Macroeconomic E ects of the Federal Reserve s MBS Purchases *

Transcription

1 Macroeconomic E ects of the Federal Reserve s MBS Purchases * Hamza Polattimur Andreas Schabert 2 TU Dortmund University PRELIMINARY VERSION April 4, 25 University of Cologne Abstract This paper studies the macroeconomic e ects of the US Federal Reserve MBS purchases during QE and QE3 programs. For this, we develop a stochastic dynamic general equilibrium model with nancial intermediation, collateralized (mortgage) loans, and sticky prices. Costs of banking activities increase in loans and decrease with holdings of reserves; the latter being supplied by the central bank only in exchange for eligible assets. We show that central bank purchases of MBS at above-market prices tend to reduce mortgage loans rates, which foster economic activity. Calibrating the model to match evidence on yield e ects of QE and QE3, we nd that MBS purchases during QE increased employment by %, consumption by.73% and GDP by.68% on impact. In comparison, the e ects during QE3 are rst smaller, but then larger than in QE due to the construction of the programs. JEL classi cation: E44, E52, G2. Keywords: quantitative easing, central bank asset purchases, mortgage-backed securities *We are grateful to Roland Winkler. TU Dortmund University, Department of Economics, Vogelpothsweg 87, Dortmund, Germany, hamza.polattimur@udo.edu. 2 University of Cologne, Center for Macroeconomic Research, Albertus-Magnus-Platz, 5923 Köln, Germany, schabert@wiso.uni-koeln.de.

2 Introduction In November 28, when the rst round of quantitative easing (QE) started, the US Federal Reserve (Fed) announced that it would purchase mortgage-backed securities (MBS) worth up to $5 billion issued by the government-sponsored enterprises Fannie Mae and Freddie Mac. Fed chairman Bernanke argued in his talk at the Federal Reserve System Conference on Housing and Mortgage Markets on December 4, 28, that housing and housing nance played a central role in precipitating the current crisis and concluded that steps that stabilize the housing market will help stabilize the economy as well. In 22 the Fed decided to purchase agency MBS (as part of the QE3 program) at the amount of $4 billion each month. In his press conference on September 3, 22 Bernanke explained how the program is intended to work: The program of MBS purchases should increase the downward pressure on long-term interest rates more generally, but also on mortgage rates speci cally, which should provide further support to the housing sector by encouraging home purchases and re nancing. The aim of this paper is to provide analysis that quanti es the e ects of MBS purchases of the Fed in the QE programs on macroeconomic variables. While there is some empirical research on how these purchases in uenced MBS yields and mortgage rates (e.g. Hancock and Passmore 2, 22), we aim at measuring their e ects on real activity and aggregate goods prices in a general equilibrium model with sticky prices and nancial frictions. In particular, we try to provide an assessment on how GDP, consumption and employment reacted to MBS purchases in a macroeconomic framework that is calibrated to match Hancock and Passmore s (2, 22) estimates for the e ects on MBS yields and mortgage rates. The analysis is conducted in a framework that consists of two types of households, patient and impatient ones, as in Kiyotaki and Moore (997). While patient households hold deposits at nancial intermediaries, these grant loans to impatient households, which have to be collateralized by their housing, as in Iacoviello (25). Banks face costs increasing in the amount of loans they supply and decreasing in the amount of reserves, similar to Curdia and Woodford (2). To allow for a meaningful role of asset purchases, we follow Schabert (25) and assume that the central bank supplies money only in exchange for eligible assets. The central bank can then in uence the yield of an eligible asset (like treasury bills or MBS) when it purchases it at an above-market price. With reduced MBS yields and thereby lower mortgage rates, the borrowing constraint on impatient agents is alleviated. Thus, house prices tend to rise, while consumption, employment and GDP can be stimulated. This model is calibrated to US data from 99Q to 28Q3. For this period, the Fed

3 supplied money essentially according to a "treasury only" regime. With the announcement of QE also MBS became eligible. Given the calibration, which is aimed to match empirical evidence on yield e ects of MBS purchases, we show how this temporary eligibility of MBS, i.e. the purchases in QE and later in QE3, a ected output, consumption and employment. Speci cally, our results indicate that MBS purchases during QE increased employment by %, which means about.4 million new jobs, consumption by.73%, and GDP by.68% in the rst quarter of the intervention. In comparison, the e ects during QE3 are smaller in the rst quarter, employment raises by.7%, consumption by.47%, and GDP by.46%, but become larger afterwards compared to the quarters of QE. This is due to the constuction of the two programs with QE containing a big impact purchase shrinking over time and QE3 depicting almost constant purchases. The rest of the paper is organized as follows. In section 2, the model with a household sector consisting two types of agents, a banking sector, a rm sector with monopolistic competition and the public sector consisting of the treasury and the central bank is described. In section 3, the model is calibrated, MBS purchases during QE and QE3 brie y described and their e ects analyzed. The last section concludes. 2 The Model In this section, we present the model. We follow Kiyotaki and Moore (997) and consider two types of agents (patient and impatient ones), which will save and borrower in equilibrium. Financial intermediaries collect deposits from savers and grant loans to borrowing households. We assume that debt contracts are not enforceable and are collateralized by housing (see Iacoviello, 25). Mortgage loans are assumed to be tradable and will be equivalent to mortgage backed securities (MBS). The treasury issues one-period bonds which are held by nancial intermediaries and the central bank. Following Schabert (25), we assume that the central bank supplies money only against eligible assets, here, treasuries and MBS. The central bank sets the policy rate and can further control the amount of money supplied against eligible assets, e.g. it can increase the supply of reserves by purchasing MBS. 2. Households There is a continuum of households of mass consisting of two types, patient ones which are indexed with p and have a share of < s < of total population, and impatient ones (indexed with i and share ( s)). They only di er with regard to their subjective discount factors: > p > i >. Both types of households derive utility from consumption c ;t, housing h ;t and disutility from labor n ;t ( = i; p) and maximize the in nite sum of 2

4 expected utility. Their objective is given by X E ;t u(c ;t ; h ;t ; n ;t ). () We consider the following CRRA-speci cation of the utility function t= c u(c ;t ; h ;t ; n ;t ) = c ;t c + zh ;t h h h ;t h n n+n ;t + n, (2) where z h ;t is a housing demand shock with log z h ;t = h log z h ;t + "h ;t, where " h ;t n:i:d: ; 2 h and < h <. We rst describe the problem of a representative patient household and then the one of a representative impatient household. Patient Households A patient household p enters a period t with deposits D p;t held at nancial interediaries and real housing h p;t. Neglecting borrowing from nancial intermediaries (which would never occur in equilibrium), its budget constraint is given by P t c p;t + P t p h;t (h p;t h p;t ) + D p;t =R D t = D p;t + P t w t n p;t + P t p;t + P t p;t ; (3) where the left hand side contains expenditures for consumption, P t c p;t, and housing, P t p h;t h p;t, and new holdings deposits D p;t at the price =R D t, while the right hand side shows deposits from the preceding period as well as labor income, P t w t n p;t, transfers from the public sector, P t p;t, and pro ts of rms and retailers P t p;t, due to the assumption that patient households are the owners of rms and retailers. A patient household chooses the values of c p;t, h p;t, n p;t and d p;t = D p;t =P t to maximize () subject to the budget constraint (3) a non-negativity constraint D p;t, leading to the rst order conditions z h p;t h h h p;t = p h;t p;t p E t p;t+ p h;t+; (4) n n n p;t = w t p;t ; (5) R D t = p E t p;t+ p;t t+. (6) Equation (4) describes housing demand of a patient household, (5) its labor supply and (6) is the Euler equation for deposits. Impatient Households Since an impatient household values current consumption more than a patient one, it will become borrower in equilibrium. We assume that its debt is non-enforceable and is collateralized by housing. Assume that a household i can borrow from intermediaries in nominal terms an amount B M i;t =RL t < in period t and has to pay back Bi;t M in period t, where RL t is the gross nominal interest rate on these loans. We follow Iacoviello (25) and assume that an impatient household s borrowing is limited by 3

5 a fraction of the expected end of period housing wealth B M i;t =R L t z t E tp t+ p h;t+ h i;t, (7) where denotes the (exogenous) pledgeable fraction of housing, z t a shock on it with log z t = log z t + " t, where " t n:i:d: ; 2 and < < and p h;t the price of housing in t. An impatient household i enters a period t with a mortgage loan Bi;t M < and real housing h i;t. It has expenditures for consumption, P t c i;t, and housing, P t p h;t h i;t, gets transfers, P t i;t, and takes a new mortgage loan Bi;t M for which it has to pay the interest rate Rt L. Neglecting deposits held at nancial intermediaries, the budget constraint of an impatient household i reads P t c i;t + P t p h;t [h i;t h i;t ] + B M i;t =R L t = B M i;t + P t w t n i;t + P t i;t, (8) where P t w t n i;t is labor income with the real wage rate w t. An impatient household i chooses the values of c i;t, h i;t, n i;t, and b M i;t = BM i;t =P t to maximize () subject to the budget constraint (8) and the borrowing constraint (7) leading to the rst order conditions z h i;t h h h i;t = i;t p h;t i E t i;t+ p h;t+! t z t E t t+ p h;t+ (9) n n n i;t = w t i;t () i;t R L t = i E t i;t+ t+ +! t R L t,! t = i;t i E t i;t+ t+ R L t () with! t being the multiplier on the collateral constraint and the complementary slackness conditions! t (b M i;t =RL t + z t E t t+ p h;t+ h i;t ) = ; (b M i;t =RL t ) + z t E t t+ p h;t+ h i;t ; and! t. Equation (9) describes housing demand of an impatient household. In the optimum, marginal utility of current housing zi;t h h h h i;t equals marginal utility of foregone consumption i;t at the price of housing p h;t less the discounted marginal utility of next period s expected consumption i E t i;t+ achieved from selling the house at the price p h;t+ and less the term! t z t E t t+ p h;t+ that stems from the collateral constraint. Equation () is a modi ed Euler equation for mortgage debt taking into account the collateral constraint. Furthermore, the transversality conditions must hold. 2.2 Financial Intermediaries There is a continuum of identical perfectly competitive nancial intermediaries (banks) of mass indexed with b. Their budget constraint is given by P t B b;t + D b;t + (B G b;t =RG t ) + (B M b;t =RL t ) + M H b;t + I b;t (R m t ) + P t t B M b;t ; Q b;t (2) = D b;t R D t + B G b;t + BM b;t + M H b;t. 4

6 where B b;t denotes pro ts. The term I b;t (Rt m ) in (2) denotes costs associated with new money injections and the term t (Bb;t M; Q b;t) denotes costs of supplying loans as an increasing function of the volume of loans t (Bb;t M; Q b;t)=@bb;t M > and a decreasing function of reserves, given by Q b;t = Mb;t H + I t (Bb;t M; Q b;t)=@q b;t <. A bank b collects deposits from (patient) households D b;t = sd p;t, pays an interest Rt D on them and holds money Mb;t H. Further, it holds government bonds BG b;t which deliver an interest of Rt G and supplies mortgage loans, Bb;t M = (s )BM i;t at the interest rate RL t. We assume that mortgage loans can in principle be traded frictionlessly and at any amount between di erent nancial intermediaries. Thus, in this model, they are equivalent to mortgage backed securities (MBS). Both types of bank assets, i.e. treasuries and MBS, are assumed to be eligible and can therefore be used to get new money injections from the central bank. In each period the central bank decides by setting its instruments G t and M t how much money is supplied in exchange for these assets. It further sets the price of money in terms of collateral Rt m, which we therefore call the repo rate or the policy rate. Notably, the federal funds rate, which actually served as the policy rate of the US federal reserve, closely relates to the rate on treasury repurchase agreements before the nancial crisis, see e.g. Bech et al. (22). New money injections I b;t that a bank receives from the central bank are then limited by the collateral constraint I b;t G t Bb;t G Rt m + M t B M b;t R m t. (3) P A bank b maximizes the present value of future pro ts max E # t;t+k B b;t+k subject to (2) and (3), where # t;t+k denotes the stochastic discount factor of banks. The rst order conditions with respect to deposits, government bonds, MBS, money holdings and injections k= E t # t;t+ = E t t+ R D t R G t R L t = E t # t;t+ t+ + t+ G t+ R m t+ = E t # t;t+ t+ + t+ M t+ R m M b;t (4) (5) t+ E t # t;t+ = E t H (7) b;t R m t b;t t (8) where # t;t+ = # t;t+k+ =# t;t+k and t denotes the multiplier on the collateral constraint of 5

7 banks and the complementary slackness conditions! t i b;t+k + G b G b;t+k t+k t+k Rt+k m + M b M b;t+k t t+k Rt+k m = ; b G b;t+k i b;t+k + G t+k t+k Rt+k m b M b;t+k + M t+k t+k Rt+k m ; t : Note that the stochastic discount factor of banks will in equilibrium equal the one of patient households (see 6 and 4). We further get the following relationships between the interest rates R D t, R G t and R L t : If government bonds are not eligible or the collateral constraint (3) is not binding, the interest rates on deposits and bonds are equal: Rt D = Rt G. Otherwise, we get Rt G < Rt D. Moreover, we have Rt G < Rt L for G t+ M t+ due to the fact that mortgage loans increase credit costs. To further compare the interest rate on deposits and mortgage loans, we insert (4) in (6) leading to =Rt L = =Rt D ( + Et t+ M t+ =Rm t+ t=@b M b;t. Hence, t =@b M b;t > E t t+ M t+ = RD t Rt+ m, i.e. marginal costs of mortgage loans are su ciently high, then the interest rate on loans is higher than the one on deposits, R D t < R L t. 2.3 Firms A continuum of perfectly competitive identical rms indexed with j produce the intermediate good according to IO j;t = z p t n T j;t where 2 (; ) and z, p t is a productivity shock with log (z p t ) = p log z p t + " p t and "p t n:i:d: ; p 2 and < p <. The rm hires labor from both types of households n T j;t = sn p;j;t + ( s)n i;j;t at a common rate w t to produce its output IO j;t, which it sells to the retailers at the price P J;t. Hence a rm j solves max P J;t z p t nj;t T P t w t n T j;t leading to the rst order condition P J;t z p t nt j;t = Pt w t (9) and to pro ts of ( ) P J;t z p t nj;t T that are distributed to the patient households which are assumed to own these rms. Price stickiness is introduced as follows. A continuum of monopolistically competitive retailers indexed with k buy intermediate goods at the price P J;t, re-package them according to IO t = R IO j;t dj, di erentiate them into y k;t = IO k;t and sell the distinct goods y k;t at the price P k;t to perfectly competitive bundlers. They bundle them to the nal good R y t = y k;t dk, where >, which is sold at the price P t. Hence a retailer k faces the demand function y k;t = (P k;t =P t ) y t and sets its own price P k;t according to this and taking P J;t as given. Following Calvo (983), we assume that each period only a fraction 6

8 of retailers is allowed to change his price. The other fraction 2 [; ) adjusts the price according to full indexation to the steady state in ation rate: P k;t = P k;t. De ning Z e t = Pk;t =P t with the optimal price of retailers Pk;t, the optimal real price can be written recursively as Z e t = Z ;t=z 2;t, with Z ;t = p;t y t mc t + p E t+ t Z;t+ and Z 2;t = p;t y t + p E t+ t Z2;t+ (for details see Appendix 5..). Due to perfectly competitive bundlers the aggregate price level P t for nal goods is given by P ( ) P k;t + P t R = P k;t dk (zero pro t condition) and can be written as Pt = t, = ( ) Z e t + t (see Appendix 5..2). More- R over, aggregate output is given by y t = zp t (n T t ) v t where v t = of price dispersion, which can be written recursively as v t = ( Appendix 5..3). (P k;t =P t ) dk is a measure ) Z e t + t vt (see Pro ts of retailers are as well distributed to patient households in a lump-sum way. All pro ts of intermediate goods producing rms and retailers that are distributed to a households p are collected in the term P t p;t. 2.4 Public Sector The treasury issues one-period bonds which are held by nancial intermediaries B G b;t and the central bank BC;t G. Hence, total demand for government bonds in period t is given by R BG b;t db + BG C;t. Bonds are supplied following a simple rule BG T;t = BG T;t, where > p is the growth rate of total government bonds. The treasury receives seigniorage revenues from the central bank P t m t and pays lump-sum transfers P t t = sp t p;t + ( s) P t i;t to households to balance its budget B G T;t + P t t = (B G T;t=R G t ) + P t m t. with the assumption that transfers are equal for both types: p;t = i;t implying t = p;t = i;t. The central bank supplies money outright Mt H = R M b;t H db and through repurchase agreements Mt R = R M b;t R db and holds treasuries BG C;t and MBS BM C;t that yield interest earnings. New money injections in each period are given by I t = Mt H Mt H + M t R. The budget constraint of the central bank reads (B G C;t=R G t ) + (B M C;t=R L t ) + P t m t = B G C;t + B M C;t + R m t M H t M H t + (R m t ) M R t : Assuming that the central bank transfers all its earnings given by P t m t = =Rt G B G C;t + =Rt L B M C;t + (Rt m ) Mt H Mt H + (R m t ) Mt R to the treasury, we get the fol- 7

9 lowing relationship between the evolution of assets held and money supplied by the central bank BC;t G BC;t G + BM C;t BC;t M = M t H Mt H. Assuming that initial assets and liabilities satisfy, BC; G + BM C; = M H, the balance sheet of the central bank reads BC;t G + BM C;t = M t H. Finally, with B G;T b;t = R BG b;tdb and the market clearing condition for government bonds as well as BC;t M = M t Bb;t M, the balance sheet becomes B G T;t B G;T b;t + M t B M b;t = M H t. The policy rate is set by the central bank following a feedback rule given by Rt m = maxf; Rt m R (R m ) R ( t =) ( R ) (y t =y) y ( R ) exp " r;t g, where R m > is an average policy rate chosen by the central bank, R, and y and " r;t n:i:d: with E t " r;t =. For simplicity, we assume that outright money supply is a constant fraction > of money supplied via repos: Mt R = Mt H : (2) Equilibrium conditions are summarized and the equilibrium is de ned in Appendix MBS Purchases in QE and QE3 In this section, we use the model to evaluate the FED s MBS purchases. First, we study purchases during the QE program, announced on November, 25, 28, which was the rst time in its history that the FED conducted these types of unconventional policy measures. markedly. During that time credit markets were disrupted and MBS yields had risen Second, we focus on the purchases in the QE3 program in 22, when the economic situation was far better than in 28. But before these analyses the model has to be calibrated. 3. Calibration For calibration purposes, we de ne a functional form for the cost function of banks given e by t b M t;b ; q b t = k M t;b. Moreover, we rewrite the collateral constraint (3). m H t =t+it Until 28 only government bonds were eligible, what is also called a "treasuries only regime". After the crisis, also MBS became eligible with the QE programs. Therefore, we model MBS purchases of the central bank as shocks, i.e. these purchases are assumed to be transitory. This means that government bonds are always, while MBS are only 8

10 temporarily eligible. Hence, we rewrite the collateral constraint (3) in real terms as i b;t G t b G b;t t R m t + zt i bm b;t t Rt m, (2) where z i t is a shock on MBS purchases with log z i t = i log z i t + " i t, where " i t n:i:d: ; 2 i and < i <. In the steady state z i t is set to such that only government bonds are eligible. One time period is assumed to be a quarter. To calculate the long run values of the rates we consider, we use quarterly US data for the time period 99Q to 28Q3. The reason for beginning in 99Q is that according to Iacoviello (24) after the mid- 98s the housing nance system was restructured and the mortgage market got better integrated into the capital market. As we interpret the rate r L as MBS yield, for which we have no data, we calculate its mean as follows. Hancock and Passmore (2) argue that the mortgage rate is a mark-up over the MBS yield, and report a mean MBS yield for the time span July 2 to March 24, which is 5.95%. In this time span the mean 3 year xed mortgage rate is 6.57% leading to a mark-up of about %. Given this mark-up and the mean mortgage rate for 99Q to 28Q3 (7.34%), the resulting MBS yield is 6.64% annually and.62% quarterly for the time period 99Q to 28Q3. The reason why we nish the period by 28Q3 is that after 28Q4 there were large jumps in the interest rates due to the nancial crisis and the announcement of the FED s intervention. In the US, the mean net in ation rate and the means of the empirical counterparts of the net interest rates considered in the model between 99Q and 28Q3 are given by = :46%, r m = :6%, r G = :9%, r D = :%, r L = :62% The discount factor of patient households follows from p = R D = :9937. Following Campbell and Mankiw (989), we set the fraction of constrained consumers to :4, i.e. s = :6. The utility parameters are set to values that are standard in the literature, i.e. n =, c = 2, and h = 2. n and h are calibrated such that total working time is n T = :33 and housing wealth to quarterly GDP is p h =y = 6 since total housing wealth to annual GDP is about :5 in the US (see Iacoviello (29)). Iacoviello (25) summarizes some estimates of the discount factors for poor households. Since these ones are more likely to be borrowers, he uses the estimates to calibrate i. He concludes that i = :95 is an appropriate value for impatient households discount rate, which is adopted here. Moreover, following him the pledgeable fraction of housing and the parameters concerning the production sector are set to = :55, = 2=3, = :75, " = 2. The total housing stock is normalized to and the value of G is set to 9

11 Table : Baseline Parameter Calibration. Description Source/Target Parameter Value Discount factor patient households and R D p :9937 Discount factor impatient households Iaco. 25 i :95 Pledgeable fraction of housing Iaco. 25 :55 Share of the patient households C&M 989 s :6 Inverse of Frisch elasticity n Inverse of IES in consumption c 2 Inverse of IES in housing h 2 Weight of housing in u p h =y = 6 h :83 Weight of labor in u n T = :33 n 3:4 Production function Iaco. 25 2=3 Price rigidity Iaco. 25 :75 Elasticity of substitution Iaco. 25 " 2 Fraction of bonds accepted by CB fully eligible G Fraction of outright money supply C&S24 Fixed housing supply normalized H Banking cost function r L = :62% k :45 Banking cost function 27 bp drop in r L e :66 implying that government bonds are fully eligible. Following Christo el and Schabert (24), is set to, roughly re ecting the ratio that can empirically be observed. The parameters of the cost function, k and e, are set as follows. We calibrate k such that in the steady state the mean MBS yield of r L = :62% is matched. The calibration of the elasticity e is done following empirical results of Hancock and Passmore (2). According to them, annual MBS yields fell between the announcement of QE in November 28 and May 29, where "the program had successfully returned mortgage markets to normal", by about. percentage points. End of May 29 the value of MBS held by the FED was at about $43 billion close to the initially announced $5 billion. Due to the lag between announcement and implementation, this time span is 2 quarters. However, signi cant MBS purchases by the FED, that made up more than % of total US Agency MBS outstanding, began after February 29. Due to this and the fact that in the model we have no lag between announcement and implementation, we de ne March, to May, 3, 29 as the rst quarter of QE and simulate the drop in the MBS yield described above within period. A purchase of $43 billion made up about 8.3% of total US Agency MBS outstanding in 28Q3 leading to zt i = :83. A decline in the annual rate by. percentage points means a decline in the quarterly rate, we consider in the model, by.27 percentage points. Hence, we calibrate e such that an increase of zt i to :83 leads to a decline in r L by 27 basis points from :62% to :35% within one quarter. The resulting parameters values for the banking cost function are then k = :45 and e = :66.

12 IRFs The impulse responses displayed in the gures in Appendix 5.3 show deviations of each variable x t from its steady state value x, i.e. x obs t = log x t =x, in response to shocks of one standard deviation. We consider shocks that are su ciently small to ensure a binding borrowing constraint for impatient households (7). The standard deviations for the shocks are set to h = :5, = :5, p = :, i = :83, and the parameters of autocorrelation of the shock processes to = :7 for = h; ; p; i. parameters of the monetary policy rule are R = :8, :5 and y = :8. Moreover, the Figure 4 shows responses to a positive shock to total factor productivity, " p t >. While total employment decreases, total output and total consumption increase following a hump-shaped path. A tfp shock raises consumption and housing of patient agents, while reducing their labor supply. For impatient agents, consumption and housing fall together with their working time following a tfp shock. Moreover, house prices increase, while the wage rate and in ation decrease. Further, with a decrease in in ation the policy rate falls leading to higher reserves and a lower MBS yield. Figure 5 shows the responses to a positive housing demand shock, " h t >. First of all, this shock leads to an increase in house prices. Since impatient households have an additional bene t from housing through the borrowing constraint, they value housing more than patient ones and increase their housing. Due to the fact that housing is normalized to, hence patient households reduce their housing. Spending more on housing, an impatient agent works more and consumes less, while a patient household does the opposite. In total, consumption, employment, and output decrease as well as the wage rate and in ation. Again, the decrease in in ation lets the policy rate fall leading to higher reserves and a lower MBS yield on impact. The responses to a positive policy rate shock, " r t >, are shown in gure 6. A higher policy rate reduces reserves, which increases the MBS yield and hence reduces private borrowing. The policy rate shock further has contractionary e ects on total consumption, total employment, and total output, and lowers the wage rate as well as in ation. The new shock in this paper is the shock to the collateral constraint (2), " i t >, that shows the e ects of MBS purchases of the central bank. The responses to this shock are shown in gure 7. First of all, this shock leads to higher reserves and reduces the MBS yield due to lower intermediation costs. This reduction in the MBS yield loosens the borrowing constraint of impatient agents, the multiplier! i falls, highers their borrowing and their housing demand. This bids up house prices. Since an impatient agent has to spend less on interest payments, it can a ord to work less and consume more. A patient household also consumes more but has to work more. The wage and in ation rate increase which leads to a higher policy rate. In the aggregate, MBS purchases of the central bank

13 Table 2: FED MBS Purchases And Shock Processes in QE. Quarter of QE Total MBS Purchases in: $ billions % of total Ag. MBS outst. :83 :37 :43 :33 :5 Shock Process for zt i :83 :54 :35 :23 :5 have expansionary e ects on consumption, employment, and output. Finally, gure 8 shows the responses of a shock to the pledgeable fraction of housing, " t >, which can be interpreted as lower lending standards and higher leverage. We see that this increases borrowing although it leads to a reduction in house prices. The MBS yield rises. Due to lower house prices and higher borrowing impatient agents housing and consumption increase and their working time decreases. In contrast, patient agents consume less, reduce their housing and work more. In the aggregate, this shock has expansionary e ects on consumption, employment, and output. 3.2 Results QE As explained before, we de ne March, to May, 3, 29 as the rst quarter of MBS purchases in QE. Table 2 shows the MBS purchases of the FED in billions of dollars as well as relative to total agency MBS outstanding. We approximate the real process by an AR() process for z i t given by log z i t = i log z i t + " i t, where " i t n:i:d: ; :83 2 and i = :65. Figure shows the reaction of selected variables to MBS purchases for two cases. The blue solid lines depict the reaction for the case where the zero lower bound is binding and the red dashed lines where not. The gure is constructed as follows. First, we simulate a crisis with a negative credit shock (z t #) for eight periods and then let the economy return to its steady state. In a second simulation we use the same crisis scenario and add the MBS purchase shock in the midst of the crisis. The lines in the gure show the di erence between the reactions of the selected variables for the simulation with MBS purchase compared to the simulation without it. Hence, we can interpret the shown reaction as merely stemming from the MBS purchase shock. To get the e ects of MBS purchases at the zero lower bound, shown by the blue lines in Figure, we also shock the policy rate to bring it to zero. For the simulations at the zero lower bound, we use the occbin toolkit by Guerrieri and Iacoviello (24), which facilitates the analysis of an occasionally binding constraint. As we can see, there is hardly any di erence in the reaction to MBS purchases between the case at the binding zero lower bound and the case where the policy rate can react and 2

14 Level. Output 2. Inflation ZLB no ZLB Total Consumption 4. Consumption Borrower Total Employment Employment Borrower House Prices 8. Housing Borrower x 3 9. MBS Yield 2. MBS Time Time Figure : E ects of MBS purchases in QE with and without binding zero lower bound. go down. The numerical results however show a slightly larger reaction at the zero lower bound. Our results indicate that in the scenario with non-binding zero lower bound GDP increases by.68% in response to the MBS purchase shock that mimics the real program of the FED in QE. Total consumption increases by about.73% and total employment even by about %. Given that in the US about 4 million persons were employed in 28Q3, this means that the MBS purchase program led to about.4 million new jobs in the rst quarter of the intervention. Taking a look at the constrained agents, the borrowers, we can see that they bene t at most from the intervention. Their consumption goes up by.3% and their housing by 2.2% thanks to lower interest payments ( 57 basis 3

15 Table 3: FED MBS Purchases And Shock Processes in QE3. Quarter of QE Total MBS Purchases in: $ billions % of total Ag. MBS outst. :6 :25 :24 :23 :26 :8 : Shock Process for zt i :6 :24 :26 :24 :2 :5 : points), higher house prices (+.93%) and hence higher borrowing (+2.%). The increase in total employment can be totally attributed to patient households (except for the rst quarter), while both types contribute to the increase in total consumption. Moreover, another important e ect of MBS purchases can be seen in gure : the increase in the gross in ation rate by.44%. This means that for the given calibration ( = :46) and the given crisis scenario the net in ation rate increases by.34 percentage points from its long run average at.46% to.8% in the rst quarter of the MBS purchase program. Hence, central bank asset purchases seem to be a powerful tool to boost in ation and prevent de ation. QE3 We de ne the beginning of MBS purchases in QE3 as 22Q4. In table 3 again absolute and relative MBS purchases by the FED are shown. Here, we approximate the real process by an AR(2) process for zt i given by log zt i = i; log zt i + i;2 log zt i + " i t, where " i t n:i:d: ; :6 2 as well as i; = :5 and i; = :62. The results for QE3 shown in gure 2 are qualitatively similar to the ones for QE. The MBS purchase program in QE3 increased GDP by.46%, total consumption by.47% and total employment by.7% in the dirst quarter of the intervention. Again the zero lower bound makes no large di erence for the reaction of the variables to the MBS purchases. According to the simulation results of our model, the MBS purchase program worked how it was intended to work: it reduced MBS yields in reducing banking costs and thus had expansionary e ects on employment, consumption and GDP. Comparison: QE vs. QE3 Comparing the results for MBS purchases during QE and QE3 (Figure 3) we nd that in the rst quarters of the interventions the rise in GDP and total employment is larger in QE, but after the second periods the QE3 program has larger e ects on GDP and employment. Noting that MBS purchases in QE3 amounted to only about 3/4 of the purchases during QE (829 billion $ in QE3 vs. 2 billion $ in QE), we can conlude that not only the size of the purchase program but also its design and its timing are important for its e ectiveness. In the aftermath of the crisis, when credit markets were disrupted and MBS yields had risen markedly, the FED reacted with a program containing a few MBS purchases of large size to rapidly counteract the 4

16 Level. Output 2. Inflation.4.2 ZLB no ZLB Total Consumption 4. Consumption Borrower Total Employment 6. Employment Borrower House Prices Housing Borrower x 3 9. MBS Yield 2. MBS Time Time Figure 2: E ects of MBS purchases in QE3 with and without binding zero lower bound. rise in MBS yields and help the housing market to stabilize. The QE3 program in 22 was conducted when the economic situation was far better than in 28. Therefore, the design of the program was di erent. It spans a longer period and shows roughly constant purchases. How this program design a ects the results becomes clearer, if we look at MBS yields. While in QE the drop in the MBS yield is large in the rst quarter ( 57 basis points) and then becomes smaller ( 42 BP in the second quarter, 27 BP in the third and 7 BP in the fourth quarter), in QE3 the drop is smaller ( 5 BP in the rst quarter), becomes larger ( 2 BP in the second quarter), remains in this range for 4 quarters and then becomes smaller. The roughly constant purchases each quarter in QE3 lead to a reduction in the MBS yield that is more smooth and hump-shaped compared to 5

17 Level.6. Output QE QE Inflation Total Consumption 4. Consumption Borrower Total Employment Employment Borrower House Prices 8. Housing Borrower x 3 9. MBS Yield 2. MBS Time Time Figure 3: Comparison: E ects of MBS purchases in QE and QE3. the initial large drop and quick return in QE. Hence, it seems to be more adequate to purchase smaller but constant amounts of MBS spread over several periods considering the economic situation in 22 than to make one or few bigger purchases as it was done in 28 just after the crisis. 4 Conclusion After the nancial crisis, the FED was forced to conduct for the rst time in its history unconventional policy measures, like the purchase of mortgage-backed securities. Hence, the discussion about the e ectiveness of these types of interventions is vivid. This paper tries to provide analysis that quanti es the e ects of MBS purchases of the Fed in the 6

18 QE programs on macroeconomic variables. Our results suggest that, MBS purchases by the FED had non-negligible e ects on macroeconomic variables. Speci cally, our results indicate that MBS purchases during QE (QE3) increased employment by % (.7%), consumption by.73% (.47%), and GDP by.68% (46%) in the rst quarter of the intervention. Reserves that were supplied to banks in exchange for MBS reduced banking costs and hence the MBS yield. This in turn led to higher consumption, employment and GDP. Moreover, the MBS purchases led to a notable increase in in ation proving them to be a powerful tool to prevent de ation Further questions one could address with this type of model are the ones concerning tapering. How should the tapering be done and what is the best time to begin with it, are currently the most interesting issues, that is left for future research. 7

19 References Bech, M. L., E. Klee and V. Stebunovs. (22) "Arbitrage, liquidity and exit: the repo and federal funds markets before, during, and emerging from the nancial crisis." Finance and Economics Discussion Series, Board of Governors of the Federal Reserve System Bernanke, Ben S. (28) "Housing, Mortgage Markets, and Foreclosures." Speech at FED Conference on Housing and Mortgage Markets, December 4, 28. Retrieved from on September 9, 24. Bernanke, Ben S. (22) FED FOMC Press Conference, September 3, 22. Retrieved from les/fomcpresconf2293.pdf on September 9, 24. Campbell, John Y. and N. Gregory Mankiw. (989) "Consumption, income, and interest rates: reinterpreting the timeseries evidence." NBER Macroeconomics Annual 989, 4, Christo el, Kai and Andreas Schabert. (24) "Interest Rates, Money, and Banks in an Estimated Euro Area Model." Working Paper. Curdia, Vasco and Michael Woodford. (2) "The central-bank balance sheet as an instrument of monetary policy." Journal of Monetary Economics, 58, Guerrieri, Luca and Matteo Iacoviello. (25) "Occbin: A Toolkit to Solve Models with Occasionally Binding Constraints Easily." Journal of Monetary Economics, 7, Hancock, Diana and Wayne Passmore. (2) "Did the Federal Reserve s MBS purchase program lower mortgage rates?" Journal of Monetary Economics, 58, Hancock, Diana and Wayne Passmore. (22) "The Federal Reserve s Portfolio and its E ects in Mortgage Markets." Working Paper. Hoermann, Markus and Andreas Schabert. (23) "A Monetary Analysis of Balance Sheet Policies." The Economic Journal, forthcoming. Iacoviello, Matteo. (24) "Consumption, house prices, and collateral constraints: a structural econometric analysis." Journal of Housing Economics, 3, Iacoviello, Matteo. (25) "House Prices, Borrowing Constraints, and Monetary Policy in the Business Cycle." American Economic Review, 95, Iacoviello, Matteo. (29) "Housing in DSGE Models: Findings and New Directions." Book Chapter for a Springer-Verlag manuscript edited by Banque de France. Kiyotaki, Nobuhiro and John Moore. (997) "Credit Cycles." Journal of Political Economy, 5, Schabert, Andreas. (25) "Optimal Central Bank Lending." Journal of Economic Theory, 57,

20 5 Appendix 5. Details to the Firms Problem 5.. Retailers Price The price setting problem of a retailer reads X E t s q t;t+s ( s Pk;t P k;t P J;t+s ) y t+s) k;t P t+s s= with the stochastic discount factor q t;t+s = ( p ) s c c p;t+s E t p;t condition for the optimal price Pk;t P t P t+s leading to the rst order Inserting q t;t+s = ( p ) s c c p;t+s E t p;t (22) can be written as P Pk;t = E t s q t;t+s ( s ) Pt+sy t+s P J;t+s s=. (22) P E t s q t;t+s ( s ) Pt+s y t+s P k;t = P t P t+s s= and de ning marginal costs of retailers as mc t = P J;t P t, P E t ( p ) s p;t+s (s ) Pt+sy t+s mc t+s s=. (23) P E t ( p ) s p;t+s (s ) Pt+s y t+s s= Following Hoermann and Schabert (23), we de ne e Z t = P k;t =P t such that we can rewrite the denominator and the numerator of (23) recursively as Z e t = Z ;t=z 2;t, with Z ;t = p;t y t mc t + p E t+ t Z;t+ and Z 2;t = p;t y t + p E t+ t Z2;t Aggregate Price Level The aggregate price level P t for nal goods is given by P t R = P k;t dk due to perfectly competitive bundlers (zero pro t condition). Since all retail prices were set according to (22) in the past, we can write this as Z Pt = P k;t dk = ( ) X s= s s Pk;t s = ( ) Pk;t + ( ) P k;t + ( ) 2 2 Pk;t 2 + ::: {z } = P t = ( ) Pk;t + Pt. 9

21 Hence the aggregate price level is given by t = ( ) Pk;t + Pt. (24) P Finally, we can rewrite (24) by dividing through P t 5..3 Aggregate Output = ( ) e Z t + t. Aggregate output of intermediate goods producers who behave identically is given by IO t = z p t n T. R R t Moreover, IOt = IO k;t dk = y k;t dk holds due to market clearing. Equalizing these equations for intermediate output and inserting y k;t = (P k;t =P t ) y t, we get z p t n T R t = (P k;t =P t ) y t dk, y t = zp t (n T t ) R v t where v t = (P k;t =P t ) dk is a measure of price dispersion. With the assumption that there is no initial price dispersion as in Yun (25) and Schmidt-Gorhé and Uribe (26), we can write v t v t = ( ) Pk;t =P t + ( ) P k;t =P t + ( ) 2 2 Pk;t 2 =P t + ::: X = ( ) l l Pk;t l t =P, l= where P k;t is the optimal price set by the price adjusting rms at time t. This term can further be simpli ed with v t v t = ( ) with e Z t = P k;t =P t and P = ( ) = ( ) l= s= X l l Pk;t l= l =P t X l l Pk;t l =P t l (Pt l= X l= l l ez t l ly s= t+ s l =P t ) lq t+ s = t + t t + t t t + ::: = (P t =P t ) + (P t =P t ) (P t 2 =P t ) + (P {z } t =P t ) (P t 2 =P t ) (P t 3 =P t 2 ) +::: = {z } (P t 2 =P t) (P t 3 =P t) P (P t l =P t ). l= 2

22 Taking di erences, we can write v t in a more compact recursive way: v t tv t = ( ) X l= l l ez t l ly X t+ s t ( ) l l ez = ( ) e Z t + ( ) e Z t t + ( ) 2 2 e Z t 2 t t + ::: 5.2 Equilibrium s= ( ) Z e t t ( ) 2 2 Z e t 2 t t :::, v t = ( ) Z e t t + vt Summary of Equilibrium Conditions l= ly t l t s= s Patient households Impatient households z h i;t h h h i;t z h p;t h h h p;t = p h;t p;t p E t p;t+ p h;t+ (25) n n n p;t = w t p;t (26) p;t R D t = p E t p;t+ t+, E t p;t t+ p p;t+ = R D t. (27) = i;t p h;t i E t i;t+ p h;t+! t z t E t t+ p h;t+ (28) n n n i;t = w t i;t (29) i;t = i;t Rt D i;t R L t = i E t i;t+ t+ +! t R L t i E t i;t+ t+ (3),! t = i;t i E t i;t+ t+ R L t b M i;t R L t z t E t t+ p h;t+ h i;t (3) c i;t + p h;t [h i;t h i;t ] + d i;t R D t + bm i;t R L t = d i;t t + bm i;t t + w t n i;t + i;t (32) As seen above, in the steady state the non-negativity constraint for deposits of impatient households is binding: d i =. Moreover, if R L R D also the borrowing constraint of impatient agents is binding in the steady state: b M i = R L p h h i. 2

23 Banks # t;t+ = t+ R D t R G t R L t = E t p p;t+ p;t = # t;t+ t+ + E t t+ G t+ R m t+ = # t;t+ t+ + E t t+ M t+ R m M b;t (33) (34) (35) # t;t+ = R m t t+ H (36) b;t t (37) i b;t G t b G b;t t R m t + M t b M b;t t R m t (38) Firms z p t nt t mct = w t Z s n T t = ez t = Z ;t = p;t Z 2;t = p;t y t = zp t v t = ( Z n p;t dp + s Z ;t=z 2;t ; n i;t di y t mc t + p t+ E t y t + p t+ E t n T t v t = ( ) e Z t ) Z e t t + v t + t Z;t+ Z2;t+ Public Sector B G T;t B G;T B G T;t = B G T;t b;t + M t B M b;t = M H T;t with B G;T b;t = Z B G b;t db and M H T;t = Z M H b;t db Aggregate resource constraint R m t = R m t R (R m ) R ( t =) ( R ) (y t =y) y ( R ) exp " r;t 22

24 y t = t B M t ; Q t + Z Z c p;t dp + c i;t di and market clearing in the housing market with xed supply H Z Z h p;t dp + h i;t di = H: Steady State Patient households Impatient households h h h i n n n i b M i h h h p = p h p ( p ) (39) n n n p = wcp c (4) p = RD (4) = i p h i! i p h (42) = w i i = i R D! i = i i c i c = i i R L R D >, i i R L = i >, > i R L (43) p i > (44) if RL R D i p (45) R L = p hh i (46) c i = b M i R L + wn i + i (47) In the steady state the non-negativity constraint for deposits of impatient households is binding: i >, d i =. Moreover, if R L R D also the borrowing constraint of impatient agents is binding in the steady state: b M i = R L p h h i. 23

25 Banks # = R D = p (48) R G = # + G R m (49) R L = # + R M (5) # H (5) b R = b i b G bg b R m + M bm b R m (53) Firms n T mc = w (54) Z s n T = Z n p dp + s n i di (55) ez = Z =Z 2 ; (56) Z = c p c y mc p (57) Z 2 = c p c y p (58) nt y = (59) v v = ( ) Z e (6) = ( ) Z e + (6) Public Sector b G T = (62) b G;T b + M t b M b;t = mh T (63) R m = R m > (64) Aggregate resource constraint 24

26 y = b M ; q Z + Z Z H = h p dp + Z c p dp + c i di (65) h i di. (66) and market clearing in the housing market with xed supply H. 5.3 Impulse Response Functions The IRFs of selected variables for the 5 shocks considered in the model are plotted below. 25

27 .2 Output.2 Consumption.2 Employment Cons. Lender Housing Lender Empl. Lender Cons. Borrower Housing Borrower Empl. Borrower House Prices Wage. 2 4 Inflation MBS Policy Rate.2 5 x MBS Yield ω i Outright Money Injections Figure 4: Impulse response functions to a TFP shock. 26

28 . Output. Consumption.2 Employment x 3 Cons. Lender x Housing Lender 3.5 Empl. Lender Cons. Borrower Housing Borrower Empl. Borrower House Prices Wage.5 5 x 3 Inflation MBS.4 2 x Policy Rate x 4 MBS Yield ω i Outright Money Injections Figure 5: Impulse response functions to a housing demand shock. 27

29 2 x 4 Output 2 4 x 4 Cons. Lender x Cons. Borrower 4 x 4 House Prices x 4 MBS 2 x 4 Consumption x Housing Lender x Housing Borrower x 4 Wage Policy Rate 2 x 4 Employment x 4 Empl. Lender x Empl. Borrower x 4 Inflation 2 4 x 4 MBS Yield x 4 ω i x Outright Money x 3 Injections 2 4 Figure 6: Impulse response functions to a policy rate shock. 28

30 Output Consumption 2 Employment Cons. Lender Housing Lender 5 Empl. Lender Cons. Borrower 5 Housing Borrower Empl. Borrower House Prices 5 Wage Inflation MBS.5 Policy Rate MBS Yield ω i 4 Outright Money 4 Injections Figure 7: Impulse response functions to a MBS purchase shock. 29

31 .2 Output. Consumption.2 Employment Cons. Lender Housing Lender Empl. Lender Cons. Borrower Housing Borrower. 2 4 Empl. Borrower House Prices Wage. 5 x 3 Inflation MBS x 3 Policy Rate x 3 MBS Yield 2 4 ω i Outright Money Injections Figure 8: Impulse response functions to a credit shock. 3