The Size of the Central Bank s Balance Sheet: Implications for Capital Formation and the Yield Curve

The Size of the Central Bank s Balance Sheet: Implications for Capital Formation and the Yield Curve Juan C. Medina and Robert R. Reed University of Alabama April 2014 Abstract The tools used by central banks have evolved considerably in recent years. One important aspect of nonconventional policy has been the move of central banks to aggressively purchase longer-term government securities along with expanding the size of its balance sheet. Both fall under the category known as Quantitative Easing. In this paper, we construct a rigorous modeling framework to investigate the impact of such policies. The inclusion of long-term assets and an independent central bank allows us to focus on a Treasury bond purchase program as a monetary policy tool. Our findings suggest that the central bank s balance sheet expansion through a Treasury debt purchase program may positively affect capital accumulation in the economy and has a significant impact on the term structure of interest rates. In so doing, we also explain how the transmission channels of monetary policy between unconventional and conventional policies have important consequences for risk-sharing in the economy. Preliminary Draft - Please do not cite or quote without explicit permission from the authors. Juan C. Medina, Department of Economics, Finance, and Legal Studies, University of Alabama, Tuscaloosa, AL 35487; Email: jmedina@crimson.ua.edu; Phone: (205) 348-7590. Robert R. Reed, Department of Economics, Finance, and Legal Studies, University of Alabama, Tuscaloosa, AL 35487; Email: rreed@cba.ua.edu; Phone: (205) 348-8667. 1

1 Benchmark Model of Treasury Purchases We begin by establishing a benchmark model that will allow us to analyze conventional monetary policy instruments within a well-known random relocation framework as that in Schreft & Smith(1997). By doing so, we intend to show the potential of this structure to exploit important institutional features of monetary policy as well as its limitations to analyze policies of the unconventional type. In this manner, it will become more palpable what obstacles we need to overcome so as to adapt and expand this framework into the analysis of LSAP s. 1.1 The Environment Consider an economy in which time is discrete t = 1, 2,... and two geographicallyseparate locations coexist. Each is populated by a unit-mass continuum of twoperiod lived agents that exert labor effort when young and value consumption of a homogeneous good when old. Given an initial population of old, the economy has an infinite stream of individuals within each location such that an overlapping generations structure prevails. Agent s preferences are described by u(c t ) = log c t and we assume they are endowed with, and inelastically supply a unit of labor for which they earn a wage, w t. In every period, each young agent has a probability π (0, 1) of moving to the other location. Invoking the law of large numbers, this implies that a fraction π of young individuals is randomly relocated every period so that consumption when old takes place in their new location. The financial sector is represented by a market for deposits in which private banks accept agent s labor income in exchange for a return at a future date. These institutions offer competitive returns on deposits as a result of them engaging in Bertrand-like competition. That is, in a Nash equilibrium, rates of return maximize depositors surplus by exhausting all economic profit. Following Schreft & Smith (1997), limited communication among banks between the two locations as well as limited asset portability confronts moving agents with an anonymity problem that eliminates the possibility of trading privately issued liabilities. In constrast, fiat money is the only asset accepted in both locations (independently of its origin) and which can be exchanged for the consumption good. Thus, movers must cash out all asset holdings and carry money balances to their new site despite money being return-dominated by all other assets in the economy. Ex-ante then, individuals are subject to reloca- 1

tion risk, which is meant to represent liquidity preference as in Diamond and Dybvig (1983). Financial intermediation is performed by private banks who provide insurance against the aforementioned risk by pooling income resources and offering returns on deposits contingent on relocation status. In doing so, the banking sector is able to provide optimal risk sharing and individuals are better off participating in financial markets relative to autarky. There exists fiscal and monetary authorities in charge of economic policy. The former generates its stream of revenue from issuing risk-free bonds B t in financial markets as well as from supplementary transfers by the monetary authority. Letting the real value of bonds be b t B t /P t, where P t is the unit price of the consumption good, a holder of a unit of government debt at t 1 has a sure claim to R t units of consumption in period t. Hence, debt payouts constitute the liability side of the fiscal authority. The monetary authority or central bank (CB) implements policy based on two instruments. First, it controls the supply of money in the economy according to the rule M t = M t 1 where M t denotes the period t nominal money supply per agent and > 0 the rate of money growth between adjacent periods. Real money balances m t M t /P t, offer a rate of return that depends on the economy s relative price level P t 1 P t. The second instrument derives from the CB being a major buyer of government debt obligations and actively engaging in a bond purchase program. As per the bond s maturity structure in this framework, these purchases can be seen as outright open market operations. Investment demand in government debt is such that every period the CB along with private banks exhaust the supply of bonds in financial markets. A constant fraction λ of net revenue that the CB generates gets transferred to the fiscal entity in every period. We consider this value as an institutional parameter pertaining to the monetary authority which can be interpreted as the degree of CB resource independence. 1.2 Factor Markets Productive inputs in the economy are comprised of labor L t and capital K t. Labor effort is exclusively provided by young agents and capital is an asset that offers a rate of return of r t and completely depreciates at the end of each period. Output of a homogeneous consumption good is realized through a constant returns to scale production technology F (K t, L t ). Notationally, we implement it in its standard capital-intensive form f(k t ), where k t K t /L t, f (k t ) > 0 and 2

f (k t ) < 0. The market for factors of production is perfectly competitive and hence each input is paid its marginal contribution to the output process. In this manner, w(k t ) = f(k t ) k t f (k t ) for labor and capital, respectively. r t = f (k t ), (1) 1.3 Timing of Actions Agents work when young. Factors are paid. Banks make allocation decisions. Relocated agents withdraw from banks. Consumption occurs. t t+1 Production occurs. Young agents make bank deposits. Relocation shock is realized. Relocation occurs. Figure 1: Depositor s timing of actions Figure 1 above illustrates the development of events that take place within an agent s two-period life span. During their young stage, those who are born at the beginning of period t provide labor effort so that along with capital, production of the homogeneous consumption good takes place. Afterwards, factors of production get paid according to (1.2) and (1) from which young workers deposit all of their labor income in private banks. Because the fraction of young individuals that are going to relocate is publicly known, banks use this information to determine the optimal allocation of deposits into currency and other assets during this phase. At the end of this initial period and when the relocation shock takes place, every young depositor who needs to migrate withdraws their deposit. However, given the constraints to communication among banks and asset portability, 3

movers will only demand cash. At the end of this first period then, relocation occurs. When period t+1 arrives old nonmovers withdraw deposits whereas their moving counterparts exchange cash for the consumption good with banks but in their new location. Finally, all old agents consume and exit the economy at the end of the period. 1.4 Financial Intermediation Among and between locations, banks and their market structure are identical. This assumption allows us to focus on a single representative bank that offers returns on deposits. A bank can participate in the market for deposits by offering rates of return to its clients as dictated by a perfectly competitive financial sector. Together with the assumption of zero transaction costs, the profit maximization program for this institution translates into one of surplus maximization for its depositors. Consequently, banks compete for deposits w(k t ) announcing an overall rate of return d m,t for the π movers and d n,t for the remaining 1 π non-movers. Through financial intermediation the consumption available to an agent becomes c m,t = w(k t )d m,t (2) if a mover, and c n,t = w(k t )d n,t, (3) for a nonmover. Banks allocate income deposits into the three different assets in the economy: real money balances in the amount m t, capital per depositor i t and government issued bonds b P t, such that m t + i t + b P t w(k t ). (4) Moreover, banks face additional constraints in order to guarantee that the actual returns are in line with those offered to the moving and nonmoving populations. That is, for those who relocate, it must be that p t πw(k t )d m,t m t (5) p t+1 4

and (1 π)w(k t )d n,t r t i t + R t b P t (6) otherwise. Under this scenario, the representative bank s optimization program is subject to (4) - (6). max {π ln [w(k t )d m,t ] + (1 π) ln [w(k t )d n,t ]} d m,t,d n,t The solution to this maximization problem yields the following first order conditions: R t r t = 0. (7) and m t /w t π = 0. (8) Equation (7) exemplifies the no-arbitrage condition that makes bank s investment allocations indifferent between bonds and capital. The efficient risk-sharing condition in (8) allows banks to insure agents against liquidity risk by allocating an amount of cash balances equal to the fraction of moving individuals. In this manner, banks contribute to the agent s welfare by reducing their consumption volatility relative to the absence of financial intermediation. 1.5 The Central Bank Monetary policy is the sole responsibility of the monetary authority represented by the CB. As previously mentioned, its instruments derive from the ability to control the money supply as well as from acquiring government debt obligations. In any given period, the CB follows a constant money growth rule so that inflation ) allows for seignorage revenue: mt. The CB also participates in financial ( 1 markets by purchasing government bonds in the amount b M t. Another operational feature of the CB is that it transfers a fraction λ of its net revenue to the fiscal authority in every period. Through this resource injection, the CB can have an indirect influence on the impact of fiscal debt in the economy. Recalling R t as the return on government securities, the CB s resource constraint can be depicted as ( ) 1 Rt 1b b M t 1 + m t b M t > 0. (9) 5

The first two terms in (9) represent the total revenue inflow from the return to bonds and seignorage revenue for t 0. The third term depicts expenses incurred in the purchase of government debt during the same time period. This resource constraint is strictly positive since the CB can t impose losses on contributors. 1.6 The Fiscal Authority The government in this economy is the exclusive issuer of riskless short-term bonds b t in financial markets and honors these liabilities by offering a rate of return to its holders. As pointed out above, demand for these assets comes from private banks and from the CB and it is such that it exhausts the supply of bonds: b t = b P t + b M t, for all t > 0. (10) In such an array, the government obtains its income stream from the sale of debt obligations as well as from CB transfers. Specifically, the latter represents an injection of a fraction (λ) of the CB s net receipts as depicted in (9). Similarly, the expense side of the fiscal authority s budget constraint consists of payments disbursed to honor its debt obligations at maturity. In equilibrium, government revenue equals expenses so that: ( ) 1 R t 1 b t 1 = b t + λ[r t 1 b M t 1 + m t b M t ], for all t > 0. (11) 1.7 Steady State Equilibrium We now attempt to arrive at the minimal set of equilibrium conditions that will capture the dynamics of the economy. First, currency as well as bond markets clear by equations (8) and (10), respectively. Because i t is defined as the share of deposits that is invested in productive capital, we also have i t = k t. Consequently, using the private bank s balance sheet it must be that k t = w(k t ) m t b P t for t 0, (12) which highlights that in private bank s portfolios, investment in productive capital declines as demand for liquidity and/or government debt increases. These asset-substitution dynamics can potentially lead to Tobin-type effects as banks seek to maximize returns for their clients. Similary, there is also potential for 6

crowding out as government debt can distort the allocation of the productive asset in the economy. Next, using the fiscal authority s budgetary constraint (11) along with (8) and (10), we can solve for the amount of government bonds purchased by private banks, b P t which yields b P t ( ) 1 = λ m t (1 λ)b M t, (13) I t where I t P t+1 P t R t. A closer look at these results allows us to see two important aspects related to the dynamics of fiscal debt. First, the CB can absorb government debt that otherwise would be acquired by private banks. This implies that the monetary authority has a tool that can act as a buffer for government debt distortions. Second, and net of CB transfers, the rate of money growth allows the fiscal entity to issue even more liabilities so that we have inflation-financed debt. Next, we use (13) into (12) and this will allow us to capture part of the steady state dynamics into (I, k) space as follows: [ ] ( 1) k = w(k) (1 π) λπ + (1 λ)b M. (14) I Equation (14) shows the steady state level of capital in the economy depending explicitly on the rate of money growth and bond purchases by the CB. Additionally, the stock of capital is negatively affected by the nominal return on government debt. As increasing debt levels imply higher borrowing costs for the fiscal authority, a potential pathway for a crowding out effect remains. We also have a no-arbitrage condition as described by (7) that needs to be satisfied in steady state. As in this type of equilibrium it is trivially true that = P t+1 /P t, using the capital market condition implied by (1) and in sight of (7), we can embed the no-arbitrage condition through the gross nominal return to government debt which results in I t = f (k t ) for t > 0. (15) Together then, (14) and (15) capture the dynamics of the economy in a steady state equilibrium. The following proposition provides the conditions for multiple steady states to exist. Proposition 1. Let ˆk be such that ˆk = w[1 π(1 λ)]+(1 λ)b M and ˆ = 1/f(ˆk). 7

Then if > ˆ two steady states exist in which money is valued and where the government is a net lender in the first and a net borrower in the second. We offer proof of the above proposition in the appendix. Figure 1 below illustrates the existence of multiple steady state equilibria with distinctly opposing features. In equilibrium A for instance, a highly distorted economy prevails as characterized by a low level of capital and high nominal interest rates. Considering (13) in steady state, and because I > in this regime, there is a positive amount of bonds outstanding b > 0 which makes the government hold a net debtor position in financial markets. In contrast, the economy represented by the equilibrium shown in B, exists in a more efficient financial sector per a low interest and high capital environment. Furthermore, if the equilibrium I has an upper bound I <, where = ( 1)m/b M, then b < 0 and the government holds a net lending position which we assume is the case. We can think of this last scenario as that pertanining to an advanced economy whose financial sector is characterized by low borrowing costs that allow for a high stock of capital. Consequently, our equilibrium in A can be representative of a less developed economy with a low capital stock. I (15) (13) I 0 A 1 B (13) k ˆk k Figure 2: SS Equilibria 1.8 Comparative Statics 1.8.1 Money Growth Rule We begin our analysis by illustrating how the effects of a money growth rule policy is transmitted to the real sector. Figure 2 shows two differing outcomes 8

depending on which type of equilibrium the economy is situated in. For the low capital case, an increase in the rate of money growth enhances the government s ability to issue debt. However, due to (10), the bond market clears so that an increased bond supply gives way to higher nominal yields which ultimately crowd out capital accumulation. In contrast, in our high-capital, low-interest steady state regime, an increase in contributes to capital formation. Because the government is subsidizing economic activity through its net lending status, an increase in the rate of money growth erodes the real return on bonds which by (7) implies a reduction in the cost of the productive asset. In turn, this effect translates into a net increase in the stock of productive capital albeit a higher nominal interest rate level. Figure 2 below illustrates these effects. Proof of these results are shown in the appendix. I I 0 Original paths I 1 0 1 k 1 k 0 ˆk0 k Figure 3: SS Equilibria & Money Growth Rule 1.8.2 Treasury Purchase Program When the CB engages in a bond purchase program, capital formation experiences nontrivial effects. Unlike a money growth instrument, government debt acquisition by the CB unambiguously increases capital formation in both equilibrium steady states. In the low capital economy, this policy action reduces the supply of goverment debt available to private investors. In turn, financial intermediaries are now able to allocate greater resources into productive capital. This allows a lower rate of marginal productivity to drive down nominal interest rates throughout the economy in the absence of no-arbitrage opportunities. Within the high capital economy and given that the cost of borrowing is lower 9

than the inflation rate, government security purchases by the CB represent lending (or subsidy?) to the fiscal authority at a low interest rate. As the government stands as a net lender in financial markets, this low cost borrowing translates into additional resources available for private investment in the productive asset. Figure 3 below illustrates these findings. I I 0 Original paths b m I 1 1 k 0 k 1 ˆk0 ˆk1 k Figure 4: SS Equilibria & Treasury Purchases (this is FIGURE 3) 2 Conclusion The effect of bond purchases on capital work differently when compared to the policy outcomes such as the use of a money growth rule. While the stage of economic activity matters for inflationary policies, it is not relevant for a government bond-purchase policy. These results suggest that no matter the development level of the economy, a CB taking on a bond purchase program has a positive impact on capital formation. The monetary authority s involvement in bond markets allows financial intermediaries to alleviate the crowding out effects of government debt and are able then to channel more resources into productive investment. Since this policy reduces the value of government debt by putting downward pressure on interest rates, it is an important tool that can be used to stimulate economic activity. Still, our benchmark model is limited in that it does not reflect a more realistic term structure of interest rates. This feature is crucial in one wants to start any discussion of unconventional monetary policy through the expansion of the CB s balance sheet. Since one of the main objectives of the latter is to 10

target interest rates at different time horizons, we must allow for a different OLG structure in our modeling framework. 3 A 3-Period Model Having shown the reach and limitations of the benchmark model, we now proceed to expand it. Recall that the main objective of this analysis is to offer a framework to analyze unconventional monetary policy as framed by the CB engaging in a longer-term government securities purchase program. Hence, in our framework, we allow for the existence of long term assets by including another period in the life cycle of agents. That is, we use a three-period overlapping generations structure that allows individuals to hold short-term, one-period maturity assets as well as long-term, or two-period maturity assets. The role of the central bank remains the same in the sense of it being an independent monetary authority that complements the fiscal authority s income every period. 3.1 The Environment Relative to our benchmark model, we extend the life-cycle of the unit-mass continuum of depositors in the economy so that their lifespan comprises 3-periods. Now, individuals are born into a young phase after which a middle-age stage follows and then, their last existing period develops as implied by an old stage. We assume the existence of an initial population of each type out of which the old agents hold M 0 units of currency. At any given t > 0, it is the case that individuals still provide labor effort exclusively when young and value consumption when old according to u(c t ) = log c t. In this case however, relocation risk is experienced by a fraction π of middle-aged depositors. Because anonymity and limited asset portability restrictions remain, individuals still experience a trading friction. As a consequence the role of financial intermediaries as providers of risk-pooling services exist under the same competitive market structure. The physical features of the economy remain the same as in our benchmark model except for the inclusion of an additional long-term asset whose details follow. The fiscal authority issues debt obligations with different maturity structures. For instance, while it continues to issue one-period or short-term bonds which we now denote as b 1 t and with rate of return Rt 1, it also sells longer-term, two-period 11

maturity bonds in amount b 2 t. Specifically, a holder of a unit of the latter type of bond at period t is entitled to R 2 t+2 units of consumption in period t + 2. The CB maintains its monopoly over the supply of fiat currency in the economy under the aforementioned money growth rule. It also buys short and long term government debt in quantities b 1M t and b 2M t,respectively. In this manner, the CB is able to implement policy by controlling the supply of money, conducting open market operations and pursuing a long-term government bond purchase program. The monetary authority continues to complement government revenues by ceding a fraction λ of its net revenues in every period. 3.2 Factor Markets We preserve the production technology studied in the base model as well as the competitive structure in factor markets. However, an important modification is introduced: investment in capital takes two periods to materialize offering a long-term (two-period) return of r units of the consumption good in period t + 2 for every unit of k rented at period t. As capital is completely consumed after production of the consumption good takes place, this implies that return to inputs follow r t (k t 2 ) = f (k t 2 ) (16) and w(k t ) = f(k t 2 ) k t 2 f (k t 2 ). (17) 3.3 Timing of Actions The sequence of events and actions in the economy is now explained. Following the timeline described in Figure 4, agents provide labor effort when young. After production takes place, labor earns wage income and returns to capital (invested two periods before) are realized. These young workers participate in financial markets by depositing all their income in banks. As intermediaries, banks pool these resources and invest them in public and private assets that offer a return at a future date. As the second or middle aged period arrives, agents learn their relocation status. Due to the same communication and portability issues in the economy, movers at this middle stage liquidate their investments from banks and exchange 12

Agents work when young. Factors are paid. Banks make allocation decisions. Relocated agents withdraw from banks. Consumption occurs. t t+1 t+2 Production occurs. Young agents make bank deposits. Relocation shock is realized. Relocation occurs. Figure 5: Depositor s timing of actions them for cash with previous generation s old movers. For the nonmoving population, intermediares maintain their investments in financial markets until old. Finally, when the latter period arrives, agents realize consumption based on returns to assets held and finally exit the economy. 3.4 Financial Intermediation After private banking institutions receive income deposits, they create private investment portfolios consisting of allocations in capital by amount i t, government bonds of one period b 1P t and two period b 2P t maturities. 1 These assets earn a nominal return of I 1,t Rt 1 (P t+1 /P t ) if short-term and I 2,t R 2 t (P t+2 /P t ), otherwise. The distribution of asset investments among depositors depends upon the liquidity preferences of its clients. For instance, the bank sustains returns to relocated individuals only through short-term debt claims b 1P m,t and whose return allows them to purchase money balances. Similarly, nonmover s investment portfolios consist in part of government bonds b 2P n,t and capital i 2P n,t. Moreover, they have a preference for a more liquid, short-term debt asset b 1P n,t whose returns are reinvested for an additional period when consumption takes place. In this manner, a representative bank offers a rate of return on deposits of d m,t if relocating and d n,t otherwise. Therefore, consumption for movers and nonmovers becomes: 1 The letter P in superscript denotes private bond holdings. c m,t = d m,t w(k t ) (18) 13

and c n,t = d n,t w(k t ), (19) respectively. The private bank s balance sheet denotes the feasibility of investments so that w(k t ) b 1P m,t + b 1P n,t + b 2P t + i t. (20) Because return to deposits is contingent on relocation status, banks must make sure that their investment allocations sustain such payouts. In this manner, returns to movers depend on realized short term bond claims which must be exchanged for cash balances that earn P t /P t+1. This condition is represented in (21). πw(k t ) d m,t ( ) Rt+1b 1 1P P t m,t. (21) P t+1 For the nonmoving agents, the promised bank returns are sustained by investment profits arising from long term bonds, capital and the reinvestment of short term bonds: (1 π)w(k t ) d n,t ( ) Rt+1b 1 1P n,t R 1 t+2 + Rt+2b 2 2P n,t + r t+2 i t. (22) Based on these conditions along with (18) and (19), banks offer rates of return for movers d m,t and nonmovers dn,t that solve { max π ln [w(kt ) d m,t ] + (1 π) ln [w(k t ) d n,t ] } (23) d n,t d m,t, subject to (18) - (22). In turn, the solution to the bank s problem yields the following first order conditions: R 2 t+2 r t+2 = 0 (24) and R 1 t+1r 1 t+2 r t+2 = 0 (25) Where (24) establishes that optimal bank allocations in long term assets and capital eliminate arbitrage profit opportunities. Likewise, (25) upholds this fea- 14

ture on short-term asset reinvestments relative to capital. Combining these two conditions allows us to determine the relationship between bond yields and their different maturities: R 1 t+1r 1 t+2 = R 2 t+2. (26) Equation (26) represents the basis for a term structure of interest rates through which a bond yield curve can be derived. By solving (23), the private bank s optimal allocation of short term bonds for its moving clientele is determined by the fraction of middle aged individuals who have to relocate b 1P m,t = πw(k t ), (27) which gives not only the private bank s short-term bond demand for movers but also the optimal risk sharing rule. As per our timing of actions, a sudden demand for liquidity by individuals during their middle age obligates these now movers to cash out redeemed bond holdings. Along this event, a middle aged mover in period t + 1 demands real money balances in the amount m t+1 = R 1 t+1b 1P m,t, (28) which will be exchanged for the consumption good when old in their new location. 3.5 The Central Bank The CB retains the policy tools just as in our baseline model. That is, it still controls the rate of money growth which allows for seignorage revenue as well as the implementation of open market operations. In this richer setup however, the CB is able to access a third policy instrument that emanates from its ability to purchase long-term government securities in financial markets. Specifically, the CB can considerably reduce the available supply of outstanding long term debt by an amount b 2M t. Additionally, and as previously mentioned, the CB transfers a fraction λ of its net receipts to the fiscal authority in every period. Given thesee characteristics, the CB s resource constraint is depicted as 15

R 1 t 1b 1M t 1 + R 2 t 2b 2M t 2 + ( 1 ) m t b 1M t b 2M t > 0. (29) Revenues in (29) come from two sources: income from the return to government debt as illustrated by the first two terms and seignorage revenue by the third. The last two terms represent expenses incurred in the purchase of short and long-term government bonds. We again establish this constraint as strictly positive refraining from the CB imposing any losses to contributors. 3.6 The Government As introduced, fiscal debt obligations in the form of riskless bonds can be redeemed within one or two periods depending on their maturity structure. The sum of outstanding short term bonds b 1 t and long term bonds b 2 t constitutes the period t total supply of bonds in the economy. From the demand side, the CB as well as private banks exhaust the supply of fiscal debt at the given short term real rate of return Rt 1 and similarly but for long term bonds, Rt 2. It follows that at any t > 0 and for all maturities, bond markets clear such that: b 1 t = b 1P m,t + b 1P n,t + b 1M t (30) and b 2 t = b 2P t + b 2M t. (31) The government continues to have another income source coming from the fraction λ of the CB s net revenues described by (29). Similarly, but on the expense side, the government is responsible for fully honoring its total debt obligations at the prevailing interest rates. At any time period, the government budget is balanced so that [ b 1 t + b 2 t + λ Rt 1b 1 1M t 1 + Rt 2b 2 2M t 2 + ( 1 ) ] m t b 1M t b 2M t = R 1 t 1b 1 t 1 + R 2 t 2b 2 t 2. (32) 16

4 Steady-State Analysis Using the definition of i t as the share of capital allocated in bank s porfolios, we let i t = k t. Also, by recurring to the private bank s short-term bond demand (27) we can rewrite the bank s balance sheet constraint (20) in steady-state as k = w(k)(1 π) b 1P n b 2P. (33) Not surprisingly, similar to our baseline model there is still a potential displacement of productive capital in bank s portfolios when opting for an increased investment mix of government securities. Bond market equilibrium conditions (30) and (31), positions the CB as an important buyer of public debt. To see how such acquisitions affect private bond demand, we focus on the constraint of the fiscal authority (32). First we note that given (26), (27)and our interest rate definitions, it holds that I 1,t 1 I 1,t = I 2,t. It follows that in a steady state (I 1 ) 2 = I 2 and thus the government s budget constraint (32) becomes: where b 1P n + (I 1 + )b 2P = δ(i 1 )w(k) (1 λ) [ b 1M + (I 1 + )b 2M], (34) δ(i 1 ) π [ I 1(1 λ( 1)/)], and I 1. (35) I 1 This expression shows that yet again asset purchases by the CB can potentially reduce private investment opportunities in government debt. We now analyze the steady-state equilibrium in the economy. Using (34) and (33) we derive the evolution of the capital stock as a function of asset purchases by the monetary authority: [ ( ) ( )] 1 I1 k = w(k) 1 πλ + (1 λ)b 1M + I 1 [ ( )] ( ) I1 + 1 λ b 2M I1 + b 2. (36) Similar to our baseline model, the steady state stock of capital is dependent on the degree of fiscal independence by the monetary authority, λ, the rate of money growth, as well as the prevailing interest rate level in the economy I 1. However, and as an improvement over our baseline model, there is an explicit role for the 17

CB s long term security purchases in the capital formation process. Moreover, we can see that the marginal effect of the different types of government liabilities seems to differ suggesting the existence of an interest rate threshold effect. Now, in a steady state and invoking (16), (24), as well as the term structure depicted in (26), it follows that I 1 = f (k). (37) Hence, we now have the no-arbitrage condition in the economy as a function of the capital stock. In this manner, we can collapse the dynamics of the economy into (I 1, k) space through the use of (36) and (37). Figure 5 shows the steady state equilibrium diagram of the economy and qualitative details are reserved for the appendix. The existence of multiple equilibria in which money is valued is illustrated by points A and B. Note that we can separate these cases by describing them in the same manner as in our benchmark model. That is, in A, a low steady-state capital stock economy with a high interest rate regime exists and viceversa for the economy in point B. Also, the same opposing features in terms of the government s lending position arise. That is, in A, the fiscal authority is a net borrower but in B, a net lending position is sustained when the CB provides enough resources coming from seignorage revenue, that is ( 1 ) m > ( I1 which we will assume is the case. ) [ b 1M + ( I1 + ) b 2M ], for I 1 <, (38) Proposition 2. Let ˆk be such that ˆk = (1 λ)b 1M + 1 [( + πλ)w(ˆk) + ( λ( + 1))b 2M + b 2 ] and ˆ 1/f (ˆk). Then if > ˆ, two steady states exist in which money is valued and where the government is a net lender in the first and a net borrower in the second. We offer proof of the above propositon in the appendix. The differing capital stocks found in our steady state equilibria allow us to discriminate the effectiveness of a given monetary policy instrument relative to the economy s stage of development. In this manner we can seek conditions under which an inflationary or a government asset purchase policy would produce similar outcomes in developed and developing economies. 18

I 1 (22) (21) A I 0 1 B (21) 1 ˆk k Figure 6: Steady state diagram 5 Monetary Policy Instruments 5.1 Money growth rule We start by considering the effects of a monetary growth policy on capital formation as exemplified on our model by an increase in. Through this instrument, the CB influences the price level in the economy and hence the value of money holdings. Figure 6 illustrates the impact on economic activity when the CB deploys this monetary instrument. The low capital economy experiences a decrease in capital formation that is accompanied by higher short term nominal interest rates as illustrated by point A. Recall that the government is a net borrower with additional (transfered) resources from the CB. This translates into greater incentives to issue yet more debt that further crowds out productive investment. At the same time, higher nominal returns on government debt have to be offered in order to clear bond markets. In the case of the high capital economy, an increase in the supply of money promotes capital accumulation. Because the government is a net lender, it actually subsidizes investment in the productive asset. Similarly, real return to bonds is diminished which by (??) translate into net investment gains in productive capital. The next proposition summarizes these findings. 19

Proposition 3. In a developing economy and under inflation-financed government debt, inflationary policies reduce capital formation while increasing shortterm interest rates. Conversely, in a developed economy, the same policy encourages capital formation while decreasing the level of interest rates. I! (22) (21) A Original paths I1 0 A A A B (22) 1 B ˆk k Figure 7: Money growth rule 5.2 Balance sheet policy: short-term bonds We analyze the effects of nonconventional monetary policy as executed via large scale asset purchases of government securities by the monetary authority. Starting with short-term or one-period bonds b 1M, Figure 7 depicts the steady state response of the capital stock and interest rates to such a program. It is clear that for both economies, the outcome aligns in the same direction: productive capital increases and the interest rate level declines. For the low capital economy, CB purchases of short term debt mitigate crowding out by absorbing government debt from private bank s portfolios. Lower returns to capital follow which via no arbitrage, cause a reduction in interest rates. In a developing economy, the same outcome on interest rates and capital prevails. The government being a net lender has its subsidizing role enhanced by the CB policy. In this case, government bond purchases are funds being lent to 20

I 1 Original paths b 1M C I1 0 C D D 1 k Figure 8: Short-term bonds the CB at a low interest rate: the cost of borrowing funds is less than the inflation rate. This condition also obeys (38). Those resources initially lent to the CB find their way to the private sector stimulating productive investment. Then, returns to capital decrease which arbitrage to lower interest rates that enhance subsidies to the real sector. These results can be seen in Figure XXX and are summarized in the following propostion: Proposition 4. In an economy under inflation-financed government debt and irrespective of its development stage, open market operations encourage steadystate capital formation an decrease the overall level of interest rates. 5.3 Balance sheet policy: long-term bonds At this stage it is evident that the impact of monetary policy on the real sector can be distorted by fiscal debt policy actions. This is especially true in the case of our unconventional policy instrument. Unlike their short-term counterparts, resources injected to the fiscal entitiy that originate from long-term securities can actually boost fiscal distortions and make monetary policy counterproductive. To see this, we start by looking at the marginal effect of b 2M on steady state capital in (36). Recalling that I 1 = R 1 we can rewrite this efffect as: 21

1 λ(r 1 + 1). (39) Given the term structure of interest rates, it holds that R 1 + 1 = R2 1 R 1 1 which depicts an interest compunding factor. This latter term together with λ in (??) reduces the effect of b 2M on capital formation. This follows as the fiscal authority receives interest-dependent income flows from assets whose maturity is beyond a single period. Moreover, the magnitude of R 1 can be such as to completely reverse the effects of long-term asset purchases by the CB. If the interest rate level is too high, CB purchases of long-term government securities can actually detriment capital formation. From this marginal effect we can determine this threshold value in steady state (SS) as I SS 1 < ( 1 λ A proposition follows based on this result: λ ). (40) Proposition 5. Irrespective of the development stage of the economy, purchases of long-term government securities encourage steady-state capital formation and ease overall financial conditions as long as the prevailing interest rate level of the economy I SS 1 satisfies (40). This condition posits that unconventional monetary policy is only effective in a low interest rate environment. Furthermore, lower proportions of CB resource transfers to the fiscal authority (lower λ) allow us to analyze a broader range of interest rate levels that satisfy (40). In what follows, we offer an analysis along this idea and explore the different cases that our model offers. We start by taking λ L to be the threshold proportion of CB resources ceded to the fiscal authority such that long-term asset purchases are neutral in the low capital economy. The same description applies to λ H but for the developed economy and it follows that 0 < λ L < 1/2 < λ H < 1 by (40). Note that the value of these parameters depend on the nominal interest rate pertaining in each economy. From this, we can describe three possible cases that may emerge according to varying degrees of resource transfer from the CB to the fiscal authority. 2 2 The intention of proceeding in this fasion has to do with analyzing the effectiveness of unconventional monetary policy under different interest rate regimes and not so much as to claim that CB transfers determine interest rates and hence the success of this policy. (I need to adjust the writing for this) 22

First, as illustrated in Figure 8, the proportion of these transfers allows our two economies to satisfy (40). As a result, the stock of capital increases, the return to capital declines and no-arbitrage restrictions result in lower nominal interest rates. In the low capital economy E, interest rates are low enough that government debt stimulated by CB transfers of long-term assets do not posses enough force to offset subsidized investment in the productive asset. The same direction of this effect arises in the developed economy corresponding to F. In this case, however, recall that the CB is lending to the fiscal authority given that I 1 < and the latter relying on high enough seignorage revenue by the CB as depicted by (38). Whereas this scheme is enough for the previously analyzed open market operations to work, successful unconventional policy requires returns to long term flows to be further disciplined by 40 and for the reasons already discussed. I 1 λ (0, λ L ) E Original paths b 2M I 0 1 E F 1 F I1 = ( ) 1 λ λ k Figure 9: Long-term bonds: full positive impact In Figure 9, the low capital economy in L violates (40) and a reduction in capital formation must take place when the CB engages in this asset purchase policy. This follows as the CB is transfering too much long-term, high-return resources to the fiscal authority which now has additional means to issue more debt crowding out productive allocations. Also, because of this additional borrowing, higher interest rates are needed to clear bond markets. The high capital economy in M, has low enough interest rates for monetary 23

I 1 λ = 1 2 (λ L, λ H ) L Original paths b 2M I 1 = L M 1 M Figure 10: Long-term bonds: differential impact k policy to work in the same manner as discussed in the first case above. Finally, Figure 10 illustrates a setting in which the cost of borrowing is at such a high level that this monetary policy fails irrespective of the development stage. It also shows the only situation in which policy, whether conventional or not, fails in the developing economy and hence it is worth discussing. Here, the government s reliance on a scheme of low cost borrowing to subsidize productive investment is compromised. Specifically, seignorage revenue transfers cannot handle the same volume of overall borrowing as before which reduces the subsidizing ability towards productive investment. It follows that a decline in the stock of steady state capital causes an increase in interest rates through our no arbitrage condition. 5.4 The Yield Curve & Long-Term Asset Purchases Below are the figures I mentioned to you. For now I just want to highlight that in our results, the effects on capital formation are stronger for open market operations relative to those of long-term bonds. Since the NA locus does not shift, this probably gives the same conclusion for impacts on the yield curve. I m assuming I will talk about this in this part of the essay but any feedback is more than appreciated. 24

I 1 λ (λ H, 1) G Original paths b 2M I 0 1 G H I 1 H Figure 11: Long-term bonds: full negative impact k λ (0, λ L ) I 1 E Original paths b 2M E F F I E 2 I E 2 I F 2 I F 2 k Figure 12: Long-term bonds: full positive impact 25

λ (0, λ L ) I 1 G Original paths b 2M G I 1 H H I 2 I G 2 I G 2 I H 2 I H 2 k Figure 13: Long-term bonds: differential impact λ (0, λ L ) I 1 L Original paths b 2M L M M I 1 I 2 I L 2 I L 2 I M 2 I M 2 k Figure 14: Long-term bonds: full negative impact 26