AEA Continuing Education Program. DSGE Models and the Role of Finance. Lawrence Christiano, Northwestern University

Transcription

1 AEA Continuing Education Program DSGE Models and the Role of Finance Lawrence Christiano, Northwestern University January 7-9, 2018

2 Two Period Version of Gertler Karadi, Gertler Kiyotaki Financial Friction Model Lawrence J. Christiano Summary of Christiano Ikeda, 2012, Government Policy, Credit Markets and Economic Activity, in Federal Reserve Bank of Atlanta conference volume, A Return to Jekyll Island: the Origins, History, and Future of the Federal Reserve, Cambridge University Press.

3 Motivation Beginning in 2007 and then accelerating in 2008: Asset values (particularly for banks) collapsed. Intermediation slowed and investment/output fell. Interest rates spreads over what the US Treasury and highly safe private firms had to pay, jumped. US central bank initiated unconventional measures (loans to financial and non financial firms, very low interest rates for banks, etc.) In 2009 the worst parts of began to turn around.

4 Collapse in Asset Values and Investment Log, real Stock Market Index, real Housing Prices and real Investment March, 2006 October, June, log September, S&P/Case-Shiller 10-city Home Price Index S&P 500 Index Gross Private Domestic Investment March, month

5 Spreads for Risky Firms Shot Up in Late 2008 Interest Rate Spread on Corporate Bonds of Various Ratings Over Rate on AAA Corporate Bonds BB B CCC and worse 2008Q mean, junk rated bonds = mean, B rated bonds = mean, BB rated bonds = 1.75

6 Must Go Back to Great Depression to See Spreads as Large as the Recent Ones Spread, BAA versus AAA bonds 5 March, October, 2007 August,

7 Economic Activity Shows (anemic!) Signs of Recovery June, 2009 Percent of Labor Force Unemployment rate Month Log, Industrial Production Index September, Log Month

8 Banks Cost of Funds Low Federal Funds Rate 6 5 Annual, Percent Rate September, Month

9 Characterization of Crisis to be Explored Here Bank Asset Values Fell. Banking System Became Dysfunctional Interest rate spreads rose. Intermediation and economy slowed. Monetary authority: Transferred funds on various terms to private companies and to banks. Sharply reduced cost of funds to banks. Economy in (tentative) recovery. Seek to construct models that links these observations together.

10 Objective Keep analysis simple and on point by: Two periods Minimize complications from agent heterogeneity. Leave out endogeneity of employment. Leave out nominal variables: just look behind the veil of monetary economics Models: Gertler Kiyotaki/Gertler Karadi In two period setting easy to study an interesting nonlinearity that is possible: Participation constraint may be binding in a crisis and not binding in normal times.

11 Two period Version of GK Model Many identical households, each with a unit measure of members: Some members are bankers Some members are workers Perfect insurance inside households everyone consumes same amount. Period 1 Workers endowed with y goods, household makes deposits, d, in a bank Bankers endowed with N goods, take deposits and purchase securities, d, from a firm. Firm issues securities, s, to produce sr k in period 2. Period 2 Household consumes earnings from deposits plus profits, π, from banker. Goods consumed are produced by the firm.

12 Problem of the Household period 1 period 2 budget constraint c d y C R d d problem max c,c,d u c u C Solution to Household Problem Solution to Household Problem u c R d c C y u u c C R d R R d c C y d R d R d u C u c c1 u c c1 1 c c y y R d R1 d 1 Rd 1 R d Rd R d 1

13 Solution to Household Problem u c u C R d c C R d y R d u c c1 1 c y R d 1 Rd R d 1 Household budget constraint when gov t buys Solution to Household Problem private assets using tax receipts, T, and gov t u c R d c C y u C R d R gets the same rate of return, R d, as d households: No change! y (Ricardian Wallace u c c1 R c d Irrelevance) 1 1 c C R d y T TRd 1 Rd y R d R d R d

14 Problem of the Household period 1 period 2 budget constraint c d y C R d d problem max c,c,d u c u C Solution to Household Problem Solution to Household Problem u c R d c C y u u c C R d R R d c C y d R d R d u C u c c1 u c c1 1 c c y y R d R1 d 1 Rd 1 R d Rd R d 1

15 Household Supply of Deposits For given π, d rises or falls with R d, depending on parameter values. But, in equilibrium π=r k (N+d) R d d. Substituting into the expression for c and solving for d: R d d Rd 1 N y R k y R d 1 R k Upward sloping deposit supply d

16 Household Supply of Deposits For given π, d rises or falls with R d, depending on parameter values. But, in equilibrium π=r k (N+d) R d d. Substituting into the expression for c and solving for d: R d d Rd 1 N y R k y R d 1 R k N decreases d

17 Properties of Equilibrium Household Supply of Deposits Deposits increasing in R d. Shifts right with decrease in N because of wealth effect operating via bank profits, π. rise in deposit supply smaller than decrease in N. 0, 1 d N R k R d 1 R k

18 Efficient Benchmark Problem of the Bank period 1 period 2 take deposits, d pay dr d to households buy securities, s N d receive sr k from firms problem: max d sr k R d d

19 Bank demand for d R d Supply of d by households Demand for d by banks R k Equilibrium d d

20 Equilibrium in Absence of Frictions Properties: Interior Equilibrium: R d,,d,c,c (i) c,d,c 0 (ii) household problem is solved (iii) bank problem is solved (iv) goods and financial markets clear Household faces true social rate of return on saving: R k R d Equilibrium is first best, i.e., solves max c,c,k, u c u C c k y N, C kr k

21 Friction bank combines deposits, d, with net worth, N, to purchase N+d securities from firms. bank has two options: ( no default ) wait until next period when N d R k arrives and pay off depositors, R d d, for profit: N d R k R d d ( default ) take N d securities, refuse to pay depositors and wait until next period when securities pay off: N d R k Bank must announce what value of d it will choose at the beginning of a period.

22 Incentive Constraint Recall, banks maximize profits Choose no default iff no default N d R k R d d default N d R k Next: derive banking system s demand for deposits in presence of financial frictions.

23 Result for a no default equilibrium: Consider an individual bank that contemplates defaulting. It sets a d that implies default,,, or R k N d R d d R k d N what the household gets in the other banks R d what the household gets in the defaulting bank 1 R k d N d A deviating bank will in fact receive no deposits. An optimizing bank would never default

24 Problem of the bank in no default, interior equilibrium Maximize, by choice of d, subject to: R k N d R d d If interest rate is REALLY low, then bank has no incentive to default because it makes lots of profits not defaulting or, R k N d R d d R k N d 0, 1 R k N R d 1 R k d 0. Note that 0 < d < requires 1 R k if not, then d R d if not, then d 0 Rk.

25 Problem of the bank in no default, For R d = R k interior equilibrium, cnt d a bank makes no profits on d so absent default considerations it is indifferent over all values of 0 d Taking into account default, a bank is indifferent over 0 d N(1 θ)/θ For (1 θ)r k < R d < R k Bank wants d as large as possible, subject to incentive constraint. So, d = R k N(1 θ)/(r d (1 θ)r k )

26 Bank demand for d R d Bank demand for d R k 1 R k R d 1 R k N (1 θ)r k 1 N d

27 Interior, no default equilibrium R d Household supply R k Bank demand In this equilibrium, R d = R k and first best allocations occur. Banking system is highly effective in allocating resources efficiently. d

28 Collapse in Bank Net Worth Suppose that the economy is represented by a sequence of repeated versions of the above model. In the periods before the crisis, net worth was high and the equilibrium was like it is on the previous slide: efficient, with zero interest rate spreads. In practice, spreads are always positive, but that reflects various banking costs that are left out of this model. With the crisis, N dropped a lot, shifting demand to the right and supply to the left.

29 Effect of Substantial Drop in Bank Net Worth R d Initial, efficient equilibrium Household supply R k Bank demand Equilibrium after N drops is inefficient because R d < R k. d

30 Government Intervention Equity injection. Government raises T in period 1, provides proceeds to banks and demands R k T in return at start of period 2. Rebates earnings to households in 2. Has no impact on demand for deposits by banks (no impact on default incentive or profits). Reduces supply of deposits by households. d+t rises when T rises (even though d falls) because R d rises. Direct, tax financed government loans to firms work in the same way. An interest rate subsidy to banks will shift their demand for deposits to the right.no impact on supply curve when subsidy financed by period 2 lump sum tax on households.

31 Equity Injection and Drop in N R d Household supply R k Tax financed injection of equity into banks or direct loans to non financial firms shift household supply left. Bank demand d

32 Basic idea: Recap Bankers can run away with a fraction of bank assets. If banker net worth is high relative to deposits, friction not a factor and banking system efficient. If banker net worth falls below a certain cutoff, then banker must restrict the deposits. Bankers fear (correctly) that otherwise depositors would lose confidence and take their business to another bank. Reduction in banker demand for deposits: makes deposit interest rates fall and so spreads rise. Reduced intermediation means investment drops, output drops. Equity injections by the government can revive the banking system.

33 Is the Model Narrative Consistent with the Evidence? Model says that reduced intermediation of funds through the financial system reflected reduced demand for credit by financial institutions. Prediction: interest rate to financial institutions fall.

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35 Model prediction for decline in cost of funds to financial institutions seems verified. But, other risk free interest rates fell even more. Interest rates on US government debt fell more than interest rate on financial firm commercial paper.

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37 Assessment Fact that interest rates on US government debt went down more than cost of funds to financial institutions suggests that a complete picture of financial crisis may require two additional features: Risky Banks: Banks in the model are risk free. Default only occurs out of equilibrium. Increased actual riskiness of banks is perhaps also an important part of the picture. Liquidity: Low interest rates on US government debt consistent with idea that high demand for liquidity played an important role in the crisis.

38 Macro Prudential Policy In recent years there has been increased concern that banks may have a tendency to take on too much debt. Has accelerated thinking about debt restrictions on banks. There are several models of financial frictions in banks, but they do not necessarily provide a foundation for thinking about debt restrictions on banks. A CSV model of banks implies they issue too little debt. (See Christiano Ikeda). The running away model of banks does not rationalize debt restrictions. (See next).

39 Optimal Debt Restriction in Two Period Running Away Banking Model Debt restriction on banks: d d What is the socially optimal level of? To answer this, must take into account structure of private economy The way households choose debt in competitive markets The fact that banks will not choose a debt level that violates incentive constraints. d

40 Social Welfare Function u c u C u y d c u earnings on deposits R d d C bank profits R k N d R d d u y d u R k N d.

41 Optimization: Household Saving u y d R d u C plus budget constraint and definition of profits (see above) implies: or d 1 Rd R k n y y R d 1 R k R d 1 d n y d Rk f d

42 Implementability Constraint Let d * denote the value of deposits that a benevolent planner wishes the banks would choose. Planner must take into account: banks will not choose a level of d which implies a violation of the incentive constraint. market arrangement in which households make their deposit supply decision. these considerations restrict d as follows: 1 N d R k f d d 0

43 Planning Problem d * is solution to the following problem: max u y d u R k N d 1 N d R k f d d d Fonc u y d /R d by households u y d u C R k 1 R k f d d f d 0 0, 1 N d R k f d d 0, 1 N d R k f d d 0.

44 Planning Problem d * is solution to the following problem: max u y d u R k N d 1 N d R k f d d d Fonc u y d R k f d 1 1 Rk f d d f d 0 Complementary Slackness 0, 1 N d R k f d d 0, 1 N d R k f d d 0

45 Planning Problem First order conditions: u y d Solving the problem: R k f d 1 1 Rk f d d f d 0 Complementary Slackness 0, 1 N d R k f d d 0, 1 N d R k f d d 0 Try 0 and solve ( saving supply crosses horizontal line at R k ) R k f d Check incentive constraint. If satisfied, Otherwise, conclude 0 and R k f d 1 N d R k f d d 0 ( Savings supply crosses incentive constraint ).

46 No Borrowing Restrictions Desired Deposits selected by government coincide with equilibrium deposits when there is no borrowing restriction. So, according to the model, restriction on bank borrowing not necessary. Model is not a good laboratory for thinking about leverage restrictions on banks, if you re firmly convinced that leverage restrictions are required.

47 Rollover Crisis in DSGE Models Lawrence J. Christiano Northwestern University

48 Why Didn t DSGE Models Forecast the Financial Crisis and Great Recession? Bernanke (2009) and Gorton (2008): By 2005 there existed a very large and highly-levered Shadow Banking system. It relied on short-term debt to fund long-term liabilities. So, it was vulnerable to a run. The overwhelming majority of academics, regulators and practitioners simply did not recognize this development, or understand its significance. The widespread belief (baked into DSGE models) was that if a country had deposit insurance, bank runs were a thing of the past.

49 Integrating Rollover Crisis into DSGE Models Will talk, at an intuitive level, about Gertler-Kiyotaki (AER2015). More full-blown models by Gertler-Kiyotaki-Prestipino

50 This is what a bank run looked like in the 19th century: Diamond-Dybvig run. Bank runs in 2007 and 2008 were different and did not look like this at all (Gorton)! It was a rollover crisis in a shadow (invisible to normal people) banking system.

51 Rolling over Consider the following bank: Assets Liabilities 120 Deposits: 100 Banker net worth 20 This bank is solvent : at current market prices could pay off all liabilities. Suppose that the bank s assets are long term mortgage backed securities and the liabilities are short term (six month) commercial paper. The bank relies on being able to roll over its liabilities every period. Normally, this is not a problem.

52 Rolling over Now suppose the bank cannot roll over its liabilities. In this case, the bank would have to sell its assets. If only one bank had to do this: no problem, since the bank is solvent. But, suppose all banks face a roll over problem. Now there may be a big problem! In this case, assets must be sold to another part of the financial system, a part that may have no experience with the assets (mortgage backed securities).

53 Rollover crisis (Nash) equilibrium Suppose an individual depositor, Jane, believes all other depositors will refuse to roll over. Suppose Jane believes that the fire sales of assets will wipe out bank net worth. Then, Jane can expect to lose money on the deposit she made with the bank in the previous period. But, that loss is sunk, and nothing can be done about it. Need some other friction to guarantee that Jane will herself refuse to roll over her deposit.

54 Rollover crisis (Nash) equilibrium Absent other frictions, Jane would just renew her own deposit and the rollover crisis would not be a Nash equilibrium. So, Gertler-Kiyotaki assume that bankers can run away with a fraction of bank assets. With zero net worth, banks would definitely run away. This is why Jane would choose not to roll over her deposit, if she believed everyone else would also choose not to roll over. The logic of the rollover crisis equilibrium is a little different from the bank run equilibrium: Suppose Jane thinks everyone else will take their money out of the bank. Then, it makes sense for Jane to run faster than everyone else, to get to the front of the line.

55 The Drama of a Roll Over Crisis Brought to Life in Some Great Movies!

56 Why firesales? A rollover crisis: when all banks in an industry (e.g., mortgage backed securities industry) are unable to roll over their liabilities. The only buyers of the securities have no experience with them, so they won t buy without a price cut (firesale). Interestingly, the buyers of the securities will all complain at how complex they are and how non-transparent they are. But, the real problem is that buyers in a fire sale are simply inexperienced. The rollover crisis hypothesis contrasts with the Big Short hypothesis: assets were fundamentally bad (Mian and Sufi).

57 Rollover crisis When the whole industry has to sell, then bank balance sheets could suddenly look like this: Fire sale value of assets: Assets Liabilities 90 Deposits: 100 Banker net worth -10 Multiple equilibrium: balance sheet could be the above, with run, or the following, with no run: A run could happen, or not. Assets Liabilities 120 Deposits: 100 Banker net worth 20 This is exactly the sort of financial fragility that regulators want to avoid! Under rollover crisis hypothesis, this was the situation in summer 2007.

58 Rollover Crisis: Role of Housing Market What matters is the actual value of assets and their firesale value. If bank is solvent under (firesale value), then probability of run is zero. Pre-housing market correction Assets Liabilities 120 (105) Deposits: 100 Banker net worth 20 (5) Post-housing market correction Assets Liabilities 110 (95) Deposits: 100 Banker net worth 10 (-5) Rollover Crisis Hypothesis: pre-2005, no crisis possible, post-2005 crisis possible.

59 Trigger: house prices stopped rising in May 2006 Housing Price Correction Need Not Have Led to House Price Collapse and Collapse in Economy.

60 How to think about regulation when the risk is of a rollover crisis. One possibility: model the rollover crisis directly. Serious model of rollover crisis at this time: Gertler-Kiyotaki (AER2015). They adapt the rollover crisis model of sovereign debt created by Cole-Kehoe (JIE1996). Cole-Kehoe related to Diamond-Dybvig.

61 Possible states: s = 1, 2, 3,, T+2. Bank run, s = 1. No bank run in s > 1. In each no-run state there is a chance of a run in the next state, unless s = 2. Steady state s=t+2 s=4 s=3 s=2 Run, Run state s=1 s = 1.

62 One Hundred Year Stochastic Simulation Price of Capital, Q Probability of a bank run in t+1, p Bank net worth, N percent deviation from steady state GDP

63 Policy Use of Model Investigate the impact on financial stability of leverage restrictions. But, this analysis is hard! Clearly, it is only in its infancy At the heart of the analysis: Assume that people know what can happen in a crisis, together with the associated probabilities. This seems implausible, given the fact that a full-blown crisis is a two or three times a century rare event. Safe to conjecture that factors such as aversion to Knightian uncertainty play an important role driving fire sales in a crisis. Still, research on various types of crises is proceeding at a rapid pace, and we expect to see substantial improvements in DSGE models on the subject.

64 Conclusion Models of rollover risk seem important in light of the crisis. These models are in their infancy, a long way from being operational for quantitative policy analysis. Possibility: assume that governments will always act as lender of last resort. Use toy models to illustrate the idea of rollover crisis. For quantitative analysis, use models that do not allow rollover crisis, but do capture moral hazard implications of bailouts. Monitor the Shadow Banking system closely.

65 Notes on Financial Frictions Under Asymmetric Information and Costly State Verification by Lawrence Christiano

66 Incorporating Financial Frictions into a Business Cycle Model General idea: Standard model assumes borrowers and lenders are the same people..no conflict of interest Financial friction models suppose borrowers and lenders are different people, with conflicting interests Financial frictions: features of the relationship between borrowers and lenders adopted to mitigate conflict of interest.

67 Discussion of Financial Frictions Simple model to illustrate the basic costly state verification (csv) model. Original analysis of Townsend (1978), Bernanke Gertler. Integrating the csv model into a full blown dsge model. Follows the lead of Bernanke, Gertler and Gilchrist (1999). Empirical analysis of Christiano, Motto and Rostagno (2003,2012).

68 Simple Model There are entrepreneurs with all different levels of wealth, N. Entrepreneur have different levels of wealth because they experienced different idiosyncratic shocks in the past. For each value of N, there are many entrepreneurs. In what follows, we will consider the interaction between entrepreneurs with a specific amount of N with competitive banks. Later, will consider the whole population of entrepreneurs, with every possible level of N.

69 Simple Model, cont d Each entrepreneur has access to a project with rate of return, Here, is a unit mean, idiosyncratic shock experienced by the individual entrepreneur after the project has been started, 1 R k 0 df 1 The shock,, is privately observed by the entrepreneur. F is lognormal cumulative distribution function.

70 Banks, Households, Entrepreneurs ~ F, 0 df 1 entrepreneur entrepreneur Households Bank entrepreneur entrepreneur entrepreneur Standard debt contract

71 Entrepreneur receives a contract from a bank, which specifies a rate of interest, Z, and a loan amount, B. If entrepreneur cannot make the interest payments, the bank pays a monitoring cost and takes everything. Total assets acquired by the entrepreneur: total assets net worth loans A N B Entrepreneur who experiences sufficiently bad luck,, loses everything.

72 Cutoff, gross rate of return experience by entrepreneur with luck, 1 R k interest and principle owed by the entrepreneur ZB 1 R k A ZB Z 1 R k B N A N Z 1 R k leverage L A 1 N A N Z 1 R k total assets A L 1 L Cutoff higher with: higher leverage, L higher Z/ 1 R k

73 Expected return to entrepreneur from operating risky technology, over return from depositing net worth in bank: Expected payoff for entrepreneur 1 R k A ZB df N 1 R For lower values of, entrepreneur receives nothing limited liability. gain from depositing funds in bank ( opportunity cost of funds )

74 Rewriting entrepreneur s rate of return: 1 R k A ZB df N 1 R 1 R k A 1 R k A df N 1 R df 1 R k 1 R L Z 1 R k L 1 Z L L 1 R k Gets smaller with L Entrepreneur s return unbounded above Larger with L Risk neutral entrepreneur would always want to borrow an infinite amount (infinite leverage).

75 Rewriting entrepreneur s rate of return: 1 R k A ZB df N 1 R 1 R k A 1 R k A df N 1 R df 1 R k 1 R L Z 1 R k L 1 L L Z 1 R k Entrepreneur s return unbounded above Risk neutral entrepreneur would always want to borrow an infinite amount (infinite leverage).

76 Expected entrepreneurial return, over opportunity cost, N(1+R) Equilibrium spread in numerical example developed in these notes. Entrepreneur would prefer to be a depositor for (leverage, interest rate spread), combinations that produce results below horizontal line leverage Interest rate spread, Z/(1+R), = , or 0.63 percent at annual rate Return spread, (1+R k )/(1+R), = , or 2.90 percent at annual rate 0.26

77 Expected entrepreneurial return, over opportunity cost, N(1+R) High leverage always preferred eventually linearly increasing Z/(1+R)=1.05, or 20 percent at annual rate leverage Interest rate spread, Z/(1+R), = , or 0.63 percent at annual rate Return spread, (1+R k )/(1+R), = , or 2.90 percent at annual rate 0.26

78 If given a fixed interest rate, entrepreneur with risk neutral preferences would borrow an unbounded amount. In equilibrium, bank can t lend an infinite amount. This is why a loan contract must specify both an interest rate, Z, and a loan amount, B.

79 Simplified Representation of Entrepreneur Utility Utility: Where df 1 R k 1 R L 1 Γ 1 Rk 1 R L Γ 1 F G Easy to show: G 0 df 0 Γ 1 Γ 1 F 0, Γ 0 limγ 0, lim 0 Γ 0 lim 0 G 0, lim G 1 Share of gross entrepreneurial earnings kept by entrepreneur

80 Banks Source of funds from households, at fixed rate, R Bank borrows B units of currency, lends proceeds to entrepreneurs. Provides entrepreneurs with standard debt contract, (Z,B)

81 Banks, cont d Monitoring cost for bankrupt entrepreneur with Bankruptcy cost parameter 1 R k A Bank zero profit condition fraction of entrepreneurs with 1 F quantity paid by each entrepreneur with ZB quantity recovered by bank from each bankrupt entrepreneur 1 0 df 1 R k A amount owed to households by bank 1 R B

82 Zero profit condition: Banks, cont d 1 F ZB 1 0 df 1 R k A 1 R B 1 F ZB 1 0 df 1 R k A B 1 R The risk free interest rate here is equated to the average return on entrepreneurial projects. This is a source of inefficiency in the model. A benevolent planner would prefer that the market price savers correspond to the marginal return on projects (Christiano-Ikeda).

83 Banks, cont d Simplifying zero profit condition: 1 F ZB 1 0 df 1 R k A 1 R B 1 F 1 R k A 1 0 df 1 R k A 1 R B share of gross return, 1 R k A, (net of monitoring costs) given to bank 1 F 1 0 df 1 R k A 1 R B 1 F 1 0 df Expressed naturally in terms of Expressed naturally in terms of,l,l 1 R B/N 1 R k A/N 1 R L 1 1 R k L

84 Bank zero profit condition, in (leverage, - bar) space parameters: 1 Rk 1 R , 0. 21, bar leverage Our value of 1 Rk, 290 basis points at an annual rate, is a little higher than the 200 basis point value adopted in 1 R BGG (1999, p. 1368); the value of is higher than the one adopted by BGG, but within the range, defended by Carlstrom and Fuerst (AER, 1997) as empirically relevant; the value of Var log is nearly the same as the 0.28 value assumed by BGG (1999,p.1368).

85 Expressing Zero Profit Condition In Terms of New Notation share of entrepreneurial profits (net of monitoring costs) given to bank 1 F 1 df 1 R L R k L Γ G 1 R L 1 1 R k L L Rk Γ G 1 R

86 Equilibrium Contract Entrepreneur selects the contract is optimal, given the available menu of contracts. The solution to the entrepreneur problem is the that maximizes, over the relevant domain (i.e., 0,1.13 in the example): profits, per unit of leverage, earned by entrepreneur, given leverage offered by bank, conditional on log df 1 R k 1 R 1 1 Rk 1 R 1 Γ G higer drives share of profits to entrepreneur down (bad!) higher drives leverage up (good!) log 1 Γ log 1 Rk 1 R log 1 1 Rk 1 R Γ G

87 Entrepreneur Objective entrepreneurial utility entrepreneurial objective as a function of bar entrepreneurial utility entrepreneurial objective as a function of bar bar bar 0 derivative of log entrepreneurial objective derivative of log entrepreneurial objective relevant range of s bar bar

88 Computing the Equilibrium Contract Solve first order optimality condition uniquely for the cutoff, : elasticity of entrepreneur s expected return w.r.t. 1 F 1 Γ elasticity of leverage w.r.t. 1 R k 1 F 1 R F 1 1 Rk Γ G 1 R Given the cutoff, solve for leverage: L Rk 1 R Γ G Given leverage and cutoff, solve for risk spread: risk spread Z 1 R 1 Rk 1 R L L 1

89 Result Leverage, L, and entrepreneurial rate of interest, Z, not a function of net worth, N. Quantity of loans proportional to net worth: L A N N B N B L 1 N 1 B N To compute L, Z/(1+R), must make assumptions about F and parameters. 1 R k 1 R,, F

90 Numerical Example Parameters: 1 R k 1 R Percent of average product of entrepreneurial Projects, absorbed by monitoring costs: 0.06% , 0. 26, 0.21 (Micro) equilibrium quantities: cutoff fraction of gross entrepreneurial earnings going to lender bankruptcy rate: 0.56% average among bankrupt entrepreneurs 0.50, Γ , F , G , leverage L 2.02, interest rate spread ZR 0.62 (APR) , avg earnings of entrepreneur, divided by opportunity cost 1 Γ 1 Rk 1 R L Note: on average, entrepreneur better off leveraging net worth and investing in project, rather than depositing net worth in bank.

91 Effect of Increase in Risk, Keep 0 df 1 But, double standard deviation of Normal underlying F. Impact on log normal density of doubling standard deviation density Doubled standard deviation Increasing standard deviation raises density in the tails

92 Jump in Risk replaced by 3 cutoff fraction of gross entrepreneurial earnings going to lender bankruptcy rate: 1.08% average among bankrupt entrepreneurs 0. 12, Γ 0. 12, F , G , leverage L , interest rate spread ZR 1.66 (APR) , avg earnings of entrepreneur, per unit of net worth 1 Γ 1 Rk 1 R L Comparison with benchmark: cutoff fraction of gross entrepreneurial earnings going to lender bankruptcy rate: 0.56% average among bankrupt entrepreneurs 0.50, Γ , F , G , leverage L 2.02, interest rate spread ZR 0.62 (APR) , avg earnings of entrepreneur, divided by opportunity cost 1 Γ 1 Rk 1 R L

93 Simple New Keynesian Model without Capital Lawrence J. Christiano January 5, 2018

94 Objective Review the foundations of the basic New Keynesian model without capital. Clarify the role of money supply/demand. Derive the Equilibrium Conditions. Small number of equations and a small number of variables, which summarize everything about the model (optimization, market clearing, gov t policy, etc.). Look at some data through the eyes of the model: Money demand. Cross-sectoral resource allocation cost of inflation. Some policy implications of the model will be examined. Many policy implications will be discovered in later computer exercises.

95 Outline The model: Individual agents: their objectives, what they take as given, what they choose. Households, final good firms, intermediate good firms, gov t. Economy-wide restrictions: Market clearing conditions. Relationship between aggregate output and aggregate factors of production, aggregate price level and individual prices. Properties of Equilibrium: Classical Dichotomy - when prices flexible monetary policy irrelevant for real variables. Monetary policy essential to determination of all variables when prices sticky.

96 Households Households problem. Concept of Consumption Smoothing.

97 Households There are many identical households. The problem of the typical ( representative ) household: ( max E 0 β t log C t N1+ϕ t t=0 1 + ϕ + γz tlog ( Mt+1 P t ) ), s.t. P t C t + B t+1 + M t+1 W t N t + R t 1 B t + M t +Profits net of government transfers and taxes t. Here, B t and M t are the beginning-of-period t stock of bonds and money held by the household.

98 Household First Order Conditions The household first order conditions: where C t N ϕ t 1 1 = βe t C t C t+1 = W t. P t m t = ( Rt R t 1 m t M t+1 P t. R t π t+1 (5) ) γc t (7), All equations are derived by expressing the household problem in Lagrangian form, substituting out the multiplier on budget constraint and rearranging. The last first order condition is real money demand, increasing in C t and decreasing in R t 1.

99 Figure: Money Demand, Relative to Two Measures of Velocity Notes: (i) velocity is GDP/M, (ii) With the MZM measure of money, the money demand equation does well qualitatively, but not qualitatively because the theory implies the scatters in the 2,1 and 2,2 graphs should be on the 45 0.

100 Consumption Smoothing: Example Problem: max c1,c 2 log (c 1 ) + βlog (c 2 ) subject to : c 1 + B 1 y 1 + rb 0 c 2 rb 1 + y 2. where y 1 and y 2 are (given) income and, after imposing equality (optimality) and substituting out for B 1, c 1 + c 2 r = y 1 + y 2 r + rb 0, 1 = βr 1, c 1 c 2 second equation is fonc for B 1. Suppose βr = 1 (this happens in steady state, see later): c 1 = y 1 + y 2 r r + r B 0 r

101 Consumption Smoothing: Example, cnt d Solution to the problem: c 1 = y 1 + y 2 r r + r B 0. r Consider three polar cases: temporary change in income: y 1 > 0 and y 2 = 0 = c 1 = c 2 = y r permanent change in income: y 1 = y 2 > 0 = c 1 = c 2 = y 1 future change in income: y 1 = 0 and y 2 > 0 = c 1 = c 2 = y 2 r 1+ 1 r Common feature of each example: When income rises, then - assuming r does not change - c 1 increases by an amount that can be maintained into the second period: consumption smoothing.

102 Goods Production We turn now to the technology of production, and the problems of the firms. The technology requires allocating resources across sectors. We describe the efficient cross-sectoral allocation of resources. With price setting frictions, the market may not achieve efficiency.

103 Final Goods Production A homogeneous final good is produced using the following (Dixit-Stiglitz) production function: Y t = [ˆ 1 0 ] ε Y ε 1 ε 1 ε i,t di. Each intermediate good,y i,t, is produced by a monopolist using the following production function: Y i,t = e a t N i,t, a t exogenous shock to technology. Before discussing the firms that operate these production functions, we briefly investigate the socially efficient allocation of resources across i.

104 Efficient Sectoral Allocation of Resources With Dixit-Stiglitz final good production function, there is a socially optimal allocation of resources to all the intermediate activities, Y i,t. It is optimal to run them all at the same rate, i.e., Y i,t = Y j,t for all i, j [0, 1]. For given N t, allocative efficiency: N i,t = N j,t = N t, for all i, j [0, 1]. In this case, final output is given by Y t = [ˆ 1 0 (e a t N i,t ) ε 1 ε di ] ε ε 1 = e a t N t. One way to understand allocated efficiency result is to suppose that labor is not allocated equally to all activities. Explore one simple deviation from N i,t = N j,t for all i, j [0, 1].

105 Suppose Labor Not Allocated Equally Example: N it 2 N t i 0, N t i,1, Note that this is a particular distribution of labor across activities: 0 1 Nit di N t N t N t

106 Labor Not Allocated Equally, cnt d Y t 1 1 Yi,t di Y i,t e at di N i,t 1 1 Yi,t di 1 2 di 1 1 Ni,t 1 di 1 1 e at 2 2 N t 1 1 di Nt e at 2 N t di e at N t e at N t f di 1 di 1 1

107 f Efficient Resource Allocation Means Equal Labor Across All Sectors f

108 Final Good Producers Competitive firms: maximize profits P t Y t ˆ 1 0 P i,t Y i,t dj, subject to P t, P i,t given, all i [0, 1], and the technology: Foncs: Y t = [ˆ 1 0 ] ε Y ε 1 ε 1 ε i,t dj. cross price restrictions {}}{ ( ) ε (ˆ ) 1 1 Pt Y i,t = Y t P t = P (1 ε) 1 ε P i,t di i,t 0

109 Intermediate Good Producers The i th intermediate good is produced by a monopolist. Demand curve for i th monopolist: ( ) ε Pt Y i,t = Y t. P i,t Production function: Y i,t = e a t N i,t, a t exogenous shock to technology. Calvo Price-Setting Friction: { P i,t = P t with probability 1 θ with probability θ P i,t 1.

110 Marginal Cost of Production An important input into the monopolist s problem is its marginal cost: s t = dcost doutput = dcost dworker doutput dworker = (1 ν) W t P t e a t = (1 ν) C tn ϕ t e a t after substituting out for the real wage from the household intratemporal Euler equation. The tax rate, ν, represents a subsidy to hiring labor, financed by a lump-sum government tax on households. Firm s job is to set prices whenever it has the opportunity to do so. It must always satisfy whatever demand materializes at its posted price.

111 Present Discounted Value of Intermediate Good Revenues i th intermediate good firm s objective: E i t j=0 period t+j profits sent to household { }} { revenues total cost {}}{{}}{ β j υ t+j P i,t+j Y i,t+j P t+j s t+j Y i,t+j υ t+j - Lagrange multiplier on household budget constraint Here, E i t denotes the firm s expectation over future variables, including the future probability that the firm gets to reset its price.

112 Decision By Firm that Can Change Its Price Let p t = P t P t. The firm s profit-maximizing choice of P t satisfies: p t = E t j=0 (βθ)j Y t+j C t+j ( Xt,j ) ε E t j=0 ε ε 1 s t+j Y t+j C t+j (βθ) j ( ) 1 ε, X t,j the present discounted value of the markup, ε/ (ε 1) over real marginal cost.

113 Decision By Firm that Can Change Its Price Recall, p t = E t j=0 (βθ)j Y t+j C t+j ( Xt,j ) ε ε ε 1 s t+j E t j=0 (βθ)j Y t+j C t+j ( Xt,j ) 1 ε = K t F t The numerator has the following simple representation: K t = E t (βθ) j Y t+j ( ) ε ε Xt,j C j=0 t+j ε 1 s t+j = ε (1 ν) Y t N ϕ t ε 1 e a t + βθe t ( 1 π t+1 after using s t = (1 ν) e τ tc t N ϕ t /ea t. Similarly, F t = Y ( ) t 1 1 ε + βθe t F t+1 (2) C t π t+1 ) ε K t+1 (1),

114 Moving On to Aggregate Restrictions Link between aggregate price level, P t, and P i,t, i [0, 1]. Potentially complicated because there are MANY prices, P i,t, i [0, 1]. Link between aggregate output, Y t, and N t. Potentially complicated because of earlier example with f (α). Analog of f (α) will be a function of degree to which P i,t = P j,t. Market clearing conditions. Money and bond market clearing. Labor and goods market clearing.

115 Important Calvo result: Divide by P t : Aggregate Price Index 1 = P t = = ( [ Rearrange: p t = (ˆ 1 0 P (1 ε) i,t di ( (1 θ) P 1 ε t (1 θ) p 1 ε t ] 1 1 θ( π t ) ε 1 1 ε 1 θ ) 1 1 ε ) + θp 1 ε 1 1 ε t 1 ( 1 + θ π t ) 1 ε ) 1 1 ε

116 Aggregate Output vs Aggregate Labor and Tech (Tack Yun, JME1996) Define Y t : Y t ˆ 1 0 Y i,t di demand curve {}}{ = Y t ˆ 1 = Y t P ε t (P t ) ε 0 ( = ( Pi,t P t ˆ 1 0 e a t N i,t di = e a t N t ) ) ε di = Y t P ε t where, using Calvo result : [ˆ ] 1 1 ε [ Pt P ε i,t di = (1 θ) P t ε Then 0 Y t = p t Y t, p t = ( P t P ˆ 1 0 (P i,t ) ε di + θ ( Pt 1 ) ] 1 ε ε ) ε.

117 Gross Output vs Aggregate Labor Relationship between aggregate inputs and outputs: Y t = p t Y t or, Y t = p t e a t N t. Note that p t is a function of the ratio of two averages (with different weights) of P i,t, i (0, 1) So, when P i,t = P j,t for all i, j (0, 1), then p t = 1. But, what is p t when P i,t = P j,t for some (measure of) i, j (0, 1)?

118 Tack Yun Distortion Consider the object, where P t = (ˆ 1 0 P ε i,t di ( pt P ) ε = t, P t ) 1 ε, P t = (ˆ 1 0 P (1 ε) i,t di ) 1 1 ε Follows easily from (intuition) and Jensen s inequality: p t 1.

119 Law of Motion of Tack Yun Distortion We have Dividing by P t : P t = ( pt P ) ε t = P t [ [ (1 θ) P t ε + θ ( Pt 1 ) ] 1 ε ε (1 θ) p ε t + θ πε t p t 1 ] 1 = (1 θ) [ 1 θ ( π t ) ε 1 1 θ ] ε 1 ε + θ πε t p t 1 1 (4) using the restriction between p t and aggregate inflation developed earlier.

120 Evaluating the Distortions Tack Yun distortion: pt = (1 θ) ( 1 θ π (ε 1) ) ε ε 1 t + θ πε t 1 θ p t 1 1. Potentially, NK model provides an endogenous theory of TFP. Standard practice in NK literature is to set pt = 1 for all t. First order expansion of pt around π t = pt = 1 is: p t = p + 0 π t + θ (p t 1 p ), with p = 1, so p t 1 and is invariant to shocks.

121 Empirical Assessment of Tack Yun Distortion Do back of the envelope calculations in a steady state when inflation is constant and p is constant. Can also use pt = (1 θ) ( 1 θ π (ε 1) ) ε ε 1 t + θ πε t 1 θ p t 1 to compute times series estimate of p t. But, results very similar to what you find with steady state calculations. 1.

122 Cost of Three Alternative Permanent Levels of Inflation p = 1 θ πε 1 θ ( 1 θ ) ε ε 1 1 θ π (ε 1) Table: Percent of GDP Lost Due to Inflation, 100(1 p t ) ε steady state inflation markup, ε s: 8% proposal for dealing with ZLB: 4% recent average: 2%

123 Tack Yun Distortion The magnitude of the distortion is typically small. Explains why standard literature abstracts from the distortion by linearizing about zero inflation. To first order approximation, p = 1 at zero inflation (see later). Could have p = 1 to first order around positive inflation if price indexation is assumed, as in CEE. But, prices don t appear to be indexed. Caution: distortion may be small because of simplicity of the model. Distortions at least two times bigger when production occurs in networks of firms. See this and this. Distortions may be bigger when there are intermediate good firm-specific idiosyncratic shocks to demand and supply of intermediate good firms.

124 Government Government budget constraint: expenditures = receipts purchases of final goods subsidy payments gov t bonds (lending, if positive) {}}{{}}{{}}{ P t G t + νw t N t + = + money injection, if positive {}}{ M t µ t + where µ t denotes money growth rate. B g t+1 transfer payments to households {}}{ T trans t tax revenues {}}{ Tt tax +R t 1 B g t Government s choice of µ t determines evolution of money supply: M t+1 = (1 + µ t ) M t, µ t money growth rate.

125 Government The law of motion for money places restrictions on m t : for t = 0, 1,.... m t M t+1 = M t+1 M t P t 1 P t M t P t 1 P t ( ) 1 + µt m t = m t 1 (8), π t

126 Market Clearing We now summarize the market clearing conditions of the model. Money, labor, bond and goods markets.

127 Money Market Clearing We temporarily use the bold notation, M t, to denote the per capita supply of money at the start of time t, for t = 0, 1, 2,.... The supply of money is determined by the actions, µ t, of the government: M t+1 = M t + µ t M t, for t=0,1,2,... Households being identical means that in period t = 0, M 0 = M 0, where M 0 denotes beginning of time t = 0 money stock of the representative household. Money market clearing in each period, t = 0, 1,..., requires M t+1 = M t+1, where M t+1 denotes the representative household s time t choice of money. From here on, we do not distinguish between M t and M t.

128 Other Market Clearing Conditions Bond market clearing: Labor market clearing: Goods market clearing: B t+1 + B g t+1 = 0, t = 0, 1, 2,... supply of labor {}}{ N t = demand for labor {}}{ ˆ1 0 N i,t di demand for final goods supply of final goods {}}{{}}{ C t + G t = Y t, and, using relation between Y t and N t : C t + G t = p t e a t N t (6)

129 Next Collect the equilibrium conditions associated with private sector behavior. Comparison of NK model with RBC model (i.e., θ = 0) Classical Dichotomy: In flexible price version of model real variables determined independent of monetary policy. Fiscal policy still matters, because equilibrium depends on how government deals with the monopoly power, i.e., selects value for subsidy, ν. In NK model, markets don t necessarily work well and good monetary policy essential. To close model with θ > 0 must take a stand on monetary policy.

130 Equilibrium Conditions 8 equations in 8 unknowns: m t, C t, p t, F t, K t, N t, R t, π t, and 3 policy variables: ν, µ t, G t. K t = ε (1 ν) Y t N ϕ t + βθe t π t+1 ε ε 1 A K t+1 (1) t F t = Y t + βθe t π C t+1 ε 1 F t+1 (2), t pt = (1 θ) 1 1 = βe t C t m t = C t+1 ( R t K t F t = [ 1 θ π (ε 1) ) ε ε 1 t + θ πε t 1 θ 1 θ π (ε 1) t 1 θ p t 1 1 (4) (5), C t + G t = pt e a t N t (6) π t+1 ( ) 1 + µt m t 1 (8) γc t (1 1 Rt ) (7), m t = π t ] 1 1 ε (3)

131 Classical Dichotomy Under Flexible Prices Classical Dichotomy: when prices flexible, θ = 0, then real variables determined regardless of the rule for µ t (i.e., monetary policy). Equations (2),(3) imply: F t = K t = Y t C t, which, combined with (1) implies ε (1 ν) ε 1 Marginal Cost of work {}}{ CN ϕ t = Expression (6) with p t = 1 (since θ = 0) is C t + G t = e a t N t. marginal benefit of work {}}{ e a t Thus, we have two equations in two unknowns, N t and C t.

132 Classical Dichotomy: No Uncertainty Real interest rate, R t R t/ π t+1, is determined: R t = 1 C t β 1 C t+1. So, with θ = 0, the following are determined: R t, C t, N t, t = 0, 1, 2,... What about the nominal variables? Suppose the monetary authority wants a given sequence of inflation rates, π t, t = 0, 1,.... Then, R t = π t+1 R t, t = 0, 1, 2,... What money growth sequence is required? From (7), obtain m t, t = 0, 1, 2,.... Also, m 1 is given by initial M 0 and P 1. From (8) 1 + µ t = m t m t 1 π t, t = 0, 1, 2,...

133 Classical Dichotomy versus New Keynesian Model When θ = 0, then the Classical Dichotomy occurs. In this case, monetary policy (i.e., the setting of µ t, t = 0, 1, 2,... ) cannot affect the real interest rate, consumption and employment. Monetary policy simply affects the split in the real interest rate between nominal and real rates: R t = R t π t+1. For a careful treatment when there is uncertainty, see. When θ > 0 (NK model) then real variables are not determined independent of monetary policy. In this case, monetary policy matters.

134 Monetary Policy in New Keynesian Model Suppose θ > 0, so that we re in the NK model and monetary policy matters. The standard assumption is that the monetary authority sets µ t to achieve an interest rate target, and that that target is a function of inflation: R t /R = (R t 1 /R) α exp [(1 α) φ π ( π t π) + φ x x t ] (7), where x t denotes the log deviation of actual output from target. This is a Taylor rule, and it satisfies the Taylor Principle when φ π > 1. Smoothing parameter: α. Bigger is α the more persistent are policy-induced changes in the interest rate.

135 Equilibrium Conditions of NK Model with Taylor Rule K t = ε (1 ν) Y t N ϕ t + βθe t π t+1 ε ε 1 A K t+1 (1) t F t = Y t + βθe t π C t+1 ε 1 F t+1 (2), t pt = (1 θ) ( K t F t = [ 1 θ π (ε 1) ) ε ε 1 t + θ πε t 1 θ 1 θ π (ε 1) t 1 θ p t 1 1 (4) ] 1 1 ε 1 1 R t = βe t (5), C t + G t = pt e a t N t (6) C t C t+1 π t+1 R t /R = (R t 1 /R) α exp [(1 α) φ π ( π t π) + φ x x t ] (7). (3) Conditions (7) and (8) have been replaced by (7).

136 Equilibrium Conditions of NK Model The model represents 7 equations in 7 unknowns: C, p t, F t, K t, N t, R t, π t After this system has been solved for the 7 variables, equations (7) and (8) can be used to solve for µ t and m t. This is rarely done, because researchers are uncertain of the exact form of money demand and because m t and µ t are in practice not of direct interest.

137 When θ = 0, then ε (1 ν) ε 1 Natural Equilibrium Marginal Cost of work {}}{ C t N ϕ t = marginal benefit of work {}}{ e a t so that we have a form of efficiency when ν is chosen to that ε (1 ν) / (ε 1) = 1. In addition, recall that we have allocative efficiency in the flexible price equilibrium. So, the flexible price equilibrium with the efficient setting of ν represents a natural benchmark for the New Keynesian model, the version of the model in which θ > 0. We call this the Natural Equilibrium. To simplify the analysis, from here on we set G t = 0.

138 Natural Equilibrium With G t = 0, equilibrium conditions for C t and N t : Marginal Cost of work {}}{ C t N ϕ t = aggregate production relation: C t = e a t N t. Substituting, e a t N 1+ϕ t = e a t N t = 1 C t = exp (a t ) R t = 1 C t βe t 1 C t+1 = 1 = C βe t t C t+1 marginal benefit of work {}}{ e a t 1 βe t exp ( a t+1 )

139 Natural Equilibrium, cnt d Natural rate of interest: R t = Two models for a t : 1 C t βe t 1 C t+1 = 1 βe t exp ( a t+1 ) DS : a t+1 = ρ a t + ε a t+1 TS : a t+1 = ρa t + ε a t+1

140 Natural Equilibrium, cnt d Suppose the ε t s are Normal. Then, E t exp ( a t+1 ) = exp ( E t a t ) V, where V = σ 2 a Then, with r t log R t r t = log β + E t a t V. Useful: consider how natural rate responds to ε a t shocks under DS and TS models for a t. To understand how rt responds, consider implications of consumption smoothing in absence of change in rt. Hint: in natural equilibrium, rt steers the economy so that natural equilibrium paths for C t and N t are realized.

141 Conclusion Described NK model and derived equilibrium conditions. The usual version of model represents monetary policy by a Taylor rule. When θ = 0, so that prices are flexible, then monetary policy is (essentially) neutral. Changes in money growth move prices and wages in such a way that real wages do not change and employment and output don t change. When prices are sticky, then a policy-induced reduction in the interest rate encourages more nominal spending. The increased spending raises W t more than P t because of the sticky prices, thereby inducing the increased labor supply that firms need to meet the extra demand. Firms are willing to produce more goods because: The model assumes they must meet all demand at posted prices. Firms make positive profits, so as long as the expansion is not too big they still make positive profits, even if not optimal.

142 Simple New Keynesian Model without Capital, II Lawrence J. Christiano January 5, 2018

143 Standard New Keynesian Model Taylor rule: designed so that in steady state, inflation is zero ( π = 1) Employment subsidy extinguishes monopoly power in steady state: ε (1 ν) ε 1 = 1

144 Equilibrium Conditions of NK Model with Taylor Rule K t = ε (1 ν) Y t N ϕ t + βθe t π t+1 ε ε 1 A K t+1 (1) t F t = Y t + βθe t π C t+1 ε 1 F t+1 (2), t pt = (1 θ) ( K t F t = [ 1 θ π (ε 1) ) ε ε 1 t + θ πε t 1 θ 1 θ π (ε 1) t 1 θ p t 1 1 (4) ] 1 1 ε 1 1 R t = βe t (5), C t + G t = pt e a t N t (6) C t C t+1 π t+1 R t /R = (R t 1 /R) α exp [(1 α) φ π ( π t π) + φ x x t ] (7). In steady state: R = 1 β, p = 1, F = K = 1 1 βθ, N = 1 (3)

145 Natural Rate of Interest Intertemporal Euler equation in Natural equilibrium: optimal consumption in t {}}{ a t = [r t rr] + E t a t+1 where a indicates natural equilibrium and r t = log (R t ). Back out the natural rate and ignoring constant Law of motion for technology: r t = E t a t+1 a t = ρ a t 1 + ε a t.

146 NK IS Curve Euler equation in two equilibria (ignoring variance adjustment): Taylor rule equilibrium, NK model {}}{ c t = [r t E t π t+1 rr] + E t c t+1 Natural equilibrium (θ = 0, ν kills monopoly power) {}}{ c t = [r t rr] + E t c t+1 where lower case letters mean log, π t = log ( π t )and means natural equilibrium. Subtract: output gap {}}{ x t = [r t E t x t+1 r t ]

147 Output in the NK Equilibrium Aggregate output relation: { = y t = log (pt ) + n t + a 1, log (pt 0 if Pi,t = P ) = j,t all i,j 0 otherwise. To first approximation (given that we set inflation to zero in steady state): p t = 1.

148 Phillips Curve Equations pertaining to price setting: K t = ε (1 ν) Y t N ϕ t + βθe t π t+1 ε ε 1 A K t+1 t F t = Y t + βθe t π C t+1 ε 1 F t+1, t pt = (1 θ) ( K t F t = [ 1 θ π (ε 1) ) ε ε 1 t + θ πε t 1 θ 1 θ π (ε 1) t 1 θ p t 1 Log-linearly expand about zero-inflation steady state: ˆ π t = (1 θ) (1 θβ) θ where ẑ t denotes z t z)/z See this for details. 1 (1 + ϕ) x t + E t ˆ π t+1 ] 1 1 ε

149 Collecting the Log-linearized Equations x t = x t+1 [r t π t+1 r t ] r t = φ π π t π t = βπ t+1 + κx t r t = E t (a t+1 a t ), where κ = (1 θ) (1 βθ) θ (1 + ϕ).

150 The Equations, in Matrix Form Representation of the shock: s t = a t = ρ a t 1 + ε t s t = Ps t 1 + ɛ t Matrix representation of system β π t x t r t r t π t 1 x t 1 r t 1 r t 1 1 κ φ π E t [α 0 z t+1 + α 1 z t + α 2 z t 1 + β 0 s t+1 + β 1 s t ] = 0 π t x t r t r t s t s t

151 Solving the Model Linearized equilibrium conditions: E t [α 0 z t+1 + α 1 z t + α 2 z t 1 + β 0 s t+1 + β 1 s t ] = 0 Data generated using this equation, z t = Az t 1 + Bs t with A and B chosen as follows: s t = Ps t 1 + ɛ t α 0 A 2 + α 1 A + α 2 = 0, (β 0 + α 0 B) P + [β 1 + (α 0 A + α 1 ) B] = 0. Standard strategy: pick the unique A with all eigenvalues less 1 in absolute value, that solves the first equation, conditional on A, choose B to solve the second equation. Strategy breaks down if there is no such A or there is more than one.

152 Simulation Draw ɛ 1, ɛ 2,..., T Solve s t = Ps t 1 + ɛ t z t = Az t 1 + Bs t,for t = 1,..., T, and given z 0 and s 0. If ɛ t = 0 for t > 1 and ɛ 1 = 1, this is an impulse response function. If ɛ t, t = 1, 2,..., T are iid and drawn from a random number generator, then it is a stochastic simulation.

153 Impulse Response Function Figure: Dynamic Response to a Technology Shock Note: β = 0.97, φ π = 1.5, ρ = 0.2, ϕ = 1, θ = 0.75, κ =

154 Interpretation What happened in the figure? The positive technology shock creates a surge in spending because of consumption smoothing. The natural rate has to rise, to prevent excessive consumption (an aggregate demand problem ). In the efficient allocations, log, natural consumption is equal to a t and natural employment is constant. The Taylor rule response is too weak, so the surge in aggregate demand is not pulled back and the economy over reacts to the shock. The excessive aggregate demand causes the economy to expand too much, which raises wages and costs and inflation. The weak response under the Taylor rule is not a function of the value of ρ, whether the time series representation is difference or trend stationary (see). How to get a better response? Put the natural rate of interest in the Taylor rule: r t = r t + φ π π t Either measure it exactly, or put in a proxy.

155 How to Proxy the Natural Rate? How to proxy the natural rate? It is consumption growth that appears in the natural rate, so something that signals good future consumption prospects would be a good proxy. Credit growth? Stock market growth? DSGE model simulations can be helpful for thinking about a good proxy. For an example, see Christiano-Ilut-Motto-Rostagno (Jackson Hole, 2011). Why keep inflation in the Taylor rule? To have a locally unique equilibrium, must have φ π > 1, i.e., Taylor Principle.

156 Uniqueness Under Taylor Principle R LM IS(πe) π π 1 y 1 Phillips curve y y 1 y

157 Uniqueness Under Taylor Principle R Suppose inflation expectations jump and the monetary authority satisfies the Taylor principle. IS(πe ) LM IS(πe) π π 1 y 1 Phillips curve y y 1 y

158 Uniqueness Under Taylor Principle R The sharp rise in interest rate will slow the economy and bring inflation down. LM IS(πe ) LM IS(πe) π π 1 y 1 y 1 Phillips curve y In a learning equilibrium the jump in expectations for no reason go away quickly. In rational expectations, the jump would not happen in the first place. y

159 Conclusion With sticky prices, the real allocations of the economy are not determined independently of monetary policy. So, performance of the economy will depend in part monetary policy design. Put the natural rate (or some good proxy) in the Taylor rule, and policy works well. Violate the Taylor principle and you could have trouble ( sunspots ). Other potential dysfunctions associated with poorly designed monetary policy are described in Christiano-Trabandt-Walentin (Handbook of Monetary Economics 2011). Without the proper design of monetary policy, aggregate demand can go wrong. In the example described above, the response of aggregate demand to a shock is excessive. Other examples can be found in which the response of aggregate demand is inadequate, and even the wrong sign.

160 Risk Shocks Lawrence Christiano (Northwestern University), Roberto Motto (ECB) and Massimo Rostagno (ECB) Based on AER publication, January 2014.

161 Finding Countercyclical fluctuations in the cross sectional variance of a technology shock, when inserted into an otherwise standard macro model, can account for a substantial portion of economic fluctuations. Complements empirical findings of Bloom (2009) and Kehrig (2011) suggesting greater cross sectional dispersion in recessions. Complements theory findings of Bloom (2009) and Bloom, Floetotto and Jaimovich (2009) which describe another way that increased cross sectional dispersion can generate business cycles. Otherwise standard model : A DSGE model, as in Christiano Eichenbaum Evans or Smets Wouters Financial frictions along the line suggested by BGG.

162 Outline Rough description of the model. Summary of Bayesian estimation of the model. Explanation of the basic finding of the analysis.

163 Standard Model L Firms K Labor market C I Market for Physical Capital household K t 1 1 K t G I,t,I t, I t 1 Marginal efficiency of investment shock

164 Standard Model with BGG L Firms K K, ~F, t Labor market Entrepreneurs household Examples: 1. Large proportion of firm start-ups end in failure 2. Even famously successful entrepreneurs (Gates, Jobs) had failures (Traf-O-Data, NeXT computer) 3. Wars over standards (e.g., Betamax versus VHS).

165 Standard Model with BGG L Firms K K, ~F, t Labor market Observed by entrepreneur, but supplier of funds must pay monitoring cost to see it. Entrepreneurs household

166 Standard Model with BGG Risk shock L Firms K K, ~F, t Labor market Entrepreneurs household

167 Standard Model with BGG L Firms K K, ~F, t Labor market C I Capital Producers Entrepreneurs household Entrepreneurs sell their K to capital producers K t 1 1 K t G I,t,I t, I t 1

168 Standard Model with BGG L Firms K K, ~F, t Labor market C I Capital Producers Entrepreneurs household Entrepreneurial net worth now established. = value of capital + earnings from capital -repayment of bank loans

169 Standard Model with BGG Firms K K, ~F, t Labor market Capital Producers K Entrepreneurs household banks Entrepreneur receives standard debt contract.

170 density Economic Impact of Risk Shock lognormal distribution: 20 percent jump in standard deviation *1.2 Larger number of entrepreneurs in left tail problem for lender Interest rate on loans to entrepreneur increases Entrepreneur borrows less Entrepreneur buys less capital, investment drops, economy tanks idiosyncratic shock

171 Five Adjustments to Standard DSGE Model for CSV Financial Frictions Drop: household intertemporal equation for capital. Add: equations that characterize the loan contract Zero profit condition for suppliers of funds. Efficiency condition associated with entrepreneurial choice of contract. Add: Law of motion for entrepreneurial net worth (source of accelerator and Fisher debt-deflation effects). Introduce: bankruptcy costs in the resource constraint.

172 Risk Shocks We assume risk has a first order autoregressive representation: t 1 t 1 iid, univariate innovation to t ut We assume that agents receive early information about movements in the innovation ( news ).

173 Risk Shock and News Assume t 1 t 1 iid, univariate innovation to t ut Agents have advance information about pieces of u t signals or news u t t t 1... t 8 i t i i ~iid, E t i 2 2 i i t i ~piece of u t observed at time t i

174 News on Risk Shocks Versus News on Other Shocks Marginal likelihood

175 Monetary Policy Nominal rate of interest function of: Anticipated level of inflation. Slowly moving inflation target. Deviation of output growth from sspath. Monetary policy shock.

176 12 Shocks Trend stationary and unit root technology shock. Marginal Efficiency of investment shock (perturbs capital accumulation equation) Monetary policy shock. Equity shock. Risk shock. 6 other shocks. K t 1 1 K t G i,t,i t,i t 1

177 Estimation Use standard macro data: consumption, investment, employment, inflation, GDP, price of investment goods, wages, Federal Funds Rate. Also some financial variables: BAA - 10 yr Tbond spreads, value of DOW, credit to nonfinancial business, 10 yr Tbond Funds rate. Data: 1985Q1-2010Q2

178 Results Risk shock most important shock for business cycles. Quantitative measures of importance. Why are they important? What shock do they displace, and why?

179 Risk shock closely identified with interest rate premium. Role of the Risk Shock in Macro and Financial Variables

180 Percent Variance in Business Cycle Frequencies Accounted for by Risk Shock variable Risk, t GDP 62 Investment 73 Consumption 16 Credit 64 Premium (Z R 95 Equity 69 Risk shock closely identified with interest rate premium R 10 year R 1 quarter 56 Note: business cycle frequencies means Hodrick-Prescott filtered data.

181 Why Risk Shock is so Important A. Our econometric estimator thinks risk spread ~ risk shock. B. In the data: the risk spread is strongly negatively correlated with output. C. In the model: bad risk shock generates a response that resembles a recession A+B+C suggests risk shock important.

182 Correlation (risk spread(t),output(t-j)), HP filtered data, 95% Confidence Interval The risk spread is significantly negatively correlated with output and leads a little. Notes: Risk spread is measured bythedifferencebetween the yield on the lowest rated corporate bond (Baa) and the highest rated corporate bond (Aaa). Bond data were obtained from the St. Louis Fed website. GDP data were obtained from Balke and Gordon (1986). Filtered output data were scaled so that their standard deviation coincide with that of the spread data.

183 Surprising, from RBC perspective Figure 3: Dynamic Responses to Unanticipated and Anticipated Components of Risk Shock A: interest rate spread (Annual Basis Points) B: credit C: investment D: output E: net worth F: consumption G: inflation (APR) response to unanticipated risk shock, 0,0 response to anticipated risk shock, 8, I: risk, t Looks like a business cycle

184 Figure 3: Dynamic Responses to Unanticipated and Anticipated Components of Risk Shock A: interest rate spread (Annual Basis Points) B: credit C: investment D: output E: net worth F: consumption G: inflation (APR) response to unanticipated risk shock, 0,0 response to anticipated risk shock, 8, I: risk, t

185 What Shock Does the Risk Shock Displace, and why? The risk shock mainly crowds out the marginal efficiency of investment. But, it also crowds out other shocks. Compare estimation results between our model and model with no financial frictions or financial shocks (CEE).

186 Baseline model mostly steals explanatory power from m.e.i., but also from other shocks: big drop in marginal efficiency of investment Variance Decomposition of GDP at Business Cycle Frequency (in percent) shock Risk Equity M. E. I. Technol. Markup M. P. Demand Exog. Spend. Term t t I,t t, z,t, f,t, t c,t g t Baseline model CEE [ ] [ ] [39] [18] [31] [4] [3] [5] [-]

187 Baseline model mostly steals explanatory power from m.e.i., but also from other shocks: technology goes from small to tiny Variance Decomposition of GDP at Business Cycle Frequency (in percent) shock Risk Equity M. E. I. Technol. Markup M. P. Demand Exog. Spend. Term t t I,t t, z,t, f,t, t c,t g t Baseline model CEE [ ] [ ] [39] [18] [31] [4] [3] [5] [-]

188 Why does Risk Crowd out Marginal Efficiency of Price of capital Investment? Supply shifter: marginal efficiency of investment, i,t Demand shifters: risk shock, t ; wealth shock, t Quantity of capital

189 Marginal efficiency of investment shock can account well for the surge in investment and output in the 1990s, as long as the stock market is not included in the analysis. When the stock market is included, then explanatory power shifts to financial market shocks. When we drop financial data slope of term structure, interest rate spread, stock market, credit growth: Hard to differentiate risk shock view from marginal efficiency of investment view.

190 Is There Independent Evidence for Risk Shocks? Cross-sectional standard deviation of rate of return on equity in CRSP rises in recessions (Bloom, 2009). This observation played no role in the construction or estimation of the model. Compute the model s best guess (Kalman Smoother) about the cross-sectional standard deviation of equity returns, and compare with data.

191 Cross-sectional standard deviation is countercyclical