Monetary Policy and Unemployment: A Tale of Two Channels

Transcription

1 Monetary Policy and Unemployment: A Tale of Two Channels Ayushi Bajaj University of California, Irvine March 3, 2016 Abstract This paper studies the impact of monetary policy in a monetary framework that combines production, labor and a financial sector. This helps us analyze and quantify the effect of monetary policy on unemployment through the productivity channel as in Berentsen, Menzio and Wright (2011) and through the interest rate channel as in Rocheteau and Rodriguez-Lopez (2014). Inflation raises unemployment through the former channel as demand for firm s output falls, but since return on private liquidity might fall in response to inflation, the rise in unemployment is countered by the latter channel. An open market sale of bonds, crowds out private liquidity and increases unemployment through the interest rate channel, with the productivity channel further amplifying the effect. The model can generate a liquidity trap when liquidity provision is scarce. I also consider the effects of aggregate uncertainty on interest rates and hence unemployment. Equity premium translates into higher unemployment as firms paying a high interest rate open fewer vacancies. J.E.L. Classification: D82, D83, E40, E50 Keywords: Monetary Policy, Unemployment, Liquidity, Interest rate I am indebted to Guillaume Rocheteau for his extensive advice. I also thank participants at the ongoing UCI Macro Lunch Workshop for useful comments and suggestions. Ayushi Bajaj: ayushib@uci.edu.

2 1 Introduction Monetary Policy affects liquidity in the economy, with implications on other macroeconomic outcomes such as interest rates and unemployment. Broadly speaking, conventional monetary policy tools consist of controlling the amount of fiat currency and bonds outstanding, or their growth rates, often with a target for some nominal interest or inflation rate. This paper studies the impact of these different tools on liquidity, interest rates and unemployment in a monetary framework that combines production, labor and a financial sector. 1 This helps us analyze the effects of monetary policy on unemployment as in Berentsen, Menzio and Wright (2011) through the productivity channel and through changes in private liquidity as in Rocheteau and Rodriguez-Lopez (2014) by the interest rate channel. By combining the two effects, we can study the effects of monetary policy on a larger set of agents in a more general environment to see the different inter-linkages and quantify the amplifying or dampening effects of the two channels. The model also allows us to consider the effects of aggregate uncertainty on interest rates and hence liquidity and unemployment. Equity premium translates into higher unemployment as firms paying a high interest rate open fewer vacancies. The basic framework combines elements from BMW (Berentsen, Menzio and Wright, 2011) and RR (Rocheteau and Rodriguez-Lopez, 2014) and builds on the two benchmark models of labor and monetary economics: Mortensen and Pissarides (1994) and Lagos and Wright (2005) and Rocheteau and Wright (2005). There are two markets that convene simultaneously. First, a retail market where households who earn wages from firms, use their money holdings to buy goods that these firms produce, much like any common household. Second, in what I call a trader s market, traders meet to exchange financial services or goods that could be financed by money, government bonds or claims on private firms. This market aims to capture wholesale financial markets including repo markets or even households financing idiosyncratic consumption opportunities for some financial services. 1 This paper can also be extended to talk about unconventional monetary policy, through direct intervention in private liquidity market by extending credit through the purchase of a variety of assets. 1

3 The model shows how unemployment responds to monetary policy as changes in the nominal interest rate affect the demand for goods the firms produce and hence firm s entry decisions and unemployment through the productivity channel as in BMW. However, potentially there is an opposite effect of monetary policy (both changes in nominal interest and supply of government bonds) on unemployment through the interest rate channel as in RR. When liquidity supply is scarce (traders are borrowing constrained) then the interest rate falls below the rate of time preference, reflecting a liquidity premium on these assets. In response to lower interest rates, firms open more jobs so that unemployment decreases. Since the two channels are looked at in combination the inter-linkages between the two that affect the net impact of policy on macroeconomic variables can be analyzed and quantified. In particular, through the productivity channel, an increase in the nominal rate translates into an increase in unemployment. However, this positive long run relationship between unemployment and inflation in BMW can potentially be countered through the interest rate channel in two ways. First, is the direct effect of inflation or nominal interest rates on the return on private liquidity. If inflation implies a lower real return on private liquidity in accordance with a Mundell effect 2 - which will be the case if liquidity is scarce and if only money is accepted in at least some meetings in the liquidity market - then unemployment falls as more firms enter in response to lower interest payments. Thus, the short-run Phillips curve relation can prevail, as discussed in RR. Second, since more firms enter in response to higher output, liquidity supply in the traders market increases, resulting in a fall in the interest rate. Thus, the interest rate adjustment counters the initial productivity effect. This effect was missing from BMW as they assumed a fixed exchange rate. An increase in the supply of real government bonds increases the real interest rate, when liquidity is scarce. The increase in interest rate in turn implies that fewer firms enter and private supply of liquidity falls (interest rate channel). Public liquidity thus crowds out private liquidity as documented by Krishnamurthy and Vissing-Jorgensen (2013). This round of effect is the same as in RR, but since we have additional effects of firm entry in the retail market, the 2 The effect on nominal return can be unclear because of the opposing Fisher effect by which given real returns, inflation raises the nominal bond returns. 2

4 overall effect on private liquidity is amplified. Since there are fewer firms, households expect to meet fewer sellers, so they carry less real balances and demand falls, which in turn implies that fewer firms enter and overall unemployment rises (productivity channel). An open market sale also implies that traders in the financial or liquidity market carry more bonds and less money holdings. This effectively shrinks the narrow measure of liquidity (currency) and expands the broader measure (currency and bonds). This leads to a trade-off between liquidity provision and unemployment. By considering differences in acceptability of different forms of payment (money, government bonds and claims to private firms) we can define the spreads on different asset classes, and analyze how they are affected by policy. Broadly speaking, the model generates all the results from BMW and RR with effects in the former potentially dampened through the adjustment of interest rates the firms face, and in the latter amplified through effects on firm s profits through changes in demand for firm s output. Thus, it is possible to quantify the size and impact of the two channels. The model can also very naturally be used to explain liquidity traps which takes place when liquidity provision in the trader s market is scarce. In such a situation, increasing the supply of government bonds would not affect total liquidity because, at the margin, it is money that matters. This result is similar to Rocheteau, Wright and Xiao (2015) but here this comes out naturally without any additional assumptions (except of course that money is equally acceptable in exchange as bonds and claims on firms) because markets are already segmented, with one (retail) market where only money is accepted. Open market operations are also ineffective when bonds are abundant, or illiquid - cases when the interest rate on bonds is equal to the rate of time preference. We can also look at the effect of aggregate uncertainty on liquidity, interest rates and unemployment. The rate of return that firms use to discount their future payoff stream under uncertainty is higher than the risk-free return on bonds. Thus, these assets carry a risk premium. In a recent work, Hall (2015) emphasises the role of the stochastic discount factor reflecting some inherent risk premium in reconciling the excess volatility puzzle in the labor search literature. In that spirit, this paper takes the firm s discounting seriously and shows how a higher interest 3

5 rate implies that fewer firms enter, and unemployment rises. Thus, unemployment can be higher when uncertainty in the economy increases. Furthermore, since this risk premium alters the liquidity value of equity (private assets) - and more so by reducing liquidity supply - it can potentially reconcile the equity premium puzzle even without including acceptability differences between bonds and equity as in Lagos (2010) 3. The work so far has analysed the effects of conventional monetary policy on liquidity and other macroeconomic variables such as interest rates, unemployment and production. It can also be extended to analyze unconventional monetary policy in which the Central Bank directly intervenes in the market for private liquidity as studied in Williamson (2012) and Rocheteau and Rodriguez-Lopez (2014) in response to a liquidity crisis. The rationale to do so would arise when assets are heterogeneous in terms of their pledgeability for reasons such as private information on firm s returns. Thus, there exists scope for Central Bank intervention to buy these private assets if liquidity falls below a threshold. 2 Literature Review As discussed, the effect of monetary policy on unemployment has been studied in Berentsen, Menzio and Wright (2011) through what can be called a productivity channel. Households exchange their money holdings to buy the firm s goods who employs workers. This work however, ignored the effects of demand for goods and firm entry on private liquidity and hence interest rates because firms were not part of the economy s private liquidity. This interest rate channel was later explored by Rocheteau and Rodriguez-Lopez (2014) by allowing claims on these private firms to be traded along with public liquidity. This implied that these assets carry a liquidity premium which is potentially reflected in their rates of return. The latter paper however did not consider the effect of firm entry on demand for goods which in turn affects entry decision of firms. Considering the joint effects of the two channels in a broader framework is important as each side misses out the effects from the other as will be discussed. In this paper I therefore, 3 Work on calibrating the model is still pending. 4

6 combine these two effects. In doing so, we can study the effects of monetary policy on a larger set of agents in a more general environment to see the different inter-linkages and in particular how monetary policy affects the private provision of liquidity, unemployment and interest rates. As noted by Petrosky-Nadeau (2014) in a comment on the paper by Rocheteau and Rodriguez- Lopez (2014), it is important to reconcile the two diverging policy prescriptions. This paper is attempt in this direction. The relationship between discount rates and job creation is less established, except recently by Hall (2014) who documents that variation in discount rates through the stochastic discount factor plays a role in understanding the dynamics of the labor market. In the present work, the interest rate on firms reflect a liquidity premium which affects the effective rate at which the firms discount future payoffs. Rocheteau, Wright, Xiao (2015) looks in depth at open market operations, and since the focus is only that they do not have segmented markets and do not consider employment or private provision of liquidity. Williamson (2012) analyses unconventional monetary policy in terms of private asset purchase programs, but private liquidity is not provided by claims on a firm s production, but by entrepreneur s uncertain claims to future return. The additional real effects of liquidity on the firm s valuation and production is therefore not considered. While so far I have not studied these unconventional policy tools, but if this framework were to be used one could analyse other effects of these private asset purchase programs. In a recent work Tong and Xiao (2015), consider the productivity channel of BMW (they call it the consumption channel) by introducing government bonds to that framework to see the effects of a wider range of monetary policy in that framework. They also consider unconventional monetary policy by introducing long term government bonds. They, however, do not have the interest rate channel (they call it the production channel) of RR, and thus the additional effects are missing which can potentially counter or at least dampen some of the results. 5

7 3 Environment Time is discrete and continues forever. In each period three markets meet sequentially: first a labor market in the spirit of Mortensen and Pissarides (1994) denoted by MP, a decentralised goods market with frictions in the spirit of Kiyotaki and Wright (1993) called KW and a frictionless centralized market in the spirit of Arrow-Debreu (AD). The alternating market structure follows Berentsen, Menzio and Wright (2011). There are three types of agents: a large measure of firms, a unit measure of households and a continuum of traders in [0, 2] divided equally between buyers and sellers, indexed by f, h and t respectively. There are three types of perishable and non-storable consumption goods: a good that is produced, traded and consumed in the AD market by households (quantity denoted by x), another retail market good produced by firms by employing households in MP which is consumed by the households in the KW (quantity denoted by q) and a third good that is produced, traded and consumed in the KW by traders (quantity denoted by y). Traders who are called buyers work in the AD and consume the KW good, and vice versa for traders who are sellers. Firms open vacancies to enter the labor market where they employ households according to a matching process. Once matched, the firms produce the retail good to maximize profits and pay wages to employed and dividends to traders who financed them. Households demand firms output in the KW market. Thus, there are two simultaneous markets at work in the AD and KW markets, as illustrated in Figure 1. The period payoff for households is: U h (q, x, l) = u(q) + v(x) l (1) The period payoffs for traders who can be buyers or sellers are: U B (y, x, l) = (y) + v(x) l and U S (y, x, l) = c(y) + v(x) l, (2) where y is the DM good, q is the retail good, x is the CM good and numeraire and l is labor supply (used to produce AD good). Assume that v(x) = x, c(y) = y (for simplicity, no results are lost). As usual, νu and u are twice continuously differentiable with ν > 0 and u > 0 6

8 Trader's Market (interest rate) Retail Market (productivity) AD MP KW Traders portfolio choice (m, b, a) Households portfolio choice (m) Vacancies - Unemployed Buyers - Sellers Firms - Households Figure 1: Model Description and νu < 0 and u < 0. Also, νu(0) = u(0) = 0. Define the efficient y = y such that νu (y ) = 1. A similar efficient quantity can be defined for q after we discuss producer s cost function. Quasi-linearity in (1) and (2) simplifies the analysis because it leads to a degenerate distribution of assets across agents of a given type at the start of each KW market, and makes AD payoffs linear in wealth. Households and trader s discount rate after every period is ρ > 0 and β 1/(1 + ρ). Since the consumption goods are perishable and non-storable, direct barter is ruled out and agents are to some extent anonymous in DM to also rule out unsecured credit. This generates a role for assets in facilitation of intertemporal exchange. Recall that there are two sub-markets in the KW where an asset would be needed to trade. There are three assets that can be used in different exchanges: money, (real) government bonds like T-Bills and claims on firms. Their supplies are: M, B and L s a with AD prices: φ m, φ b and φ a. The supply of money and bonds is given by the government s budget constraint, while claims on firms is a private asset whose supply is endogenously determined, as will be discussed. Only money can be used in exchange for retail goods with firms (it s hard to imagine going to the grocery store with bonds or stocks), but money, bonds and claims can be accepted in exchange for KW goods between buyers and sellers (one can interpret the KW good as a financial service which is exchanged in an OTC market in exchange for bonds and stocks). However, there are differences in acceptability of the three assets: with α a probability, a buyer meets a seller who accepts all three, and with α b 7

9 accepts bonds and money and with α m only money is accepted. 4 There are search frictions in meeting between buyers and sellers given by the matching function, α(b, S) which is a function of the number of buyers and sellers. The government s budget constraint is: G + T + B(1 φ b ) γφ m M = 0, (3) where the first term is consumption of the AD good, second is a lumpsum transfer. Money growth rate is γ implying M +1 /M = 1 + γ. We also focus on stationary equilibria, in which z φ m M is constant over time which implies that money growth rate equals the inflation rate: φ m /φ m,+1 = M +1 /M = 1 + γ, where subscript +1 indicates next period. Stationarity also entails z b B (for real bonds) is constant implying that B is constant. The third term in (3) is debt service and fourth is seigniorage. Given other variables, T adjusts such that (3) holds. Finally, some further details on firms are needed before we start solving the model. Firms finance entry cost, τ (or cost of opening a vacancy) by selling claims on its future dividend to buyers at price φ a in the AD. Once firms enter, unemployed workers, 1 n are randomly matched with vacant firms, v according to the matching function h(1 n, v) in the CM. Market tightness is defined as θ v/1 n. Vacancy-filling probability for firms, σ l h(1 n, v)/v = h(θ 1, 1) = Aθ α l, if we assume a Cobb-Douglas matching function. A match between a firm and a worker is dissolved with an exogenous separation probability, δ at the end of every period. Firms that are matched produce output p, and then match with buyers in the KW market according to the matching function m(b, n). Retail goods market matching probability for firms, σ g m(b, n)/n = n ( αg) if we assume a Cobb-Douglas matching function since B = 1. To convert output p from the match to be sold in the retail market, firms incur cost c(q). Terms of trade between firms and buyers is determined by proportional bargaining. If matched, firms pay back the realised revenue, π net of wages, w to its holder at the end of DM. Thus, a claim on firms pays back ϕ = (π w)σ l to its holder till the match is destroyed. Interest Rates 4 Look at Rocheteau, Wright, Xiao (2015) for a further discussion on how to interpret differences in acceptability. 8

10 It is useful to define different interest rates. To begin with, I assume that government bonds are short-term (one-period) real government bonds that pay 1 unit of numeraire at the end of DM. Thus, real return, r b on a liquid government bond is 1/φ b, and just for completeness the nominal return is φ m /φ m,+1 φ b = (1 + γ)/φ b. Fisher equation gives the nominal return on an illiquid bond (one that is never accepted in DM): (1+ι) = (1+γ)(1+ρ). Real return is the rate of discounting. This may not be true for liquid bonds as will be discussed. Note that Friedman rule: ι = 0, or use γ as policy instrument since 1 + γ = β(1 + ι). Similarly, the real return on a liquid private asset (claims on firms) as is discussed in detail in the firm s problem is: (1 + r a ) = ϕ + (1 δ)φ f φ f. One can also define the real spread on bonds and private assets (where i {b, a}) as: S i = ρ r i 1 + r i 4 Households (BMW - Productivity Channel) Let the AD, MP and DM value functions for household be denoted by W h ( ), U h ( ) and V h ( ). A household h in the AD solves: s.t. W h e (z) = max x,z [x + (1 e)l + βu h e (z ), ] x = ew + (1 e)ui T + z (1 + γ)z. Substitute for x, W h e (z) = [ew + (1 e)(ui + l) T + z + max z [ (1 + γ)z + βu h e (z )] (4) where z = φ m m is the real balances carried each period (bonds and claims are already in real terms). And, denotes the next period values, i.e. the real balances, bonds or claims carried to the next sub-period. Choice of z is independent of employment status, as a result of quasi-linearity. 9

11 For a household h in the MP market: U h 1 (z) = V h 1 (z) + δ[v h 0 (z) V h 1 (z)], (5) U h 0 (z) = V h 0 (z) + h(1, θ)[v h 1 (z) V h 0 (z)]. (6) where δ is the job destruction rate and h(1, θ) is job-finding rate for households. Before writing the KW market value function, we have to determine the terms of trade in the KW market match between a household and a firm where terms of trade are determined by proportional bargaining. Here this assumption gives firms a share of the surplus which allows us to study the entry decision of firm. Let η be firm s bargaining power. The buyer s problem in each such match becomes: s.t. max[u(q) d] q,d (1 η)[ c(q) + d] = η[u(q) d] and, d z. or, max(1 η)[u(q) c(q)] s.t. ηu(q) + (1 η)c(q) z, (7) q where q is the output the firm produces in exchange for real money transfer d. Thus, there is notion of price of firm s good, d/q, where price is decreasing in q but increasing in η. Let z η (q) ηu(q) + (1 η)c(q) so, q = z 1 η (z), (8) gives the solution to (7) if the constraint is binding, which will be the case (See Nosal and Rocheteau (2011), pg. 83 for a discussion). We use the above and write the value function of the household in the KW retail market, (where we simplified using the linearity of W h e ): 10

12 Ve h = σ g (n)n max (1 η)[u(q) c(q)] + z + W e h (0) (9) q,d [0,z] where (q is given by (8). Note that all households (both employed and unemployed) participate in this market, but only matched firms can. The value to a household in the KW retail market is her share of the surplus when matched with a firm which happens with probability σ g (n)n (note that this is equal to m(1, n) which is an increasing function of n). There is also a possibility that the buyer does not match with a firm in which case she carries forward her asset holdings to the AD. Now we combine the three markets to get the maximization problem for households that determines their portfolio choice: U h 1 = σ g (n)n max q,d [0,z] (1 η)[u(q) c(q)] + z + (1 δ)w h 1 (0) + δw h 0 (0). We h (z) = I e + max[ (1 + γ)z + β(σ g (n)n max (1 η)[u(q) c(q)] + z )] + βew z q,d [0,z e h (0), (10) ] where I e = ew + (1 e)(ui + l) T + z. We get the following FOC: ( ) u (q) c (q) ι = σ g (n)n(1 η), (11) z η(q) where q is given by (10) from (8), so (11) gives either q or z. The FOC has been re-written using 1 + ι = (1 + γ)/β to give it a more natural economic interpretation. The left hand side, ι is the cost of holding real balances - if it were not for the liquidity property of money, agents might as well hold illiquid assets with return equal to the rate of discount. The right hand side is the benefit from holding real balances because of the liquidity services provided by money, which is adjusted by the matching probability and household s share of the surplus. If household s share goes down, reducing (1 η), they carry fewer real balances because of the hold-up problem. Note that the right hand side is increasing in number of matched firms in the retail market, 11

13 n i.e. if there are more trading opportunities, buyers carry more real balances and get more output. This establishes a link between monetary policy and employment through productivity. The other side - liquidity/interest rate effects comes because the claims on firm s profits can be traded in another market which we call the trader s market. 5 Firms For a firm f in the MP market, U f 1 = δv f 0 + (1 δ)v f 1, (12) U f 0 = σ l(1, θ)v f 1 + (1 σ l(1, θ))v f 0. (13) Firms do not carry money balances from AD. Output in a match, denoted p measured in units of the AD good. For a firm f in the KW market, V f 1 = σ gw f 1 [p c(q), d] + (1 σ g)w f 1 (f, 0), (14) V f 0 = 0. (15) Using the results from household s terms of trade determination, plug in for d = ηu(q) + (1 η)c(q). Now, we see how firms value their holdings of inventory and real balances net of wages they pay in the AD market. Note that since the claim on firms is a private asset offering a liquidity service, its return will reflect a liquidity premium and might be different from the rate of discounting. In particular, r a will be determined by trader s activity in the KW market. For a firm f in the AD market the value of holding inventory x, real balances z, and wage commitment w (< p) is: 12

14 This gives us: Thus, we get: W f 1 = x + z w + δv f 0 V f 1 = σ g(n)η(u(q) c(q)) + p w + + (1 δ)v f r a. (16) (1 δ)v f r a. (17) V f 1 = (π w)(1 + r a). (18) r a + δ where π = p + σ g (n)η(u(q) c(q)) is firm s expected revenue. Note how the KW market affects revenue, firm entry, n and hence employment - this is the BMW productivity channel. The effective rate of discounting is the rate of interest the firms pay plus the separation rate. Thus, the value of a firm at the time of entry, φ a is given by: φ a = σ l V f 1 /(1 + r a) = σ l (π w)/(r a + δ) = ϕ/(r a + δ). Here, ϕ = σ l (π w) is the profit over all firms which is paid out as dividend. The free-entry condition implies that number of vacancies, v or market tightness, θ adjusts such that the following holds 5 : Substituting for ϕ we get: τ + φ a = 0, τ = ϕ r a + δ. τ(δ + r a ) = σ l (θ)(p w + σ g (n)η(u(q) c(q))). (19) Note that σ l and σ a are decreasing functions of market tightness, θ and number of matched firms or employed in the market, n respectively. Next, we will combine the free-entry condition from (19) with a steady state condition to get a relationship between market tightness, θ and the measure of employed, n. In the steady-state number of jobs created is the number of jobs 5 We need τ(r a + δ) < (f w + η(u(q ) c(q ))), otherwise firms won t enter. 13

15 destroyed, thus uσ u = δ(1 u), or, (1 n)σ u = δn where σ u (θ) = h(u, v)/u = h(1, θ) is the job finding rate for the unemployed. n = σ u(θ) δ + σ u (θ) Since σ u (θ) is increasing in θ, so n is increasing in θ. We thus have that σ l and σ a are decreasing functions of market tightness, θ. And, (19) combined with (20) determine market tightness, θ and thus number of vacancies (v = θu). Given other variables, number of vacancies is decreasing in r a, the return on private assets. The return is determined endogenously from the demand for liquidity that comes from trader s activities. Note that σ a (1 u) = 0 gives RL or MP model with r a = ρ. One can also easily check that given other variables q is increasing in n as in BMW, r a = ρ gives BMW once we abstract from wage-bargaining. Also, from the surplus from a match in the retail market which depends positively on q as given by (8) or in terms of z from (11) which in turn is an increasing function of n, we get that q and n are positively related. Now we can get the supply of private assets/private liquidity: (20) L s a(r a ) = φ a v = φ a θu = V f 1 (1 u)δ/(1 + r a), (21) where, n = 1 u = σ l v/δ holds in the steady state and is the measure of filled jobs and V f 1 = φ a(1 + r a )/σ l (use to double check calculations), so we get: L s a(r a ) = ϕ r a + δ θ δ δ + σ u (θ). (22) where, ϕ, defined above is a function of θ and n, and they are related by the steady state condition and θ is determined by (19) and is decreasing in r as discussed. We also need r > δ to guarantee a positive value to firms. Thus, the supply is decreasing in r a. Also, note that supply of bonds for liquidity is B/(1+r b ), and since B is independent of r b, we get the liquidity supply of bonds is decreasing in r b, a result we will use after we discuss demand for liquidity. 14

16 6 Traders (RR - Interest Rate Channel) Let the AD and KW value functions for buyers be denoted by W B ( ) and V B ( ). Traders are inactive during MP. A trader d in the AD solves: s.t. W B d (z d, b, a) = max x d,l d,z d,b,a [x d l d + βvd B (z d, b, a ), ] x d = l d + z d + (1 + r b )b + (1 + r a ) (1 + γ)z d b a. Substitute for x d l d, W B d (z d, b, a) = z d + (1 + r b )b + (1 + r a )a + max z d,b,f [ (1 + γ)z d b a + βv B d (z d, b, a )], (23) where z d = φ m m is the real balances carried by trader, d each period (bonds and claims are already in real terms), wage is 1 as 1 unit of labor, l d produces 1 unit of AD good x d. And, denotes the next period values, i.e. the real balances, bonds or claims carried to the next sub-period. There is a similar value function for sellers but we can assume without loss in generality that they carry no assets to KW. If assets are priced fundamentally, they are indifferent to holding them and if assets bear a liquidity premium they strictly prefer not holding them as they do not have liquidity needs in KW. Before we move to the KW value function, we need to determine the terms of trade during bilateral matching between a buyer and seller. We assume for simplicity that the terms of trade in the KW goods market is determined by buyer take-it-or-leave-it offers. This is without loss in generality (See for a discussion on the different bargaining protocols). It could be unrestricted, as g = v(y), but I use TIOLI for simplicity. Denote l d i as the buyer s liquidity holding for each type of match, i = {a, b, m}. For example, l d a = [(1 + r a )a + (1 + r b )b + z d ] is the relevant asset holding in a match with a seller who accepts all three assets, which happens 15

17 with probability α a. Similarly, for other matches. The buyer s problem in each such match becomes: max y,g i [ν(y) + W B d (ld i g i )] s.t. y + W S d (g i) 0 and, g i l d i, where y is the output the seller produces in exchange for the asset transfer g i. Using linearity of the value function we get: Thus, max y,g i [ν(y) g i ] s.t. y + g i 0. y = min{y, l d i }, (24) g = min{y, l d i }. If the buyer holds enough liquidity the efficient level of output, y is obtained, else the buyer gets as much as she can given her asset holdings. Buyer s KW value function is as follows: V B d = i α i max [ν(y) y] + z d + (1 + r b )b + (1 + r a )a + W y,g L d d B (0). (25) i The value to a buyer in the KW is her share of the surplus when matched with the seller from whom she buys the DM good, which happens with probability α i, where i stands for the type of match. There is also a possibility that the buyer does not match with a seller in which case she carries forward her asset holdings to the AD. Now we take the KW value function one period forward and plug in to the CM value function. Ignore constants and in the benchmark specification we assume that all three assets are equally acceptable (so α b = α m = 0) to get the following maximization problem for buyers that determines their portfolio choice: max z d,b,a [ (1 + γ)z d b a + β(α max [ν(y) y] + y z d + (1 + r b)b + (1 + r a )a )]. (26) If buyers choose positive holdings of money, bonds and claims we get the following FOCs, where y = min{y, [(1 + r a )a + (1 + r b )b + z d ]}, 16

18 1 + γ = β[1 + α(ν (y) 1)], (27) 1 = β(1 + r b )[1 + α(ν (y) 1)], (28) 1 = β(1 + r a )[1 + α(ν (y) 1)]. (29) Thus, if z 2 > 0, b > 0 and a > 0, then the assets must have the same return. But this need not be the case always, as we might expect money to be dominated in return by the other two assets. The next sub-section which discusses the possibility of a liquidity trap deals with this. But, first consider the case when z d = 0, b > 0 and a > 0. From (28) and (29), r b = r a = r and we get that the real interest rate satisfies: Liquidity Premium {}}{ 1 = β(1 + r) [1 + α(ν (y) 1)]. (30) Thus, assets carry a liquidity premium (if y < y ) as the rate of return is less than the discount rate, ρ. 6 And, if the rate of return were equal to the discount rate (r = ρ), then buyers would carry enough liquidity to get the efficient output. 6.1 Liquidity Trap First re-write (27) as follows using that 1 + ι = (1 + γ)/β: ι = α(ν (y) 1)]. (31) Note that ι is the cost of holding real balances - if it were not for the liquidity property of money, agents might as well hold illiquid assets with return equal to the rate of discount. Now to see when buyers hold money, first note that buyers hold all the bonds and claims (let L s = (1+r)(B+L s a), since measure of buyers is normalised to one, we use the total supply which 6 r = ρ can be referred to as the fundamental value of the asset as this would be the interest rate if the assets had no liquidity value, i.e. for α = 0. 17

19 is multiplied by (1 + r) because that is how demand is measured). Whether or not they hold money depends. There are three possibilities. If bonds and claims are plentiful i.e. L s > L where L = y then z d = 0 and y = y. This happens when the rate of return is equal to the discount rate, then buyers would carry enough bonds and claims to get the efficient output. If bonds and claims are not as plentiful such that y < y but are still enough such that buyers get enough output, and agents choose not to bear the cost ι of holding real balances, then z 2 = 0. This happens when L s L where ι = α(ν ( L) 1)). In this case, even though agents get less than efficient output, the return, r on bonds and claims dominates the return on money and buyers do not hold money. The third case is when L s < L i.e. bonds and claims are scarce. We get z d > 0 where ι = α(ν (z d ) 1)]. In this case, total liquidity in DM goods market is independent of L because, at the margin, it is money that matters. In this case, the bonds and claims are so scarce that if buyers held only that, money would dominate the return on those assets. In other words, cost of holding money is lower than holding only bonds and claims This region depicts a liquidity trap as changes in L s do not induce any effect on output and hence total liquidity. ( z d = ũ ι ) if r < r (32) α = 0 if r r This result is similar to Rocheteau, Wright and Xiao (2015) but here this comes out naturally without any additional assumptions because markets are already segmented, with one (retail market) where only money is accepted. 7 Equilibrium and Analysis 7.1 Interest Rate Channel Liquidity Demand Consider the benchmark case illustrated above: when money, bonds and claims are equally liquid i.e. all sellers accept money, bonds or claims, then r = r b = r f as discussed. Also, by 18

20 market clearing, B = b and L s a = a. Demand for liquidity (we only discuss liquidity in terms of assets other than money, i.e. bonds and claims) is thus, L d (r) = (1 + r)(b + L s a), provided that ι α(ũ (L d ) 1)) or r r where 1 + ι = 1/β(1 + r) or (1 + γ) = 1/(1 + r) or r = γ/(1 γ) and we get the following equation: [ ( L d (r) = 0, ũ ι )] if r < r (33) α ( ) = ũ 1 (δ r) 1 + if r r < δ (1 + r)α = [y, ) if r = δ It can be seen from (33) that demand for liquidity is increasing in r: as return on an asset increases, demand increases. When the return on the asset is very low, such that money dominates return on other liquid assets (r < r), buyers do not demand other assets. This happens when the economy is in a liquidity trap as discussed above. Liquidity Supply To make supply comparable to demand, we need (1 + r)l s (r), where L s (r) is total liquidity supply. (1 + r)l s (r) = ϕ 1 + r δ + r θ δ + B, (34) δ + σ u (θ) where, θ(r) is determined by (19) and is decreasing in r as discussed. Thus, the supply is decreasing in r, since δ 1. Under the case when both bonds and claims are equally liquid, we get a unique r that equates the demand and supply of liquidity. Using (22) and (33) we get the following equation that determines the unique r: 19

21 ϕ 1 + r δ + r θ δ δ + σ u (θ) + B = = ũ 1 [ ( )] 0, ũ ι α if r < r (35) ( ) 1 + (δ r) (1+r)α if r r < δ = [y, ) if r = δ, where, θ(r) is determined by (19). Figure 2 shows the determination of the unique r. Note that the supply of liquidity can be such that the r = ρ which implies that liquidity is abundant and increasing supply of bonds has no effect on total output as the economy has already reached the efficient output level. It can also be the case that liquidity is so scarce that r = r and agents hold money balances implying that bonds and claims have no effect on the margin and increasing supply of bonds in this region also has no effect on total output. In the latter case the economy is said to be in a liquidity trap i.e. money dominates the return on other liquid assets. Shifts in the curves are discussed in the general equilibrium section when comprehensive comparative statics are considered. r ρ L d (r) r e r 1 L d L e y L d, L s L s (r) Figure 2: Equilibrium Interest Rate 7.2 Productivity Channel From the discussion above, there is a unique r that clears the market for liquidity, where market tightness θ is determined from (19) with firms producing output q given by (11). 20

22 Labor Market For given values of r, equilibrium of the labor market is given by (19) which gives a relationship between θ and q and is reminiscent of the MP curve in BMW, which is a modification of the classic MP determining u. The modification includes the demand channel where firm s revenue depends on expected gain from trade in the KW market. Re-writing the equation here: τ(δ + r a ) = σ l (θ)(p w + σ g (n)η(u(q) c(q))). The difference from BMW here is the endogenous determination of r a by the introduction of market of liquidity where trader s consume output, y given by (24) using bonds and claims on firms in exchange, which gives the return on these assets, r a liquidity premium, like RR. However, for given values of r a, since θ is increasing in n (the measure of employed, as seen by (20)), it is decreasing in u = 1 n. And, the MP curve (19) which gives a positive relationship between q and θ gives a negative relationship between q and u. Intuitively, when q is higher, profits and hence the benefit from opening a vacancy are higher, so unemployment falls. Thus, the MP curve slopes downwards in the (u, q) space. Recall that we assume that τ(r a + δ) < (f w + η(u(q ) c(q ))), otherwise firms won t enter. Retail Goods Market The other market is the retail goods market where for given values of r, the equilibrium is given by (11). Re-writing the equation: ( ) u (q) c (q) ι = σ g (n)n(1 η), z η(q) This gives a positive relationship between n and q and is reminiscent of the LW curve in BMW. LW is for Lagos-Wright (2005), because the equation is the same as in that paper, except that we are using proportional bargaining and the matching probability, σ g is not fixed, but it depends on n. The LW curve thus gives a negative relationship between q and u, and it slopes downwards in the (u, q) space. Intuitively, the higher the u, the lower is the probability that a household matches with a firm in KW matches, which lowers the demand for money and hence reduced q. And, as is well-known, given u, the LW curve implies that q is decreasing in ι. When u = 1, q = 0. 21

23 Thus, both curves slope downwards in the (u, q) space in the box B = [0, 1] [0, q ]. The analysis is the same as BMW, except that we do not have bargaining for wages and have proportional bargaining in retail market and of course r is endogenous. reproduced from BMW as shown in Figure 3. q The figure can be q MP 1 MP 2 MP 3 LW 0 1 u Figure 3: Labor Market and Goods Market Equilibrium Note that the LW curve enters B from the left at (0, q 0 ) where q 0 q and exits from the right at (1, 0). The end-point of the MP curve depends. Shifts in the two curves are discussed under the comparative statics of the general equilibrium framework. Depending on the parameter values implying different MP curves but the same LW curve, we can have different equilibria. In case of MP 1, there is a nonmonetary equilibrium and at least one monetary equilibrium. In case of MP 2 and MP 3, there exists a nonmonetary equilibrium at (1, 0) and monetary equilibria may also exist. Generally speaking equilibrium always exists but need not be unique. If monetary equilibrium is not unique, I focus on the one with the lowest u. 7.3 General Equilibrium We now combine the two channels: the first is the interest rate channel from RR and the second is the productivity channel from BMW. Using the analysis from above (the determination of r from RR using liquidity demand and supply curves, and u and q from BMW using the MP and LW curves), the full set of equilibrium equations of the model economy is a system of seven 22

24 equations in seven unknowns as follows: Definition 1. A stationary monetary equilibrium can now be defined as a list (q, z, z d, y, θ, n, r) solving (8), (11), (19), (20), (24), (32) and (35). The determination of equilibrium was discussed above, in particular we get a unique r given q from the goods market. For a complete characterisation of the equilibrium, we also include n, which is determined by (20) and gives the measure of steady-state employment. So, instead of θ, we do the analysis in terms of n or u = 1 n, as in BMW. Also, output in the trader s sector, y is obtained from (24) and real balances carried by households, z comes from (8). Finally, in case the economy is caught in a liquidity trap, z d is different from zero, in particular it is determined from (32). We can now do some comparative statics and monetary policy analysis. As the bargaining power of firms, η increases, more firms enter and the MP curve shifts to the left. And, households hold fewer real balances and retail output falls shifting the LW curve downwards. So, the final impact on q and u is unclear. And, thus on r. Liquidity demand is unchanged, and if liquidity supply shifts out, then r increases, else decreases. Consider an increase in firm s productivity, p. Firms become more valuable, and more firms enter so θ increases and MP curve shifts downwards, and u decreases and q increases (if the equilibrium is unique or in the one with the lowest u). The supply of liquidity also increases leading to an increase in r (if r < ρ), and offsetting some of the increase in θ, but overall θ increases. Next consider an increase in cost of entry for firms, τ. Fewer firms enter so θ decreases and MP curve shifts upwards, and u increases and q decreases (if the equilibrium is unique or in the one with the lowest u). The supply of liquidity also decreases leading to a decrease in r (if r < ρ), which offsets some of the decrease in θ, but overall θ decreases. Wage, w has the opposite effects as p, and same as τ. So does match separation rate, δ. Consider an increase in trader s and household s discount rate, ρ. Liquidity demand decreases as households get more impatient, this causes an increase in r and a decrease in liquidity supply and θ. This shifts the MP curve upwards. Since ι is also affected, i.e. it increases, the 23

25 LW curve shifts downwards. This implies that u increases and q decreases (if the equilibrium is unique or in the one with the lowest u). Matching probability in trader s market, α has the opposite effect as ρ in the liquidity market. Liquidity demand increases as matches get more frequent, this causes an decrease in r and an increase in liquidity supply and θ. This also shifts the MP curve downwards, but the LW curve remains unchanged, with u and q falling. This reinforces the initial effect on θ. Monetary Policy A one-time change in money supply, M is neutral as φ m adjusts in (11) to keep z = φ m M the same. Thus, an open market operation that swaps B for M has an effect only from increase in B. Increase in supply of government bonds, B leads to an increase in L s, so r increases from (35) provided L e y. This leads to an upward shift in the MP curve, while the LW curve remains unchanged, with u and qincreases. This reinforces the initial effect on θ. Thus, public liquidity crowds out private liquidity, and there is a multiplier effect from the effect from the retail market. Finally, consider a change in the growth rate of money supply γ (or ι given Fisher equation): As γ increases, money becomes costly to hold, and ι increases, the LW curve shifts downwards. Unless the economy is in a liquidity trap, there is no direct effect on the trader s market and the liquidity demand and MP curves remain unchanged. This implies that u increases and q decreases (if the equilibrium is unique or in the one with the lowest u). The additional fall in q comes because there are fewer firms and hence fewer matches for households. But, since, θ decreases, L s a and r decreases (provided r < ρ) implying that the initial effect on θ gets dampened by the decrease in r, and liquidity supply does not fall by as much if the rate of return were fixed. The last channel is missing from BMW. RR is unable to study this at all in its benchmark as they do not study a monetary economy in the benchmark specification. Overall we see that the on including both channels, we get a way to amplify the effects in RR by adding the productivity channel as firm entry creates multiplier effects. We also see a dampening of the effects in BMW, because firm entry affects the interest rate which in turn offsets some of the earlier effect. Thus, we see how money supply affects firm entry and also 24

26 interest on claims. 8 Differences in Liquidity Now consider the case when there are differences in acceptability of the three assets: with α a probability, a buyer meets a seller who accepts all three, and with α b accepts bonds and money and with α m only money is accepted. Each probability is adjusted by the overall matching probability, α. This is the more general case, but to generate liquidity traps and as a benchmark we looked at the case when all assets were equally acceptable. While once we include differences in acceptability, changes in money growth can directly affect the trader s liquidity market. One can follow the same steps as before (and use 1/(1 + r m ) = (1 + γ), where r m is return on money) to get the following FOCs for the DM goods market: 1 = β(1 + r m )[1 + α m (u (y m ) 1) + α b (u (y b ) 1) + α a (u (y a ) 1)], (36) 1 = β(1 + r b )[1 + α b (u (y b ) 1) + α a (u (y a ) 1)], (37) 1 = β(1 + r a )[1 + α a (u (y a ) 1)], (38) where y m = min{z d, y }, y b = min{z d +(1+r b )b, y } and y a = min{z d +(1+r b )b+(1+r a )a, y }. We get the following rate of return differences across assets from the above equations: r a r b = α b (u (y b ) 1)]. (39) r b r m = α m (u (y m ) 1)]. (40) We get that private claims dominate bonds in their rate of return, r a > r b provided that α g > 0 and y g < y. Thus, bonds carry a higher liquidity premium than claims as bonds are more liquid. Similarly, bonds dominate money in their rate of return, r b > r m provided that 25

27 α m > 0 and y m < y. Note that r m = γ/(1 + γ), so money has a negative rate of return as it yields no dividend; people carry it for its liquidity property. Or, we can do the analysis in terms of different interest rates and spreads. Recall that real spread is S b = ρ r b /(1 + r b ) in relation to return on an illiquid bond (similarly for private claims). Using that we get: ι = α m (u (y m ) 1) + α b (u (y b ) 1) + α a (u (y a ) 1)], (41) S b = α b (u (y b ) 1) + α a (u (y a ) 1)], (42) S a = α a (u (y a ) 1)], (43) Also, note that liquidity trap will not take place in this case, as buyers will always hold some money. Liquidity Demand and Supply with Differences in Liquidity Demand for liquidity now depends on the types of matches and is given by (36), (37) and (38) as a function of the different rates of return. We can see that demand is increasing in rate of return. We can also construct an aggregate measure of effective liquidity from the above following Friedman and Schwartz (1970) as in Rocheteau and Rodriguez-Lopez (2014). L = φ m M +(α b + α a )(1 + r b )B + α a (1 + r a )L s a, or L = α m L d m + α b L d b + α al d a. To get different rates of return the supply of actual liquidity for different matches is to be written. We get, L s m = φ m M, L s b = φ mm + (1 + r b )B and L s a = φ m M + (1 + r b )B + (1 + r a )L s a. Also the aggregate DM output is i α iy i. The productivity channel in this case will remain unchanged from the benchmark. The interest rate channel will change in that all assets have different rates of return. We can now define the equilibrium in this case. Definition 2. A steady-state monetary equilibrium is a list (q, z, z d, y i, θ, n, r b, r f ) that solves (8), (11), (19), (20), (24), (36), (37) and (38). 26

28 Monetary Policy As before, a one-time change in money supply, M is neutral as φ m adjusts in (11) to keep z = φ m M the same. Thus, an open market operation that swaps B for M has an effect only from increase in B. Increase in supply of government bonds, B leads to an increase in L s, so r b increases, and so does r a provided that L s a < y. This leads to an upward shift in the MP curve, while the LW curve remains unchanged, with u and q increases. This reinforces the initial effect on θ. Thus, public liquidity crowds out private liquidity, and there is a multiplier effect from the effect from the retail market. Money holdings, z d by buyers also falls, and so does output in matches where only money is accepted, y m. Output in matches accepting bonds, y b and y a rises. The effect on aggregate KW output is ambiguous. Since now y b and y a increases, S b and S a goes down. When buyers are unconstrained in type- a matches (L s a y ), but constrained in type-b (L s b < y ), then same effect on money but no effect on private liquidity. If unconstrained in types-b and a (L s b y ) then no effect. Finally, consider a change in the growth rate of money supply γ (or inflation or nominal interest rate, ι given Fisher equation): As γ increases, money becomes costly to hold, the LW curve shifts downwards. The other effect is on the traders, who also hold fewer real balances, z d and hence lower output from matches in which money can be used, in the above case all matches accept money. Since now y b and y a decreases, S b and S a goes up and return on bonds and claims falls. Effect on the nominal return will be unclear. The decrease in r a implies that more firms enter and the effect of inflation on the retail market is potentially countered,as u decreases. When buyers are unconstrained in type- a matches (L s a y ), but constrained in type-b (L s b < y ), then same effect on return on bonds but there is no effect on private liquidity. Hence, overall unemployment rises in response to inflation as in BMW. If unconstrained in types-b and a (L s b y ) then private assets and bonds provide no liquidity on the margin and there is no effect in the trader s market except that y m and z m decreases. 27