Technical Appendix for Financial Business Cycles

Transcription

1 Technical Appendix for Financial usiness Cycles Matteo Iacoviello Federal Reserve oard December 4, 05 This appendix contains some of the derivations for the paper Financial usiness Cycles.. The Extended Model ere I present the extended model... ousehold Savers Savers solve X max t (A p;t ( ) log (C ;t C ;t ) + ja j;t A p;t log ;t + log ( N ;t )) subject to: t0 C ;t + K ;t + D t + q t ( ;t ;t ) + ac K;t + ac D;t A K;t R M;t z K;t + K;t K ;t + R ;t D t + W ;t N ;t (.) A K;t where the adjustment costs take the following form ac K;t K ac D;t D (K ;t K ;t ) K (D t D t ) and the depreciation function is K;t K + b K 0:5 0 zk;t + 0 zk;t + 0:5 0 D where 0 is a parameter measuring the curvature of the utilization rate function. 0 implies 0 0; approaching implies 0 approaches in nity and K;t stays constant. b K + K and implies a unitary steady state utilization rate. ac measures a quadratic adjustment cost for changing the quantity i between time t and time t +. The adjustment cost is external. abits are external too. A The household problem yields, denoting with u C;t p;t C ;t C ;t and u ;t ja j;ta p;t ;t the marginal

2 utilities of consumption and housing. u C;t t R ;t u C;t+ (.) W ;t u C;t (.3) N ;t u C;t K;t A R M;t+ z K;t+ + K;t+ u C;t+ (.4) ;t A K;t+ q t u C;t u ;t + q t+ u C;t+ (.5) R M;t 0 (z K;t ) (.6) where A K;t is an investment shock, A p;t is a consumption preference shock, A j;t is a housing demand shock... ousehold orrowers They solve subject to and to max X t S (A p;t ( ) log (C S;t C S;t ) + ja j;t A p;t log S;t + log ( N S;t )) t0 C S;t + q t ( S;t S;t ) + R S;t L S;t " ;t + ac SS;t L S;t + W S;t N S;t (.7) L S;t S L S;t + ( S ) m S A M;t q t+ R S;t S;t " ;t (.8) where " ;t is the borrower repayment shock, A M;t is a loan-to-value ratio shock. The adjustment cost is ac SS;t SS (L S;t L S;t ) The rst order conditions are, denoting with u CS;t Ap;t C S;t and u S;t ja j;ta p;t S;t the marginal utilities of consumption and housing; and with S;t u CS;t the (normalized) multiplier on the borrowing S;t S;t L S u CS;t S (R S;t S S;t+ ) u CS;t+ (.9) S W S;t u CS;t (.0) N S;t q t+ q t S;t ( S ) m S A M;t u CS;t u S;t + R S q t+ u CS;t+ (.) S;t.3. ankers ankers solve subject to: X max t log (C ;t C ;t ) t0 C ;t +R ;t D t +L E;t +L S;t +ac D;t +ac E;t +ac S;t D t +R E;t L E;t +R S;t L S;t " E;t " S;t (.)

3 where " E;t is the entrepreneur repayment shock. The adjustment costs are ac D;t D ac E;t E ac S;t S (D t D t ) D (L E;t L E;t ) L E (L S;t L S;t ) Denote " t " E;t + " S;t. Let L t L E;t + L S;t : The banker s constraint is a capital adequacy constraint of the form: (L t D t " t ) D (L t D t " t ) + ( ) ( D ) (L t " t ) bank equity bank assets stating that bank equity (after losses) must exceed a fraction of bank assets, allowing for a partial adjustment in bank capital given by D. Such constraint can be rewritten as a leverage constraint of the form L S D t D (D t (L E;t + L S;t (" E;t + " S;t ))) + ( ( ) ( D )) (L Et + L S;t (" E;t + " S;t )) (.3) The rst order conditions to the banker s problem imply, choosing D; L E ; L S and letting ;t u C;t be the normalized multiplier on the borrowing D;t ;t u (R ;t D ;t+ ) u C;t+ (.4) t ( E ( D ) + D ) ;t E;t u (R E;t+ D ;t+ ) u C;t+ (.5) E;t ( S ( D ) + D ) ;t S;t u (R S;t D ;t+ ) u C;t+ (.6) S;t.4. Entrepreneurs Entrepreneurs obtain loans and produce goods (including capital). Entrepreneurs hire workers and demand capital supplied by the household sector. subject to: X max t E log (C E;t C E;t ) t0 C E;t + K E;t A K;t +q t E;t +R E;t L E;t +W ;t N ;t +W S;t N S;t +R M;t z K;t K ;t (.7) Y t + KE;t A K;t K E;t + q t E;t +L E;t + " E;t + ac KE;t + ac EE;t and to Y t A Z;t (z K;t K ;t ) (z KE;t K E;t ) ( ) E;t ( )( ) N ( ) ;t N S;t (.8) 3

4 where A Z;t is a shock to total factor productivity. The adjustment costs are ac KE;t KE ac EE;t EE (K E;t K E;t ) K E (L E;t L E;t ) Note that symmetrically to the household problem entrepreneurs are subject to an investment shock, can adjust the capital utilization rate, and pay a quadratic capital adjustment cost. The depreciation rate is governed by KE;t KE + b KE 0:5 0 EzKE;t + 0 E zke;t + 0:5 0 E where setting b KE + KE implies a unitary steady state utilization rate. E Entrepreneurs are subject to a borrowing/pay in advance constraint that acts as a wedge on the capital and labor demand. The constraint is q t+ L E;t E L E;t + ( E ) A ME;t m E;t + m K K E;t m N (W ;t N ;t + W S;t N S;t ) (.9) R E;t+ Letting u CE;t be the marginal utility of consumption and E;t u CE;t the normalized borrowing constraint, the rst order conditons for L E ; K E and E are: E;t LE;t u E (R E;t+ E E;t+ ) u CE;t+ (.0) E;t KE;t E;t E ) m K A ME;t u CE;t E ( KE;t+ + R K;t+ z KE;t+ ) u CE;t+ (.) E;t q t+ q t E;t ( E ) m A ME;t u CE;t R E q t+ ( + R V;t+ ) u CE;t+ (.) E;t+ Additionally, these conditions can be combined with those of the production arm of the rm, giving.5. Equilibrium L E Y t R K;t z KE;t K E;t (.3) ( ) Y t R M;t z K;t K ;t (.4) Y t R V;t q t E;t (.5) ( ) ( ) Y t W ;t N ;t ( + m N A ME;t E;t ) (.6) ( ) Y t W S;t N S;t ( + m N A ME;t E;t ) (.7) R K;t 0 (z KE;t ) (.8) Market clearing is implied by Walras s law by aggregating all the budget constraints. For housing, we have the following market clearing condition ;t + S;t + E;t (.9) The model endogenous variables are 4: quantities Y E S K E K N N S C C E C C S z K z KE 3: loans & deposits L E L S D 3: prices q W W S 6: interest rates R K R M R V R E R S R 3: multipliers E S 4

5 together with the de nition of the depreciation rate functions and the adjustment cost functions given in the text above..6. Shocks The following zero-mean, AR() shocks are present in the estimated version of the model " E;t " ;t log A j;t log A K;t log A ME;t log A M;t log A p;t log A z;t The shocks follow the processes given by: " E;t be " E;t + u E;t ; u E N(0; be ) " ;t bh " ;t + ;t ; u N(0; bh ) log A j;t j log A j;t + j;t ; u j N(0; j ) log A K;t K log A K;t + K;t ; u K N(0; k ) log A ME;t me log A ME;t + ME;t ; u ME N(0; me ) log A M;t mh log A M;t + M;t ; u M N(0; mh ) log A p;t p log A p;t + p;t ; u p N(0; p ) log A Z;t z log A Z;t + z;t ; u z N(0; z ) 5

6 . Steady State of Extended Model Interest Rates and Multipliers. Let s begin with them R (.) R D (.) R E ( ) D + ( D ) E R D (.3) R S ( ) D + ( D ) S R D (.4) E ER E E E (.5) S SR S S S (.6) ( R V E E ) m E E R E (.7) R K E ( ) E m K ( E ) (.8) R M ( ) (.9) This implies the following restrictions on discount factors (for multipliers to be positive). > 0 if < E > 0 if E R E < E (( ) D + ( D ) E ) R < D if 0 E < E + E S > 0 if S R S < S (( ) D + ( D ) S ) R < D Some Useful Constants. List of constants 6

7 oo j (.0) oo j S S ( S ) m S R S (.) oo 3 (.) + ( R S ) m S oo m oo 4 E (R ) + m K m N (.3) R V R E R K R E + m N E m S oo 5 S oo oo 3 (R ) (.4) R S oo 6 R M (.5) ( ) ( ) oo 7 (.6) + m N E oo 8 ( ) R M (.7) Start with Ratios, then Move to Levels. The housing consumption ratios are: q oo C (.8) q S oo C S (.9) q E Y R V (.0) The ouseholds Use C (R M ) K + (R ) D + W N (.) so D E L E + S L S E (m q E R E + m K K E R E m N (W N + W S N S )) + S m S q S R S (.) Also m C (R M ) K + E (R ) q E + m K K E m N (W N + W S N S ) R E R R + S m S R S (R ence C S and C can be rewritten as ) q S + W N + F (.3) C S W S N S ( R S ) m S q S (.4) C S W S N S oo 3 W S N S (.5) + ( R S ) m S oo m C (R M ) K + E (R ) q E + m K K E m N (W N + W S N S ) R E R R + S m S R S (R ) oo oo 3 W S N S + W N (.6) oo 6 K + oo 4 Y + oo 5 W S N S + W N (.7) 7

8 Use so that W N oo 7 Y (.8) C oo 6 K + + oo 4 W N + oo 5 W S N S (.9) oo 7 Labor Market Equilibrium. Labor supplies W C W S C S (.30) N S (.3) N S We want to solve this mess for N: The trick is to write everything as a function of W N in the budget constraint. Note that ence: W N oo 7 Y (.3) W S N S W N (.33) K oo 8 Y (.34) K oo 8 oo 7 W N (.35) ence from all of this we get, using and likewise W oo oo 8 6 oo oo 4 oo 7 + oo 5 W S oo 3 W S N S W N oo 8 z oo oo 4 + oo 5 oo 7 oo 7 z N N > N N S N (.36) S N S (.37) (.38) + z (.39) + oo 3 S (.40) The Consumption Output Ratios. consumption output ratios Given the housing consumption ratios above, using the following ( ) cy S oo 3 + E cy z oo 7 cy E + E (R ) m q E + m K K E m N (W N + W S N S ) R E R R (.4) (.4) (.43) R K one can solve for can be solved using the stu below. Note that the cy E ratio is not needed, I just 8

9 put it there in case. q oo cy Y (.44) q S oo cy S Y (.45) q E Y R V (.46) + S + E (.47) Finally, the Levels. ence Y ( ) R M ( ) R K E N N S (.48) K E Y R K (.49) K ( ) Y R M (.50) L E m q E R E + m K K E R E m N + m N E Y (.5) W + m N E ( ) Y N (.5) W S + m N E Y N S (.53) L S m S R S oo C S m S R S oo oo 3 W S N S (.54) Then solve for C ; C E and C S as well as D (can use simpler formulas now). I would use ratios de ned above for cy S ; cy and cy E and then go back to levels for bankers D E L E + S L S (.55) C (R E ) L E + (R S ) L S (R ) D (.56) 9

10 3. Derivation of Demand and Supply Curves in Figure 5 In this framework, one can plot and analyze demand and supply curves very easily. Let the adjustment costs functions and derivatives be written in this form t LE (L t L t ) L E t d t dl t LE L (L t L t ) 0 (L t L t ) esides, assume that adjustment costs are external, so that the expectation of future adjustment costs does not show up in the Euler equation. Using this functional form, the parameter LE ends up measuring the semielasticity of loan demand to the interest rates (up to rst order, as will be shown below). To be more precise, when quarterly interest rates rise by 5 basis points say (annual rates go up by 00 basis points), loan demand falls in percentage terms by 0:5 LE : A LE 0:5 implies for instance a drop in loan demand by percent following a % rise in interest rates. 3.. Entrepreneurs: Loan demand To save on notation, I let L E;t be L t in what follows. Loan demand is the schedule described by equation.0. Taking into account adjustment costs, the loan demand equation is implicitly de ned by the following equation, where L t and R t+ are loan demand and loan rate, and everything else is a shifter. LE L (L t L t ) E;t C E E;t E (R E;t+ E;t+ ) C E E;t+ (l_d) When a shock hits that a ects all variables, one can trace how loan demand reacts but plotting all combinations of L and R such that the equation above holds. I just linearize this equation around the steady state to make it more transparent. After linearization, this equation becomes: L t L t 0 ( E;t E ) LE E ( E ) 0 LE (C E;t C E;t+ ) 0 LE E R E E E (R E;t+ R E E ( E;t+ E )) (l_d_linearized) where a variable without time subscript denotes the steady state value. Using this equation, one can note that, when E is zero, for small values of E and R E close to one, we have that hence 4 LE dl t L dr t+ LE is the semielasticity of loan demand to changes in the annualized interest rate. 3.. anks: Loan Supply Loan supply is given by equation L (L t L t ) ( E ( D ) + D ) ;t C ;t RE;t+ t+ D ;t+ C ;t+ (l_s) After we rearrange, denoting with E ( D ) + D ; we have L t L t + 0 ( t )+ ( ) L 0 L C (C t C t+ )+ 0 (R t+ R E L R E D D ( ;t+ )) (l_s_linearized) For small D ; (4 L ) is the elasticity of loan supply to changes in the interest rate. 0

11 3.3. ouseholds: Deposits Supply Deposit supply is the linearized version of equation.. Starting from u C;t D;t R u C;t+ t t+ (d_s) we can rewrite this as log u C;t + log D;t log + log R t + log u t bu C;t + bdt Dt b R b t + bu C;t+ bd t Dt b brt + bu C;t+ bu C;t Therefore this simpli es to (in levels) D t D t D SS Rt R SS + u C;t+ u C;t R SS u C;SS (d_s_linearized) 3.4. anks: Deposit Demand Deposit demand is given by equation.4 0 D (D t D t ) ;t C ;t R;t t+ D ;t+ C ;t+ (d_d) After linearization, this equation becomes D t D t 0 ( ;t ) D ( ) 0 D C (C ;t C ;t+ ) 0 D R D (R ;t R D ( ;t+ )) (d_d_linearized) 3.5. Entrepreneurs: Capital Supply Capital supply is given by equation.: + 0 KE (K E;t K E;t ) ( E ) m K E;t C E E;t E ( KE;t+ + R K;t+ ) C E E;t+ (k_s) After we arrange, letting E (( E ) m K ) + 0 KE K E;t K E;t + E 0 ( E;t E ) + E ( E E ) KE 0 KE C (C E;t C E;t+ ) E E E R KE + KE (R KE;t+ R KE ( KE;t+ KE )) (k_s_linearized) If adding LTV shocks, K E;t K E;t Since + E ( 0 E;t E ) + E E m K (m KE 0 K;t m K ) + ::: KE R K;t b KE 0 Ez KE;t + 0 E zke;t KE;t KE + b KE 0:5 0 z KE;t + 0 z KE;t + 0:5 0

12 It makes sense to express capital supply as a function of which depends on R: To do so note that t b KE (z t z) R K;t R b KE + 0 (z t z) t + 0 (R K;t R) ( ) (R K;t R) which highlights how makes capital supply more or less elastic Firms: Capital Demand Capital demand is the remarkably linearized version of equation.3 (using the formula for Y t given by the production function) that this page is too small to contain.