in ation can be as high as 7 percent of GDP per year. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 81 / 85

The welfare cost of in ation The introduction of an endogenous participation decision, see "Trading frictions", can either mitigate or exacerbate the cost of in ation, depending on agents bargaining powers. As we saw earlier, in some instances, the cost of small in ation can be negative. Under competitive posting (i.e., competitive search equilibrium), search externalities are internalized and the cost of in ation is approximately the same as the one obtained without extensive margin and competitive pricing. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 82 / 85

Conclusion Under standard pricing mechanisms, the optimal monetary policy corresponds to the Friedman rule. The money growth rate must be negative so that the rate of return of money balances is approximately equal to the rate of time preference. Under this policy the cost of holding money is driven to zero and agents demand for money balances are satiated. To be implemented, the Friedman rule requires that money is withdrawn from the economy through lump-sum taxation. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 84 / 85

Conclusion If the government s coercive power is limited, the Friedman rule is not always incentive feasible. The optimality of the Friedman rule seems at odds with the usual practise of central banking. Some extensions of the model show where a deviation from the Friedman rule is optimal. Finally, in ation leads to welfare costs, but those depend crucially on the assumed pricing mechanism. Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 85 / 85