Bernanke and Gertler [1989]

Transcription

1 Bernanke and Gertler [1989] Econ 235, Spring Background: Townsend [1979] An entrepreneur requires x to produce output y f with Ey > x but does not have money, so he needs a lender Once y is realized, the entrepreneur will observe it but no one else will Source of trouble: entrepreneur will want to say sorry, y was very low, I can t pay back Lender can observe y by paying a cost γ. Everyone is risk neutral and does not discount the future By revelation principle, use contracts where entrepreneur truthfully announces y Let ŷ be the output announced by entrepreneur A contract will specify: the probability of being audited p(ŷ) a repayment function - or, equivalently, the entrepreneur s consumption c (ŷ, a) Consumption is constrained to be nonnegative (limited liability) Technically, consumption function should specify what happens if the audit reveals that entrepreneur lied. Clearly it s optimal to set c = 0 for this case, so the program below just assumes this. Special case: y {y H, y L }. π Pr [y = y H ] Notice that the entrepreneur will never want to announce ŷ = y H when y = y L. This means that we only need a truth-telling constraint for state y = y L and that we don t need to audit ŷ = y H announcements. 1

2 Program max πc H + (1 π) [pc L,a + (1 p) c L,0 ] (1) p,c( ) s.t. π (y H c H ) + (1 π) [y L p (c L,a + γ) (1 p) c L,0 ] x (2) c H p 0 + (1 p) [c L,0 + y H y L ] (3) c L,a 0 c L,0 0 (4) (1) is the entrepreneur s expected consumption (2) is the lender s break-even condition (assume entrepreneur has bargaining power, or lenders compete) (3) is the truth-telling constraint (4) is the limited liability constraint (2) will obviuosly bind. Add it back to the objective function and we get that the objective is to maximize πy H + (1 π) y L (1 π) pγ which is just equivalent to minimizing p. Just maximize surplus s.t. constraints. Two possible cases: 1. y L x. Here p = 0, c = y k meets the constraints, so it is optimal 2. y L < x. Here constraint (3) will bind. Solve by taking FOCs: 1 + λ (1 π) [c L,0 c L,a ] + µ [c L,0 + y H y L ] = 0 (p) λπ + µ = 0 (c H ) λ (1 π) (1 p) µ (1 p) + η 0 = 0 (c L,0 ) λ (1 π) p + η a = 0 (c L,a ) for p, c H, c L,0 and c L,a respectively. The FOCs imply that both η 0 and η a will be positive, so the limited liability constraints will bind: c L,0 = c L,a = 0. We can then 2

3 solve by using (2) and (3): π (y H c H ) + (1 π) [y L pγ] = x (5) c H = (1 p) [y H y L ] (6) π (y H (1 p) [y H y L ]) + (1 π) [y L pγ] x p = x y L π (y H y L ) (1 π) γ (7) More auditing is necessary when: x is high. More borrowing require higher repayment in good state more incentive to lie γ is high. Need more resources to repay audit costs require higher repayment in good state more incentive to lie y H y L is high more incentive to lie Commitment Without lotteries and many states: debt is optimal 2 The BG model Embed the costly state varification model in a bare-bones RBC model with OLG Feed iid shocks into the model, obtain serially correlated output (while the frictionless model would be iid) Balance sheet effects 2.1 Preferences η entrepreneurs and 1 η lenders They live for two periods Endowment of labour when young: L e and L. ηl e + (1 η) L = 1 Entrepreneurs born at t have utility c e,o t+1 3

4 Lenders born at t have utility ( u c l,y t ) + βc l,o t Technology There are two goods: output and capital Output is Y t = θ t F (K t, L) θ is iid, with Eθ = θ Capital is produced by entrepreneurs 1 unit of output goods at time t one (indivisible) project at time t a random amount of capital κ at t + 1, with mean κ 1 The outcome of the project κ is subject to costly state verification - a lender has to pay γ units of capital to audit it Capital fully depreciates within a period Output can also be stored. Storage has an exogenous gross rate of return of r. Denote the amount of output stored in period t by M t+1 Let I t be the fraction of entrepreneurs who choose to undertake a project The aggregate number of projects initiated is ηi t Let p t be the unconditional probability that an entrepreneur who invested is audited in period t + 1 with respect to the project he undertook in period t The measure of t projects that are audited at t + 1 is ηi t p t The resource constraint is [ ηi t + ηc e,o t + (1 η) c l,y t K t+1 = ηi t [ κ γp t ] ] + c l,o t + M t+1 θ t F (K t, L) + rm t Assumption 1. Parameters are such that a positive level of storage is always used 1 Original paper has heterogeneous entrepreneurs, with type ω requiring x (ω) units of output to create a project. This adds an extensive margin to the investment decision: which entrepreneurs end up investing and which do not. It also adds the possibility of endogenous investment-specific productivity. 4

5 2.3 Equilibrium with γ = 0 By assumption, storage is used at the margin A unit of capital at t + 1 is worth q t+1 = θ t+1 F (K t+1 ) with expected value q t+1 = θf (K t+1 ) (8) Expected value of a project: Opportunity cost of undertaking a project: r κ q t+1 In an interior solution Solving: κ q t+1 = r (9) r = κ θf (K t+1 ) which simply says that the expected marginal product of investment should equal the return on storage 2 (Assume I t = K t+1 η is interior) K t+1 does not depend on any state variables (θ is iid, so θ is constant). Output is iid. 2.4 Equilibrium with γ > 0 Each entrepreneur will have labour income (which he saves, because he only consumes in period t + 1) of s = w t 1 L e = θ t 1 F L (K t 1, 1) If he wants to do the project, he needs to borrow an amount 1 s from lenders 2 The original paper has two source of diminishing marginal product of investment: the intensive and extensive margin, but the main results do not really rely on the fact that there is an extensive margin. 5

6 Borrowing contracts are assumed to take the optimal form from section 1 renormalized so lenders expect a return of r rather than 1 and we multiply the value of the outcome by the price of capital, because it is units of capital Suppose there are two possible values of κ: κ L and κ H Assume the following timing of the resultion of uncertainty: 1. κ realized 2. contract between entrepreneur and lender settled (announcement, auditing, repayment, etc.) 3. θ realized (so IC constraints, etc. do not need to be specified separately for each realization of θ) Optimal contract: If κ L q t+1 > r [x s], then no auditing Otherwise, from (7), audit with probability p = r [1 s] q t+1 κ L π q t+1 κ (1 π) q t+1 γ where κ κ H κ L (Recall, this comes from satisfying the break-even constraint (2) and the incentive constraint (3) with equality) The entrepreneur s expected consumption will be given by πc H Use (3) to solve back for c H : Notice that πc H = π (1 p) q t+1 κ = π π q t+1 κ (1 π) q t+1 γ r [1 s] + q t+1 κ L q t+1 κ π q t+1 κ (1 π) q t+1 γ = (1 π) γ [ q t+1 κ (1 π) q t+1 γ r [1 s]] (10) πc H s = r (1 π) γ > r (11) so the return to internal funds is greater than r: they also loosen the incentive constraint / reduce the need for monitoring 6

7 In an interior equilibrium πc H = rs (12) what do we mean by interior? some entrepreneurs choose to do projects but others don t in original paper, with heterogeneity in entrepreneurs, there is exactly one entrepreneur who is indifferent; here they all are, hence (12) Does (12) contradict (11)? Difference between marginal return to a single entrepreneur and marginal return to entrepreneurs as a whole Using (12) and (10): so, using (8): (1 π) γ [ q t+1 κ (1 π) q t+1 γ r [1 s]] = rs (1 π) γ [ q t+1 ( κ (1 π) γ) r] + rs (1 π) γ = rs (1 π) γ [ q (1 π) γ t+1 κ (1 π) q t+1 γ r] + rs (1 π) γ = 0 [ ( κ (1 π) γ) θf (K t+1 ) r ] (1 π) γ + rs (1 π) γ (1 π) γ = 0 s F (K t+1 ) K t+1 Here are the steps in the chain: 1. Higher labour income / savings from entrepreneurs 2. Lower required outside funds 3. Payment in high state can be lower while still letting lenders break even 4. Lower audit probability is needed to maintain incentive compatibility 5. (Other things being equal), entrepreneurs get higher payoff from doing projects 6. At the margin, entrepreneurs decide to do projects 7. The marginal product of capital goes down until they are indifferent again Introduces serial correlation in output: 7

8 Recall that s t = θ t 1 F L (K t 1, 1) (because entrepreneurs labour income come from their wages when young) High θ t 1 means high wealth for entrepreneurs, therefore high investment Implications for the cyclical patterns of leverage Re-introducing an extensive margin / heterogeneity among investors could change these Literature notes: Carlstrom and Fuerst [1997]: Embed this mechanism in an otherwise-standard quantitative RBC model Bernanke et al. [1999]: Embed the mechanism in a New Keynesian model, so they can look at how this interacts with sticky prices, monetary shocks, etc. Financing friction affects producers of output rather that specialized conversion of output into capital sector Questions What s the point of always having an active storage margin? What types of effects does this rule out? Who benefts from a reduction in financing frictions? References Ben Bernanke and Mark Gertler. Agency costs, net worth, and business fluctuations. American Economic Review, 79(1):14 31, March Ben S. Bernanke, Mark Gertler, and Simon Gilchrist. The financial accelerator in a quantitative business cycle framework. In J. B. Taylor and M. Woodford, editors, Handbook of Macroeconomics, volume 1 of Handbook of Macroeconomics, chapter 21, pages Elsevier, September Charles T. Carlstrom and Timothy S. Fuerst. Agency costs, net worth, and business fluctuations: A computable general equilibrium analysis. The American Economic Review, 87(5): , Robert M. Townsend. Optimal contracts and competitive markets with costly state verification. Journal of Economic Theory, 21(2): , October