Some notes on monetary economics Francesco Lippi EIEF and University of Sassari

November 19, 2015

Abstract An introductory set of notes on various monetary models

Contents 1 Notes on the Samuelson-Lucas model of money 1.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The planner’s problem (“Ramsey Plan”) . . . . . . . . . . . 1.3 Decentralized equilibrium with constant money supply . . . 1.4 Lump-sum money transfers . . . . . . . . . . . . . . . . . . 1.5 Proportional money transfers . . . . . . . . . . . . . . . . . 1.6 Affine transfers (“paying interest on large money holdings”) 1.7 Welfare analysis . . . . . . . . . . . . . . . . . . . . . . . . .

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1 1 1 2 2 3 4 5

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7 8 8 9 10

A Lagos-Wright monetary economy 3.1 Coexistence of money and nominal bonds (assume θ = 1) . . . . . . . . . . 3.2 Coexistence of money and real assets (assume θ = 1) . . . . . . . . . . . . .

12 15 18

2 A deterministic model with one period cycles 2.1 Friedman rule . . . . . . . . . . . . . . . . . . . 2.2 Stationary equilibria when µ > β . . . . . . . . 2.3 A deterministic model with N-period cycles . . 2.4 A continuous time optimal control problem . . 3

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1

Notes on the Samuelson-Lucas model of money

A classic model, based on Lucas (1996).

1.1

Setup

Assumptions: two overlapping generations of equal size; produce when young, consume when old, each period young and old coexist. The (time separable) lifetime utility of the agent in the generation born at time t is: o Wt = Uty + β Ut+1 where o Uty = −nt , Ut+1 = U (ct+1 )

β is the intertemporal discount factor and U(·) is a concave utility function. Technology: the consumption good is produced using labor with the linear production function: ct = nt (1) There is NO storing technology, hence agents cannot save for their old age when they are young. The only way to consume for the old is to use the production of the young.

1.2

The planner’s problem (“Ramsey Plan”)

We describe the choice of a “planner” who cares about the utility of ALL generations giving them an equal weight, time-discounted at the rate β. His utility function is max ∞

{ct }0 {nt }∞ 0

U0o

+

∞ X

t

β Wt =

t=0

max ∞

{ct }0 {nt }∞ 0

∞ X

β t (Uty + Uto )

(2)

t=0

subject to the constraint (1). Notice that one generation is special: whereas all generation have both a utility for their youth and one for their old-time, at time zero (the first period) there is a generation of old who does NOT have the utility from youth. Substituting the constraint in the objective function gives max

{nt }∞ 0

∞ X

β t (−nt + U(nt ))

t=0

Hence the planner’s allocation rule is n∗ such that U ′ (n∗ ) = 1 Without an arrangement between young and old (i.e. without markets), the best each individual can do is not to work when young and not to consume when old (take the FOC of the agent problem to verify this. What is the expected utility of this plan? Answer: U(0)/(1 − β)). 1

1.3

Decentralized equilibrium with constant money supply

Suppose young agents (at time t) are given m money tokens by the old upon delivery of the consumption goods nt , traded at the price pt : nt pt = m Let m be a constant for the moment (e.g. the number of shells used for trade in the primitive island economy). In period t + 1, agents could use this money to buy their consumption from the young, and so on. How does this work? young agents at time t must form an expectation of the prices at t + 1: pt+1 (the price at which they buy consumption tomorrow). We consider an equilibrium with rational expectations, i.e. one where agents’ expectations are correct (in this model without uncertainty rational expectations amounts to perfect foresight, i.e. forecast errors are identically zero). From the perspective of the agent the only decision is how much to work: nt . Working leads to accumulate money m = nt · pt . This can be used next period to buy goods at the ·pt price pt+1 , which amounts to future consumption: ct+1 = nptt+1 . The problem solved by the young agent is    pt max E −nt + β U nt nt pt+1 pt where we used that nt+1 = nt pt+1 . The FOC w.r.t. n gives

  pt ′ −1 + β E U (ct+1 ) =0 pt+1 Using the equilibrium condition for the exchange m = pt nt (demand for goods equal supply of goods ) it follows that in a stationary equilibrium (where nt = n = c) the price level is constant: pt /pt+1 = 1.1 Thus, under this setting, the agent labor supply (production) decision satisfies U ′ (ˆ n) =

1 β

(3)

This optimal choice by the agent equates the marginal disutility of labor (-1) with the marginal (future) benefit: β U ′ (c), which is discounted by β because it is realized next period.

1.4

Lump-sum money transfers

Modify the setup as follows. At the beginning of period t all agents (both old and young) receive a lump-sum money transfer τt = mt−1 (µt − 1), where µ ∈ (0, +∞) (note that the size of the transfer τ is independent of the money the agent has accumulated with his work, i.e. 1

Note that we use the Hp of rational expectations by replacing this value into the above equation. Are you able to prove that a non-stationary allocation for n cannot be a decentralized equilibrium in this model?

2

its derivative w.r.t. n is zero, this hypothesis is very important). So the money stock evolves as follows mt+1 = mt + τt+1 = (µt+1 − 1) mt + mt = µt+1 mt (4) The problem solved by a young agent now is:    nt pt + τt+1 max −nt + β U nt pt+1 which gives the first order condition −1 + β

pt U ′ (ct+1 ) = 0 pt+1

pt is an endogenous variable that we must solve for. Using the market The price ratio pt+1 clearing nt pt = mt , we get that the equilibrium for the price level must satisfy the quantity t equation pt = m hence in a stationary equilibrium with constant labor supply nt = n and nt µt = µ, then mt+1 pt+1 = =µ for all t pt mt so that the optimal labor supply choice, n ¯ , is given by the value that solves µ U ′ (¯ n) = (5) β

How does this equilibrium compares to the one with constant money supply in terms of labor supply (¯ n vs. n ˆ )? The answer depends on the rate of money growth µ. If money growth is positive, µ > 1 then the concavity of the utility function implies that n ¯
1.5

Proportional money transfers

Assume the new money is given to the agents in a way that is proportional to their previousperiod money holdings. Notice the difference with lump sum transfer, where the money received is independent of the money holdings. The agent solves    nt pt · µ max −nt + β U nt pt+1 2

Notice how, for instance, the dependance of production on inflation in the equilibrium with lump sum money transfers can be used to construct a decentralized equilibrium that replicates the allocation of the allocation of the planner problem solved above, setting µ = β. This yields n ¯ = n∗ because it induces the agents to perceive a return on money, pt /pt+1 = 1/µ > 1, that exactly offsets the labor-reducing effect of the intertemporal discount β. Is this a good or a bad policy? see Section 1.7

3

which gives the first order condition −1 + β As argued above

pt+1 pt

pt µ U ′ (ct+1 ) = 0 pt+1

= µ hence the optimal labor supply choice solves U ′ (ˆ n) =

1 β

as was the case for the constant money supply. Despite the presence of inflation, the proportional money transfers lets the agent internalize that the money he will have tomorrow depends on how much he works today. This model has the property that real variables are unaffected by the growth rate of money, this property is called “super-neutrality”. This was not the case, for instance, with the model with lump sum transfers. In both models, however, the price level is proportional to the money stock. This property is called “neutrality” (real variables do not depend on the stock of money).

1.6

Affine transfers (“paying interest on large money holdings”)

We consider here a policy alternative to the lump sum transfer, first proposed by Andolfatto (2010, 2011) and Wallace (2013). The motivation for this policy comes from mechanism design theory: notice that a policy involving a negative transfer τ < 0 amounts to taxing all agents, who have to give the planner part of their assets. An individual might thus have an incentive to hide its money (not pay taxes), unless it his in his interest to participate to the planner’s mechanism. Therefore, one might want to focus on transfer schemes that are individually rational (IR). It is immediate that this rule out negative transfers. Consider then a class of affine transfer policies that satisfy τt+1 = max{0, a + bmt } where b > 0 and a ≷ 0. Transfers occur at the beginning of the period before any trade takes place. Young agent at time t get a transfer equal to zero because they hold no money when they start the period. Assuming that τt+1 = a + bmt > 0 for an old agent (we will later establish condition for this to happen), a young time-t agent solves    nt pt + a + bmt max −nt + β U nt pt+1 which gives the first order condition −1 + β

pt (1 + b) U ′ (ct+1 ) = 0 pt+1

where we used that mt = nt pt . Notice that the agent now knows that one extra unit of work delivers a higher money holding also through a higher transfer (since the transfer depends on mt = nt pt through b).3 3

This also happened in the proportional transfer scheme but not the constant a 6= 0 will change the outcome because the size of the transfer is NOT proportional.

4

As above inflation will equal the money growth rate supply choice solves µt+1 U ′ (n) = β(1 + b)

pt+1 pt

= µt+1 hence the optimal labor

As an example of a stationary equilibrium, consider a transfer policy with at+1 = αmt , this is the lump-sum component of the transfer that does NOT depend on labor effort (for each agent at time t this value is given). The money growth rate is then µt+1 =

mt + a + bmt =1+α+b mt

so that U ′ (n) =

1+b+α β(1 + b)

It is immediate that if α = 0 we go back to the proportional scheme and no real effects occur, i.e. U ′ (n) = β1 . Notice however that a value of α < 0 allows the planner to increase output above the one achieved with constant money. Since incentive compatibility requires that τ ≥ 0 this implies that b + α ≥ 0, i.e. this policy will generate inflation (contrast this with the output increasing policy obtained with lump sum transfers which is deflationary). For instance if a policy with U ′ (n) = 1 was desirable, this can be achieved by a transfer scheme with 1 α = (1 + b)(β − 1) < 0 and b≥ −1 β There is a continuum of schemes to do that; i.e. any positive inflation rate can be made consistent with this; for instance one with zero inflation α = (β − 1)/β < 0 and b = β1 − 1. Notice that the scheme with zero inflation has, in equilibrium, zero transfers (like the constant money case). Yet (and this is the interesting part) the equilibrium allocations are different form the constant money supply because of the incentives at the margin. Interestingly, this arrangement is a mechanism to “pay interest on money” , i.e. to design the transfer scheme in such a way that money accrues to money holders in a way that is proportional to their money holdings (notice that the transfer τ is given only to the old, who already own money earned in their youth – the young do not receive any money since their holdings are zero). Paying interest on money means that (1 + b)/µ = (1 + b)/(1 + b + α) > 1 which obviously requires α < 0.

1.7

Welfare analysis

Suppose you were running the central bank in this economy and had to decide how much money to print (or whether to print it at all)....what would you do? The efficiency notion we adopt is Pareto. To answer the question of what allocations are efficient we need to compute the set of Pareto efficient equilibria, that is the allocation of resources that cannot be improved upon (make one agent happier without making another agent worse off). The general strategy to compute Pareto equilibria is to solve a Planner’s problem assigning weight (Pareto weights) to the utility of each agent. Each weighting will give us an allocation. By varying the weights, we can compute the set of Pareto efficient allocations. 5

Begin by noting that the planner utility function as of time zero, equation (2), can be rewritten as −n + βU(n) U(n) + 1−β This shows you that in this economy there are 2 types of agents, with inherently different utility. At the moment the economy starts, there is the generation of old and all future generation of young agents (notice that this is the present value of individual utility −n + βU(n)). We could decide to look for a rule that treats all generations identically from the current young onwards, but the current generation of old is different, as it does not have a utility term concerning its youth.4 Let us assign a Pareto weight λ to the current old, and 1 − λ to all future full-cycle generation. The Pareto problem solves   −n + β U(n) P (λ) = max λ U(n) + (1 − λ) n 1−β for a given λ ∈ (0, 1). The frontier of Pareto efficient allocations is computed by varying the weight over its domain. The FOC for the Pareto problem gives 0 = λ U ′ (n) +

1−λ [ −1 + β U ′ (n) ] 1−β

which gives 1 U (n) = β + λ 1−β 1−λ ′

  1 ∈ 0, β

(6)

This equation indexes the set of efficient (stationary) allocations: all values of n that solve (6) for λ ∈ (0, 1). Let us see some examples: the decentralized equilibrium allocation with constant money supply solved above, which gives U ′ (n) = 1/β is obtained as one Pareto equilibrium for λ = 0 , hence it is efficient. Notice that this equilibrium maximizes the second term (the welfare of present and future young generations) but ignores the first one (the welfare of the time-zero old generation). Also, the Planner problem U ′ (n) = 1 is obtained as one Pareto equilibrium for λ = 1/2 , i.e. when the current old generation and all future young generation are given the same weight. As λ → 1 more and more weight is given to the generation of the current old (and production increases). But notice that not all allocations (i.e. not all values of n ∈ R+ ) are Pareto efficient. For instance the autarkic allocation, the one where n = 0, cannot be obtained as a Pareto equilibrium. Also note that with lump sum money transfer we can implement both (Pareto) efficient and inefficient policies. Since U ′ (n) = µ/β, then all values of µ/β ∈ (0, 1/β] produce efficient allocations. Notice that this requirement imposes that money growth µ ∈ (0, 1]. Constant money (µ = 1) gives the best allocation for the young. Deflation is also a Pareto efficient equilibrium, with more weight given to the old than to the young. Notice that inflation (µ > 1) is never efficient: in any equilibrium with inflation it is possible to increase the 4

This is inevitable in an OLG model.

6

welfare of both the young and the old generation by reducing µ towards 1, so that the utility of both young and old increases.5 .

2

A deterministic model with one period cycles

The model in this section describes an economy with two agents with production possibilities that alternate deterministically through time. Agents are ex-ante identical, discount the future at rate β and live forever. At the beginning of the period agents receive a monetary transfer τ /˜ q (in dollar terms). Without loss of generality we assume that agent type 1 is productive in even periods and unproductive in odd periods. Agent type 1 faces the following problem, max

1 ∞ 1 ,m1 {c12t ,c12t+1 ,l2t 2t+1 ,m2t+2 }t=0

∞ X

    1 l2t + τ2t − c12t p p 1 1 1 1 1 ˜ β ln c2t − l2t + γ˜2t m2t + − m2t+1 + θ2t m2t+1 q ˜ 2t t=0     ∞ X τ2t+1 − c12t+1 u 2 1 1 1 u 2t+1 ˜ − m2t+2 + θ2t+1 m2t+2 ln c2t+1 + γ˜2t+1 m2t+1 + + β q˜2t+1 t=0 2t

where the price of money q˜t and the lagrange multipliers γ˜ts (budget constraint) and θ˜ts (borrowing constraint) (with s ∈ {1, 2}), are measured in “nominal” terms (units of consumption per 1 dollar). Note that γ˜ p and γ˜ u are the co-state variables when the agent is productive and unproductive, respectively. The first order conditions give p 1 γ˜2t = c12t q˜2t p γ˜ 1 (l2t ) : 1 = 2t q˜2t γ˜ u 1 (c12t+1 ) : 1 = 2t+1 c2t+1 q˜2t+1

(c12t ) :

p p u (m12t+1 ) : γ˜2t = β˜ γ2t+1 + θ˜2t (m1 ) : γ˜ u = β˜ γ p + θ˜u 2t+2

2t+1

2t+2

2t+1

where the multipliers θ˜ are positive if the borrowing constraint binds. We construct equilibria where nominal variables are homogeneous in the aggregate level 5

As an exercise: solve the problem assuming the utility function of young agent of generation t is −nt + log(ct ), i.e. that they also like to consume when young. Determine whether the effects of inflation on output and welfare are different

7

of money: q˜t = qt /mt , γ˜ts = γts /mt , and θ˜ts = θts /mt . The first order conditions reduce to (c12t ) : 1 (l2t ) :

(c12t+1 ) : (m12t+1 ) : (m12t+2 ) :

p γ2t 1 = c12t q2t p γ 1 = 2t q2t u γ2t+1 1 = c12t+1 q2t+1 p p u γ2t µ = βγ2t+1 + µθ2t p u u + µθ2t+1 γ2t+1 µ = βγ2t+2

(7) (8) (9) (10) (11)

where we also used that monetary policy controls money growth rate µ, mt+1 = µmt . It is immediate that c12t = 1 (follows from combining equation (7) and equation (8)). Combining equation (10) and equation (11) gives p γ2t =

β u p γ + θ2t µ 2t+1

u γ2t+1 =

,

β p u γ + θ2t+1 µ 2t+2

which gives p γ2t

2.1

 2 β u β p p + θ2t γ2t+2 + θ2t+1 = µ µ

Friedman rule

Conjecture and verify that the following is an equilibrium: p u µ = β, θts = 0, c12t = c22t+1 = c¯ = 1, γ2t = γ2t+1 = q2t = q¯, ∀ t

which is easy to verify that satisfies the first order conditions, i.e. equation (7) to equation (11). and sustain the complete markets allocation. To pin down the price level q¯ (and therefore the value of the multipliers γ¯ ) we use that the agent’s total nominal assets (inclusive of transfers) equals the total nominal consumption expenditures over the one period cycle. Without loss of generality we consider the first period and we assume that the unproductive agent starts will all the money, c¯ = βm0 q˜0 where βm0 are the after-transfer money holdings of the unproductive agent. This equation can be solved to obtain, c¯ q¯ = γ¯ = β

8

2.2

Stationary equilibria when µ > β

Conjecture and verify that the following is an equilbrium: p p u = γˆ p , γ2t+1 = = 0 , c12t = 1 , γ2t θ2t

µ p γˆ , β

∀ t = 0, 1, 2, 3, ....

p u u which, using γ2t+1 = µβ γ2t+2 + θ2t+1 , implies u θ2t+1



=

µ β − β µ



γˆ p > 0 ,

∀ t = 0, 1, 2, 3, ....

u In this case the borrowing constraint of the unproductive agent binds, i.e. θ2t+1 > 0. As before, it is readily verified that the proposed solution satisfies the first order conditions, i.e. equation (7) to equation (11), even though in this case the consumption allocations are not those obtained under complete markets. In particular, consumption when unproductive is

c12t =

p γ2t+1 β = <1 , u γ2t µ

∀ t = 0, 1, 2, 3, ....

As before, to pin down the price level q¯ (and therefore the value of the multipliers γ¯ ) we impose that the agent’s total nominal assets (inclusive of transfers) equals the total nominal consumption expenditures over the one period cycle. Without loss of generality we consider the first period and we assume that the unproductive agent starts will all the money, c¯ = µm0 q˜0 where µm0 are the after-transfer money holdings of the unproductive agent. This equation can be solved to obtain, c¯ q¯ = γ¯ = µ

2.3

A deterministic model with N -period cycles

Assume agents productive status switches deterministically every N periods. The agent problem is max

{c1z ,lz1 ,mz+1 }∞ z=0

∞ X

j2N +N −1

β

j2N

j=0

+

   lt1 + τt − c1t p 1 1 1 ˜ mt + − + β − mt+1 + θt mt+1 q ˜ t t=j2N      (j2+1)N +N −1 1 X τs − cs β s ln c1s + γ˜su m1s + − m1s+1 + θ˜su m1s+1  q˜s X

t



ln c1t

lt1

s=(j2+1)N

9

γ˜tp

We now have first order conditions within the cycle and between cycles (the latter as as above). Using the homogeneity in m as above, the intra temporal FOC are : 1 γtp = c1t qt p γ (lt1 ) : 1 = t qt γu 1 (c1s ) : 1 = s cs qs

(c1t ) :

(12) (13) (14)

the intertemporal euler eq. within the cycle are (where BC never binds, i.e. θt = θs = 0) p (m1t ) : γtp µ = βγt+1 u (m1s ) : γsu µ = βγs+1

(15) (16)

while between cycles, i.e. in the last period of a productive/unproductive cycles, the intertemporal conditions are (where BC only binds for unproductive agent) u (m1t ) : γtp µ = βγt+1 p + µθsu (m1s ) : γsu µ = βγs+1

(17) (18)

Note that the consumption of the productive agent is c1 = 1 while the consumption for p γ the unproductive agent is given by cut = γtu . t Let t = 1 and t = N denote the first and last periods in a cycle. Conjecture that the borrowing constraints will only bind in the last period of an unproductive spell, so the intra period Euler equations equation (15) and equation (16) imply that the price of money grows at the rate µ/β for both agents (notice from the euler eq that this is consistent with constant consumption for each agent while intra -cycle )  k  k µ µ p p u γ1 and γ1+k = γ1u for k = 1, 2, ....N − 1 γ1+k = β β At the change between periods the borrowing constraint for the unproductive agent may u bind so θN > 0 where N denotes the last period of a cycle so that using equation (17) and equation (18) at this juncture we have         β β β β p p u u u u u γN +1 = γ1 and θN = γN − γN +1 = γN − γ1p γN = µ µ µ µ where γ1u , γ1p denote the real value of money in the beginning of a cycle for an unproductive and a productive agent, respectively. These conditions imply  N µ u γ1p γ1 = β or that the consumption of an unproductive agent is  N γtp β u ct = u = <1 γt µ 10

which is decreasing in the length of the cycle . The borrowing constraint binds for the unproductive agent !  2N β µ u θN = − 1 γp1 > 0 µ β the only variable that is left to be pinned down is γ1p which is found by using the budget constraint. Notice that in this equilibrium the real price qt grows during the cycle at the rate µ/β per period and then drops between the last period N and the beginning of the new at N + 1 one by the amount (µ/β)N and the cycle starts over again. Only in the last period the BC binds for the unproductive agent and his euler equation does not hold with equality (he is at a corner, that is why the Lagrange multiplier of the borrowing constraint binds).

2.4

A continuous time optimal control problem

Let ρ > 0 denote the time discount rate. There are 2 agents, as above, only one of which is productive at any point in time. Production possibilities switch deterministically avery T periods. Each agent chooses consumption ct , labor supply ℓt , and depletion of money balances m ˙ 1t in order to maximize her expected discounted utility, Z ∞  −ρt E0 e (¯ c ln ct − It ℓt ) dt (19) max 1 ∞ {ct ,ℓt ,m ˙ t }t=0

0

subject to the constraints

m1t

m ˙ 1t ≤ (ℓt + τt − ct ) /˜ qt 1 m ˙ t ≤ (τt − ct ) /˜ qt and ℓt = 0 ≥ 0 , ℓt ≥ 0 , ct ≥ 0 , m10 , I0 given ,

if It = 1 if It = 0

(20) (21) (22)

where c¯ > 0 is a parameter governing the marginal utility of consumption, q˜t denotes the price of money, i.e., the inverse of the consumption price level, τt denotes a government lump-sum transfer to each agent. The inverse price level q˜t is homogenous in the total money supply (i.e. is we change units of accounts from dollar to cents nothing changes except adding a 2 zeros), so that q˜t = qt /mt , where qt is the REAL price of money (i.e. units of consumption). The indicator variable It switches deterministically every T periods so that the agents alternates productive and unproductive periods deterministically. We can setup the Hamiltonian (note: this way compute an optimum, not necessarily an equilibrium) as H(ct , ℓt , ωt ) ≡ e−ρt (¯ c ln ct − It ℓt ) + ω ˜t

(ℓt + τt − ct ) q˜t

where ω ˜ t is the costate variable (lagrange multiplier) for the money holdings m ˙ it , which is homogenous in mt .6 The first order necessary conditions for optimality are: Hc = 0 , Hℓ = 0 , Hm = −ω ˜˙ 6

Note that since the value function H is defined in real terms, the costate must be homogenous w.r.t. the money supply ω ˜ t = ωt /mt (i.e. doubling of mt , or a change in unit of account, must not change the value of the problem).

11

They give:

c¯ ω ˜t ω ˜t = , e−ρt = , 0 = −ω ˜˙ ct q˜t q˜t The first and second equation imply ct = c¯. The second equation and the homogeneity of q˜ give qt m ˙t q˙t ω ˜ t = e−ρt so that the third FOC yields = −ρ + mt mt qt Notice that these equations are solved by ct = c¯, c˙t = q˙t = 0 (constant consumption and constant real value of money), and µ = −ρ (Friedman rule). The constant level of q is pinned down by imposing that total nominal assets at the beginning of a cycle, which in the stationary equilibrium are held by the unproductive agent, are equal the total nominal consumption and tax expenditures over the cycle of length T . Without loss of generality let’s consider the first cycle, starting at time 0, where the unproductive agent holds all the money supply; m0 = M0 . Using that Mt = M0 eµt , µ = −ρ, and q˜t = M1t q, we have Z T Z T c¯ T 1 − e−ρT ∼ dt + . ρMt dt = M0 , which gives q = c¯ = c¯ −ρT ˜t ρe 1 − ρT 0 q 0 where the approximation is accurate for small T . It is immediate to verify that this allocation also solves the Euler equation of the productive agent and that her money holdings are never negative. e−ρt

3

A Lagos-Wright monetary economy

This is a very compact summary of the essentials, see the paper of Lagos and Wright (2005) for more rigor and details. Agents are infinitely lived and discount the future at rate β. Timing: each period is split in 2 parts, say the day and the night. Key assumption: the exchange is bilateral and anonymous (or decentralized) in the day market (DM), and centralized (or Walrasian) in the night market (CM). Each subperiod has a particular consumption good, q and X respectively, produced linearly by labor h = q, H = X, respectively. Agent cannot produce their own consumption good, so they either meet with someone who has the good they like in the day market, with probability σ ∈ (0, 1/2), or they do not consume. Likewise, they can meet with someone who likes what they can produce with probability σ, so they have a chance to exchange. With probability 1 − 2σ there is no trade meeting in the DM and agents move on to the night market. Let Mt be the money supply at time t, an intrinsically useless, non counterfeitable, object. Environment: Preferences and technologies The agent preferences are U(q, h, X, H) = u(q) − c(h) + U(X) − H

(23)

we normalize u(0) = c = (0) = 0 (the utility and disutility of output in DM in case there is no trade). The usual Inada conditions apply. Production is linear in labor (there is no capital for now) so that q = h and X = H. Agents cannot eat their own good (we want to have a motive for trade). 12

First best allocations with complete markets (i.e. pure credit) It is immediate to see that the Arrow Debrew allocation q ∗ , X ∗ is Pareto efficient and satisfies u′ (q ∗ ) = c′ (q ∗ ) and U ′ (X ∗ ) = 1. Monetary equilibrium. Let m denote the nominal money holdings of one agent, and W (m) denote the value function of an agent in the centralized market (CM) and V (m) the value function in the decentralized market: W (m) = max U(X) − H + β V (m) ˆ X,H,m ˆ

subject to the budget constraint H + φm + T = X + φm ˆ where φ is the price of money (the inverse nominal price of X) and T is a lump sum money transfer (expressed in units of the CM good) that agents receive at the beginning of the night. Replacing the constraint into the value function it is straightforward to see that W (m) is linear in m due to the quasi linearity assumption (the way H enters in U). Let X ∗ be the argmax U(X ∗ ) = 1 (see above), then W (m) = φm + T + U(X ∗ ) − X ∗ + max (−φm ˆ + β V (m)) ˆ m ˆ

(24)

Now consider the DM. Assume that when two agents meet they have an option to trade consumption q for a money transfer d. The value function is V (m) = σ [u(q) + W (m − d)] + σ [−c(q) + W (m + d)] + (1 − 2σ)W (m) In principle the exchange of q for d (a part or all of the money holdings) may depend on the money holdings of both the buyer m and the seller m, ˜ so that the exchange involves exchanging q(m, m) ˜ units of the goods for d(m, m) ˜ units of money. Using the quasi linearity of W (m) , can be rewritten as V (m) = σ [u(q(m, m)) ˜ − d(m, m)φ] ˜ + σ [−c(q(m, ˜ m)) + d(m, ˜ m)φ] + W (m)

(25)

Trades and bargaining in the DM market. The chosen criterion for the exchange is a bargaining protocol with bargaining weight θ ∈ (0, 1) for the buyer max (u(q) + W (m − d) − W (m))θ (−c(q) + W (m ˜ + d) − W (m)) ˜ (1−θ) q,d

subject to d ≤ m (buyer cannot give more than he has). Using the linearity of W (·) yields max (u(q) − dφ)θ (−c(q) + dφ)(1−θ) q,d

13

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Assuming the constraint d ≤ m is binding (i.e. m = d so we take no FOC on d)7 , the FOC w.r.t. q gives θu′ (q)c(q) + (1 − θ)c′ (q)u(q) ≡ z(q) (27) φm = θu′ (q) + (1 − θ)c′ (q) this is the equilibrium value of real balances φm exchanged in a DM meeting.8 It appears that the quantity that is exchanged depends solely on the money holdings of the buyer, i.e. d(m, m) ˜ = m = z(q)/φ. (obviously we still have to determine the equilibrium value of φ). Note that in the case the buyer has all the bargaining power θ = 1 then φm = c(q). This is a key equation to solve for equilibrium allocations. The implicit function theorem gives φ ∂q(m) = ′ >0 ∂m z (q)

(28)

(to determine the sign you need to differentiate and analyze the properties of z(q)) . Optimal choices in DM and CM. Taking the derivative of the value function equation (25) w.r.t. m gives ∂q(m) V ′ (m) = (1 − σ)φ + σu′ (q) ∂m notice that derivative uses the fact that d(m, m) ˜ = m so that the differential of d(m, ˜ m) w.r.t. m is zero. Using equation (28),    ′ u (qt ) ′ −1 (29) V (mt ) = φt 1 + σ z ′ (qt ) where we used time subscript to emphasize that this is an intra-period condition (it will be useful soon). Likewise the optimal choice for future money holdings in CM is obtained by taking the derivative of equation (24) w.r.t. m ˆ (i.e. mt+1 ) which gives φt = βV ′ (mt+1 )

(30)

Equation (29) and (30) give the ODE   ′  u (qt ) φt = β φt+1 1 + σ −1 z ′ (qt ) Focusing on stationary equilibria where qt = q we have from the real balance equation mt φt = z(q) that φt /φt+1 = µ, or in other words that the price level 1/φt is homogenous of 7 It is possible that the constraint d ≤ m is not binding in which case q = q ∗ , the first best allocation satisfying u′ (q ∗ ) = c′ (q ∗ ) and φm = θc(q ∗ ) + (1 − θ)u(q ∗ )

which is easily obtained by the FOC w.r.t. d and the FOC w.r.t. q. This is an interesting special case of the previous one, more comments later. 8 Note: in the model the quantity of money m is immaterial as long as m > 0; halving m will double φ and real allocations are unchanged. The only relevant object with economic content is real balances φm.

14

degree 1 in the money supply. Given the money supply at time t, Mt , the price of money is φt = z(q)/Mt . We can then write  ′  µ u (q) =1+σ −1 (31) β z ′ (q) which is an equation that pins down the equilibrium quantity q. For CRRA utility function u(q) ( with u(0) = 0 as from our assumptions) and quasi convex costs c(q) the function ′ (q) is decreasing in q since ∂Ψ(q, θ)/∂q = (u′′ (q)z ′ (q) − u′ (q)z ′′ (q))/(z ′ (q))2 < 0. Ψ(q, θ) ≡ uz ′(q) This implies that the smaller the discount rate β the lower the equilibrium output. In other words, for a fixed µ is it always possible to find a low enough β such that the efficient level q ∗ cannot be sustained in equilibrium so that our assumption that the constraint d < m was binding is satisfied. Likewise, for a given β output is decreasing in inflation µ. Thus output is maximal at the lowest feasible µ, namely µ = β (the FR). You should prove it as an exercise that there is no equilibrium for µ < β (suggestion: note that in this case the constraint d < m will not bind, so that the equilibrium output will be the efficient level.... prove it). Welfare analysis. Define the nominal interest rate βµ = 1 + i and the function Ψ(q, θ) ≡ u′(q)/z ′ (q) (the function depends on θ through z(q)), and rewrite the equation as 1+

i = Ψ(q, θ) σ

(32)

Since the function Ψ(q, θ) is decreasing in q then higher nominal interest rate (i.e. higher money growth µ) decreases the output q in the DM market, i.e. this proves that ∂q <0 ∂i If µ = β we are at the Friedman rule and i = 0, then equation (32) shows that output is maximized at this level. Now we ask what is the cost of a small inflation. The ex-ante value function for a representative agent in this economy is a natural welfare criterion. In the stationary competitive equilibrium studied above, where qt = q, mt φt = z(q) and X = H = X ∗ we have E(V ) =

σ (u(q) − z(q)) + σ (−c(q) + z(q)) + U(X ∗ ) − X ∗ 1−β

or E(V ) = Which shows that

σ (u(q) − c(q)) + U(X ∗ ) − X ∗ 1−β

σ (u′(q) − c′ (q)) ∂q ∂E(V ) = ∂i 1−β ∂i

) If i = 0 (equivalently µ = β) AND θ = 1 then q = q ∗ and so ∂E(V = 0, i.e. the economy ∂i allocation is at a first best (cannot be improved upon). Small deviations from this point give small welfare changes, formally the first order effect of inflation on welfare is zero.

15

Instead, if i = 0 (equivalently µ = β) and θ < 1 then production is q < q ∗ , i.e. it is below the first best. To see this we need to show that Ψ(q, θ) is increasing in θ, so that when θ decreases then the output level that solves equation (32) falls. Then if q < q ∗ we ) have u′(q) > c′ (q) so that ∂E(V < 0. Formally, the first order effect of inflation on welfare is ∂i not zero. (it is negative). (see LW for a discussion of the empirical implications and formal conditions for Ψ(q, θ) increasing in θ). Figure 1: Examples of function Ψ(q, θ) as θ varies

10

θ θ θ θ

9

= = = =

1 0.9 0.8 0.5

8

function Ψ(q , θ )

7

6

5

4

3

2

1

0 0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

Output : q

This uses a generalized CRRA u(q) with u(0) = 0 and a quadratic cost function c(q).

3.1

Coexistence of money and nominal bonds (assume θ = 1)

We modify the above setup by assuming that in the CM market agents can trade money m AND nominal bonds b (zero coupon) that will pay their face value in the next CM period. A fraction s ∈ [0, 1] of the bond can also be traded (exchanged quid pro quo) in the DM market. The reason for this restriction is not discussed here (see chapter 10 of Nosal and Rocheteau (2011)). We want to analyze under which conditions (if any) money and nominal bonds can coexist in equilibrium. Nominal bonds are claims to money so that when they mature they are a perfect substitute with m. Therefore the value function for the CM period for an agent holding money balances m and maturing bonds b reads   W (m + b) = φ(m + b) + T + U(X ∗ ) − X ∗ + max −φm ˆ − ωˆb + β V (m, ˆ ˆb) m, ˆ ˆb

subject to

H + T + φ(m + b) = X + φm ˆ + ωˆb 16

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where ω is the nominal price of the bond and the hats denote next period values. If ω = φ bonds trade at par. Exploiting the linearity of W (·), as done above, The value function in the decentralized market is V (m, b) = σ [u(q) − φ(dm + db )] + σ [−c(q) + φ(dm + db )] + W (m + b)

(34)

where dm and db represent quantity of money and bonds, respectively, traded for the consumption good q. We now simplify the discussion of trading in the DM market by assuming the buyer can make a take it or leave it offer to the seller. This is equivalent to assigning θ = 1 to the buyer, subject to the participation constraint (non negative surplus) for the seller. The buyer’s choice solves max

q,dm ,db

u(q) − φ(dm + db )

s.t.

dm < m , db < s b ,

(35)

and the participation constraint for the seller: −c(q) + φ(dm + db ) ≥ 0. If the economy is not at the first best q ∗ then the constraint will bind, so that the buyer will give all his asset to the seller who will have a zero surplus c(q) = φ (m + s b) Note, as before that the equation implicitly defines q(m, s) as a function of m and s so that ∂q(m, s) φ = ′ > 0 and ∂m c (q)

∂q(m, s) sφ = ′ >0 ∂b c (q)

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(notice that z(q) = c(q) now since θ = 1). The agent portfolio problem. money and bonds solving max m, ˆ ˆb

or max m, ˆ ˆb

The agent in the CM chooses the optimal quantity of −(φm ˆ + ωˆb) + βV (m, ˆ ˆb)

h i ˆm −(φm ˆ + ωˆb) + β σ(u(q) − φ( ˆ + sˆb)) + W (m ˆ + ˆb)

where again we used the linearity of W (·). The first order conditions with respect to m ˆ gives   ′ u (q) ˆ − 1 + β φˆ ≤ 0 −φ + βσ φ ′ c (q) likewise the FOC for ˆb: ˆ −ω + βσ φs



 u′ (q) − 1 + β φˆ ≤ 0 c′ (q)

17

Focusing on an internal solution (where FOC hold with equality), using µ ≡ φ/φˆ for inflation and δ ≡ ωφ for the bonds’ discount rate µ −1=σ β



u′ (q) −1 c′ (q)



and

µ δ − 1 = σs β



u′ (q) −1 c′ (q)



Note that since θ = 1 in this case the FR delivers the first best outcome q = q ∗ (the reason for this was discussed before). For µ > β the output level is going to be suboptimal. For an equilibrium to exist (both eqts true) we must have that   µ µ 1 δ −1 = −1 β s β If bonds are fully sellable then s = 1 and δ = 1, i.e. bonds must trade at par with money. If bonds are partially sellable, i.e. s ∈ (0, 1), then they will trade at a discount i.e. pay an interest; the interest is increasing in s, less “liquid” bonds must pay higher interest in equilibrium (i.e. sell at a lower discount δ). This illustrates an example where money and bonds coexist: the bonds pay a premium compared to money because they are less liquid, in the sense that they are less useful for conducting transactions. At the other extreme if bonds are not sellable at all s → 0 then it is seen from the FOC for ˆb that an equilibrium with non zero bond holdings requires ω → 0 i.e. δ → 0, this means that bonds will not have value in equilibrium. It remains to be determined how the nominal interest on the bonds is paid, which is done by drawing resources from the economy using the lump sum transfer T . Recalling that it is a ˆˆ ˆˆ stationary equilibrium Rso that φB R = φB and R φM = φM , the budget constraint, aggregated over all agents so that Xdi = Hdi and m di = M gives ˆ − M) + ω B ˆ − φB = (µ − 1)φM − (δµ − 1)φB T = φ(M

To sum up: interest paying bonds and money can coexist if s < 1. This suggests a fundamental reason for coexistence: the different degree of liquidity of money and bonds (how to justify this is another story). However bonds are not essential: the same alloc could be sustained simply with money.

3.2

Coexistence of money and real assets (assume θ = 1)

We modify the above setup by assuming that in the CM market agents can trade money m AND a real asset k that is bought in the CM and delivers kR general good X in the next period. We let f (H) be the production function for the capital good k which uses the general good as an input. A fraction s ∈ [0, 1] of the asset can also be traded (exchanged quid pro quo) in tomorrow’s DM market. The value function for the CM period for an agent holding money balances m and asset k reads   ˆ + β V (m, ˆ W (m, k) = φm + k + T + U(X ∗ ) − X ∗ + max −φm ˆ − f (k) ˆ Rf (k)) ˆ m, ˆ k

18

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subject to

ˆ H + T + φm + k = X + φm ˆ + f (k)

where hats denote next period values. The value function in the decentralized market is V (m, k) = σ [u(q) − φdm − dk ] + σ [−c(q) + φdm + dk ] + W (m, k)

(38)

where dm and dk represent quantity of money and capital, respectively, traded for the consumption good q. Assume the buyer can make a take it or leave it offer to the seller. The buyer’s choice solves max u(q) − φdm − dk s.t. dm < m , dk < s k , (39) q,dm ,dk

and the participation constraint for the seller: −c(q) + φdm + dk ≥ 0. If the economy is not at the first best q ∗ then the constraint will bind, so that the buyer will give all his asset and the seller will have a zero surplus: c(q) = φ m + s k The equation implicitly defines q as a function of m and k so that ∂q φ = ′ > 0 and ∂m c (q) The agent portfolio problem. money and capital solving max ˆ m, ˆ k

or max ˆ m, ˆ k

∂q s = ′ >0 ∂k c (q)

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The agent in the CM chooses the optimal quantity of ˆ + βV (m, ˆ −φm ˆ − f (k) ˆ Rk)

h i ˆ ˆ ˆ ˆ −φm ˆ − f (k) + β σ(u(q) − φ m ˆ − sRk) + W (m, ˆ Rk)

The first order conditions with respect to m ˆ gives  ′  u (q) ˆ −φ + βσ φ ′ − 1 + β φˆ = 0 c (q)

or

“ < ” if

m=0

ˆ likewise the FOC for k: ˆ + βσs −f (k) ′

Case 1: f (k) = k, s = 1. FOC for for m: ˆ



 u′ (q) − R + βR = 0 c′ (q)

or

“ < ” if

k=0

Let φ/φˆ ≡ µ. For an internal solution we must have : u′ (q) µ−β +1= ′ βσ c (q)

19

ˆ likewise the FOC for k:

u′(q) 1 − βR +R= ′ βσ c (q)

So (1 − β)µ + βσ = 1 − βR + βσR or R=

1 − µ(1 − β) − βσ β(1 − σ)

So R is decreasing in µ. If µ = 1 then coexistence requires R = 1. Inflation µ > 1 requires depreciation R < 1; otherwise (if R ≥ 1) money is not used. Case 2: f (k) = k and s < 1 (limited pledgeability). ˆ The FOC for k: 1 − βR u′(q) +R= ′ βσs c (q) So

Internal solution

1 − βR 1−β µ+1= +R βσ βσs   1 1 1−β + µ+1=R 1− βσ σs βσs

or

or R=

1 βσs

−

1−β µ βσ

1 σs

−1

−1

=

1−(1−β)µs−βσs βσs β(1−σs) βσs

=

1 − (1 − β)µs − βσs β(1 − σs)

Note that the RHS is decreasing in s (a suff. condition is that β > σ 2 s). Thus, as in the case of nominal bonds, limited pledgeability allows for a wedge in the returns of the 2 assets: a smaller s is consistent with a higher R. For instance with s < 1 it is possible to have R > 1 even though µ = 1. Case 3: convex costs f (k) (and assume µ = 1 for simplicity). ˆ The FOC for k:  ′  u (q) ˆ βσs ′ − R + βR = f ′ (k) c (q)

Internal solution

this pins down a level of capital k ∗ > 0 that allows to implement the first best q ∗ , so that βσs + βR(1 − σs) = f ′ (k ∗ ) Exercise: analyze what capital level is the steady state optimal level of capital for the agent. Analayze whether this coincides with the capital that ensures q ∗ . Analyze the difference between the economy with a real asset and the economy with nominal bonds. Are real assets essential? do they allow the agent to reach an equilibrium preferred to the one without them (i.e. where k = 0)? 20

References Lagos, Ricardo and Randall Wright. 2005. “A Unified Framework for Monetary Theory and Policy Analysis.” Journal of Political Economy 113 (3):463–484. Lucas, Jr, Robert E. 1996. “Nobel Lecture: Monetary Neutrality.” Journal of Political Economy 104 (4):661–82. Nosal, Ed and Guillaume Rocheteau. 2011. Money, Payments, and Liquidity. MIT Press.

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