Bubbles and Credit Constraints

Transcription

1 Bubbles and Credit Constraints Miao and Wang ECON 101 Miao and Wang (2008) Bubbles and Credit Constraints 1 / 14

2 Bubbles Bubble Growth Ḃ B g or eventually they get bigger than the economy, where g is the overall growth rate of the economy. Without liquidity or dividend yield arbitrage Ḃ B = r where r is the return on other assets or discount rate. With liquidity or dividend yield q on B, y = r, so lim t e rt B (t) = lim t e rt B (0) e (r q)t = 0 Ḃ B Miao and Wang (2008) Bubbles and Credit Constraints 2 / 14

3 Household Utility Representative enterpreneur owns all unit measure of j (0, 1) firms: Max C t + V j ψ t t j dj = j 0 e rt C (t) dt j Π j tψ j tdj + w t N t where Vt j denotes firm j s stock price, ψ j t denotes holdings of firm j s stocks, Π j t denotes firm j s profits, w t is the wage, N t is labor supply. From linear utility, set ψ j t 1. Linear utility gives rvt j = Π j t + V t j Miao and Wang (2008) Bubbles and Credit Constraints 3 / 14

4 Firms Production Continuum j (0, 1) Yt j = (Kt j ) α (Nt j ) 1 α R t Kt j = Max N j (Kt j ) α (Nt j ) 1 α w t Nt j t ( ) K j α t w t = (1 α) R t = α ( K j t N j t N j t ) α 1 = α ( wt ) α 1 α 1 α D t j = R t K j t Q t ( K j t + δk j t ) dividend Miao and Wang (2008) Bubbles and Credit Constraints 4 / 14

5 Investment Opportunities Investment K j t+dt = (1 δdt) K j t + I t with prob πdt (1 δdt) K j t with prob (1 π) dt 0 I t L j t where L j t is the loan Loans L j t are fully repaid after the realization of investment returns Q t It j when Q t It j L j t : Collateral Constraint L j t V t (ζk j t ) Miao and Wang (2008) Bubbles and Credit Constraints 5 / 14

6 The value of the firm, subject to above constraints, is ) ) (( rv t (Kt j = Max I j Dt j + V t (Kt j π L j t + Q t It j t ) ( )) It j + L j t (1) and D t j = R t K j t Q t ( K j t + δk j t ) dividend (2) Here δ represents the depreciation rate of capital and D t t represents dividends excluding net investment returns. Firm j receives internal funds R t K j t and purchases (or sells) capital K j t at price Q t. When an investment opportunity arrives at the Poisson rate π, the firm borrows L t j from other firms that do not have investment opportunities, and makes investment I j returns Q t I j t. ( The change in firm value is equal to t before receiving investment L j t + Q t I j t ) ( ) It j + L j t. Miao and Wang (2008) Bubbles and Credit Constraints 6 / 14

7 Equilibrium Firms choose { } Kt j, It j, Nt j, L j t Households choose {C t } Markets clear: K t = 1 0 K t j dj, I t = 1 0 I t j dj, N t = 1 0 Nj t dj = 1, Y t = (K t ) α at prices {w t, R t } Miao and Wang (2008) Bubbles and Credit Constraints 7 / 14

8 Firm Value V ( ) Kt j = v t Kt j + b t Tobin s Q t : Bubble: Q t = e rt V t+dt K j t = e rt v t+dt e rt V t+dt K t+dt B t = e rt b t+dt = average Q t Miao and Wang (2008) Bubbles and Credit Constraints 8 / 14

9 Letdt 0. For Q > 1 invest maximally (constraint binds): ( ) It j = L j t = V ζkt j = Q t ζkt j + B t (3) Then dynamics are with R t = αk α 1 t r = Ḃ B + π(q t 1) Q t = (r + δ) Q t R t π (R t + ζq t ) (Q t 1) K t = δk t + π (R t K t + ζq t K t + B t ) Transversality: lim t e rt Q t K t = e rt B t = 0; V t (K j t ) = Q t K j t + B t, Q t = v t, B t = b t Miao and Wang (2008) Bubbles and Credit Constraints 9 / 14

10 To see this simplify Bellman eq. (1), using (2) and (3) : rv t (Kt j ) = [ ] rq t Kt j + rb t = Max I j R t Kt j Q t ( K t j + δkt j ) j t t ) π (Q t 1) (Q t ζkt j + B t [ ] + Q t Kt j + Q t K t j + Ḃ t and equate coeffi cients to get equations for Dynamics. Note: Under Kiyotaki-Moore constraint, for Q > 1, It j = L j t = Q t ζkt j. Substituting for It j the dynamic equation for the bubble, without the liquidity or collateral yield, has to grow faster, Ḃ = rb, which violates transversality. So, lim t e rt B 0 e rt = B 0 = 0, no bubble: B t 0. Miao and Wang (2008) Bubbles and Credit Constraints 10 / 14

11 Bubbleless: B t = 0 At steady state, if constraints are slack, which occurs if ζ π δ Q t = Q = 1, K = K where R = α (K ) α 1 = (r + δ) This is the planners problem: with linear utility you jump to the steady state immediately. At steady state, if constraints are not slack and firms are credit constrained because ζ is low, which occurs if ζ π δ, Q = δ ζπ, ( ) α 1 α rδ K = ζπ + δ, K < K In both bubbless equilibria, the steady states are saddles in the (K t, Q t ) dynamics (no B t ). Miao and Wang (2008) Bubbles and Credit Constraints 11 / 14

12 Bubble Equilibrium In this case Bubble provides a liquidity yield or dividend: π(q t 1). Bubble Steady State: iff (when ζ is low) B = δ ( r ) K b π ζ π + 1 Q b = r π + 1 ( r ) αk α 1 b = [(1 ζ) r + δ] π < ζ < δ r + π Note that condition ζ < δ r +π implies condition ζ π δ. Thus, if condition ζ < δ r +π holds, then there exist two steady state equilibria: one bubbleless and the other bubbly. Miao and Wang (2008) Bubbles and Credit Constraints 12 / 14

13 Results ) Bubble effects: Two steady states, (B, Q b, K b and and Q are jump vaiables ( ) 0, Q, K. B i) K > K b > K (Raise steady state K), ii) Q b < Q (Lower shadow value because K b > K, lower MPK ), iii) C > C b > C. Bubbless steady states are saddles with two roots, one (+) and one (-). Bubbly steady state indeterminate with one (+) root, two (-) roots. Global Dynamics? Miao and Wang (2008) Bubbles and Credit Constraints 13 / 14

14 Stochastic Bubbles (in brief) After Collapse economy follows bubbless equilibrium, with Value V, by jumping to the saddle path of bubbless equilibrium. Before Collapse: Probability of collapse is θ. Adjust firm values V and bubbly equilibrium path dynamics to allow jump with probability θ to bubbless saddle path. Solve similarly. (Not MIT shock: jump probability incorporated.) Miao and Wang (2008) Bubbles and Credit Constraints 14 / 14