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- 1. Asset Supply and Liquidity Transformation in HANK Yu-Ting Chiang 1 Piotr Zoch 2 1FRB of St Louis 2University of Warsaw and GRAPE May 9, 2023
- 2. Motivation Question: role of the financial sector in the transmission of macroeconomic policies. Framework: General model of the financial sector + HANK. - banks: issue deposits, hold illiquid capital and liquid asset - nesting a large class of models of financial intermediation Result: aggregate responses depend on the financial sector through a liquid asset supply function Dt rB s , rK s ∞ s=0 := deposits - liquid assets held by banks Key elasticities: ∂Dt/∂rK s Dt (cross-price) - Low cross-price: disturbance in the liquid asset market → large changes in capital price and aggregate demand. 1 / 17
- 3. Results Overview Liquid asset supply elasticities are quantitatively important Policy experiment: illiquid asset purchases financed by liquid government debt Model comparison: liquid asset supply ranging from perfectly inelastic (HANK) to perfectly elastic, with Gertler-Karadi-Kiyotaki in between Result: Aggregate output responses are by two orders of magnitude larger in HANK than with perfectly elastic supply. 2 / 17
- 4. Model
- 5. Household A continuum of households choose consumption and savings in liquid and illiquid asset to maximize utility: max ai,t,bi,t,ci,t E X t≥0 βt u (ci,t, hi,t) , s.t. ai,t + bi,t + ci,t + Φi,t(ai,t, ai,t−1) = (1 + rA t )ai,t−1 + (1 + rB t )bi,t−1 + (1 − τt) Wt Pt zi,thi,t 1−λ where zi,t is idiosyncratic shocks, Φi,t is portfolio adjustment costs, and ai,t ≥ a, bi,t ≥ b 3 / 17
- 6. Production and Labor Supply I Firms maximize profit with technology yt = kα t−1h1−α t : max kt−1,{h`,t} Ptyt − Rtkt−1 − Z W`,th`,td`. ht: CES aggregator; union ` supplies h`,t = R zi,thi,`,tdi I Union ` sets nominal wage growth πW,`,t = W`,t/W`,t−1 to maximize household welfare with Rotemberg adjustment cost. I Capital: kt = (1 − δ) kt−1 + Γ (ιt) kt−1, ιt = xt kt−1 . I Return on capital: 1 + rK t+1 = max ι̂t+1 Rt+1 Pt+1 + qt+1 (1 + Γ (ι̂t+1) − δ) − ι̂t+1 qt . 4 / 17
- 7. The Financial Sector I A representative bank issues deposits to hold capital and gov. debt. Both deposits (d̃t) and gov. debt (bB t ) are liquid. I Let dt := d̃t − bB t denote the liquid asset supply. The bank’s problem: νt+1nt = max kB t ,dt rK t+1qtkB t − rB t+1dt subject to their balance sheet and a financial constraint: qtkB t = dt + nt, qtkB t ≤ Θ rK s+1, rB s+1 s≥t nt I Net worth of banks follows: nt+1 = (1 − f )(1 + νt+1)nt + m. I Illiquid assets consist of capital and net worth of banks: at = qtkF t + nt, rA t+1 = 1 at (rK t+1qtkF t + νt+1nt). 5 / 17
- 8. Government I Government issues liquid liabilities bG t to purchase illiquid assets aG t and goods bG t − aG t = (1 + rB t )bG t−1 − (1 + rA t )aG t−1 + gt − Tt. I Tax revenue collected by the government is Tt = Wt Pt ht − Z (1 − τt) Wt Pt zi,thi,t 1−λ di. I Government sets the nominal interest rate iB t to keep the real interest rate equal to its desired level rB t . 6 / 17
- 9. Definition of Equilibrium Given {rB t , gt, bG t , τt}, an equilibrium consists of prices {Pt, qt, Rt, iB t , rK t , rA t , W`,t} and allocations {yt, kt, xt, dt, nt, kF t , kB t , aG t , ai,t, bi,t, ci,t, hi,`,t} s.t. agents optimize given constraints, gov. budget constraints hold, and markets clear: Z ci,t + Φi,t(ai,t, ai,t−1)di + xt + gt = yt, Z bi,tdi = dt + bG t , Z ai,tdi + aG t = qtkt − dt, and kt = kF t + kB t . Labor and capital rental market clearing are embedded in the notation. We focus on an equilibrium in which the financial constraint is always binding. 7 / 17
- 11. Nesting Models of Financial Intermediation Problem P: Given {rK t , rB t }t≥0, solve for {dt} such that νt+1nt = max dt rK t+1nt + (rK t+1 − rB t+1)dt, where dt+nt ≤ Θ {rK s+1, rB s+1}s≥t nt, nt+1 = (1 − f )nt(1 + νt+1) + m. Models of financial intermediation: 1. asset diversion (Gertler and Kiyotaki (2010), Gertler and Karadi (2011)) 2. costly state verification (Bernanke et al. (1999)) 3. costly leverage (Uribe and Yue, Cúrdia and Woodford (2011)) 4. regulatory constraints (Van den Heuvel (2008)) 8 / 17
- 12. Nesting Models of Financial Intermediation Lemma Suppose that {dM t } solves the bank’s problem in model M ∈ {1, . . . , 4}. There exists a function Θ such that {dM t } is the solution to Problem P. Moreover, when evaluated at the stationary equilibrium, ∂Θt ∂rK s+1 = γs−t Θ̄rK , ∂Θt ∂rB s+1 = −γs−t Θ̄rB , ∀s ≥ t, where Θt := Θ({rK s+1, rB s+1}s≥t) and Θ̄rK , Θ̄rB ≥ 0, γ ∈ [0, 1) are determined by parameters of model M and steady-state variables. 9 / 17
- 13. Liquid Asset Supply Function Let Dt({rK s ; rB s }∞ s=0) be the solution of the bank’s problem. Cross-price elasticities: ∂Dt/∂rK s Dt depend only on Θ̄rK , γ and steady state variables. Why it is useful: reduces infinite-dimensional elasticities to three parameters and allows for systematic model comparison. Comparing models: - Asset diversion: Θ̄rK , Θ̄rB , γ 0 , fully determined by the steady state. - Costly state verification costly leverage: Θ̄rK , Θ̄rB 0, γ = 0, not determined by the steady state 10 / 17
- 14. Liquid Asset Market and Aggregate Responses
- 15. A Supply and Demand Representation Approach: characterize equilibrium as an intertemporal supply and demand system. Lemma There exist functions Ct, Bt, Xt, RA t such that, given gs, Ts, rB s , bG s ∞ s=0 , the equilibrium ys, rK s ∞ s=0 solve: Ct({ys, rA s ; rB s , Ts}∞ s=0) + Xt({ys, rK s }∞ s=0) + gt = yt, Bt({ys, rA s ; rB s , Ts}∞ s=0) = Dt({rK s ; rB s }∞ s=0) + bG t , and rA t = RA t {rK s ; rB s }∞ s=0; Dt−1({rK s ; rB s }∞ s=0) . Moreover, functions Ct, Bt, Xt, RA t do not depend on Θ. Implication: aggregate responses depends on financial frictions Θ only through the liquid asset supply function Dt. 11 / 17
- 16. Liquid Asset Market and Returns on Capital Define excess liquid asset supply as t(y, rK , rB , T, bG ) := Dt(·) + bG t − Bt(·). Proposition In equilibrium, returns on capital satisfy drK = −−1 rK × [dbG + T dT + rB drB + ydy] | {z } excess liquid asset supply shift . Cross-price elasticities of liquid asset supply determine rK : I affect by how much drK need to adjust for the economy to absorb an increase in excess liquid asset supply I lower rK imply larger adjustment in drK in response to an increase in liquid asset supply 12 / 17
- 17. Aggregate Output Response Define aggregate demand as Ψt(y, rK , rB , T, g) := Ct(·) + Xt(·) + gt, Theorem Given {drB, dT, dbG, dg}, the aggregate output response is: dy = (I − Ψy − Ω y)−1 | {z } (iii) modified Keynesian cross × dg + ΨT dT + ΨrB drB | {z } (i) goods market + Ω dbG + T dT + rB drB | {z } (ii) liquid asset market , Ω = −ΨrK (rK )−1: aggregate demand response through the liquid asset market 13 / 17
- 18. A Quantitative Study of Asset Purchases
- 19. Government asset purchase and model comparison Policy: inject liquid assets and purchase illiquid assets dbG t = daG t , keep drB t = 0, adjust dTt to balance budget. Model comparison: same S.S., change cross-price elasticities 1. Perfectly inelastic (HANK): DrK = DrB = 0 2. Baseline (GKK): Θ̄rK , Θ̄rB , γ from steady-state bank balance sheets. 3. Departure: change Θ̄rK from 5 → 10. 4. Perfectly elastic: Θ̄rK → ∞. 14 / 17 Calibration
- 20. Comparative Statics: Aggregate Responses Figure 1: y-axis: % of steady-state GDP. Light red: low cross-price elasticities. Dark red: high cross-price elasticities. Blue: inelastic supply. Black: perfectly elastic supply. 15 / 17
- 21. Decomposition of Aggregate Output Response Figure 2: Decomposition of output response; y-axis: % of steady-state GDP. The decomposition uses formula from Theorem 1: dy = (I − Ψy − Ω y)−1 | {z } (iii) Keynesian cross × ΨT dT | {z } (i) goods market + Ω (dbG + T dT) | {z } (ii) liquid asset market . 16 / 17
- 22. Conclusion I Framework to analyze the role of the financial sector for macro policies: focus on liquid asset market I Key elasticities: own- and cross-price elasticities of liquid asset supply (empirics coming soon) I Low cross-price elasticity =⇒ disturbances in liquid asset market generate strong aggregate responses I Quantitatively, large differences in output responses depending on cross-price elasticities. 17 / 17
- 23. Appendix
- 24. Calibration Details I Returns: liquid asset 1.6% per annum, illiquid assets 4.4%, capital 4.1% I Preferences: u (c, h) = c1−σ − 1 1 − σ − ς h 1+ 1 ϕ 1 + 1 ϕ with σ = 2 and ϕ = 1 I Income risk: discrete-time version of the income process from Kaplan et al. (2018), which matches moments from SSA data on male earnings. I Assets: no borrowing a = b = 0 and Ψi,t(ai,t, ai,t−1) = χ1 χ2 ai,t − (1 + rA t )ai,t−1 (1 + rA t )ai,t−1 + χ0 χ2 h (1 + rA t )ai,t−1 + χ0 i , where χ0, χ1, χ2 0 are parameters that characterize the adjustment cost Back
- 25. Calibration Details I Production: The elasticity of output with respect to capital α = 0.35. Depreciation rate δ is 6% per year. Capital adjustment cost is given by Γ (ιt) = ῑ1ι1−κI t + ῑ2 with the elasticity of investment to capital price equal to 2. The slope of the wage Phillips curve is set to 0.04. I Government: Net taxes to output: T = 0.15, λ, the parameter governing the tax system’s progressivity is 0.1. Government debt is set to match the liquid asset holdings of households. Back
- 26. Calibration Households: - labor income process: matches moments from SSA data (Guvenen et al. (2015) ). - portfolio adjustment cost: generates a fraction of HtM households ≈ 22% - total holdings of liquid and illiquid assets: target FoF data (balance sheet items consolidated corresponding to the model) The financial sector: - net worth: match Call Report data, adjusted to FoF total. - “effective” leverage: ratio of assets to net worth, excluding liquid assets. Back
- 27. Balance Sheets: Data v.s. Model Households assets liabilities liquid asset 0.74 (0.50) equity 4.56 (3.80) illiquid asset 3.82 (3.30) Private Depository Institutions (Banks) assets liabilities liquid asset 0.18 (0.17) liquid liability 0.43 (0.43) capital 0.37 (0.39) equity 0.12 (0.13) Data: Financial Accounts of the United States, 2019 Q3. Model moments are in parenthesis. Items are expressed as fractions of the annual % GDP. Appendix C of our paper discusses the categorization and consolidation of balance-sheet items. Back