Supplemental Material (Technical Appendices)

Supplemental Material (Technical Appendices) Appendix H Tax on entrepreneurs In the main text we have considered the case that the total bailout money is financed by taxing workers In this Technical Appendix we consider a case that in order to finance bailout money the government taxes not only workers but also entrepreneurs who do not suffer losses from bubble investments In this case when bubbles collapse at date t bailout money λ X is financed through aggregate tax revenues from workers T u t and aggregate tax revenues from entrepreneurs who do not suffer losses T e t : with T e t λ X = T u t + T e t = τ ( ) q t α H Zt H r t Bt H α H ( θ)p τ (α L θα H ) ϕ(λ) + (α H α L )p σkσ t if 0 λ λ = τ( θ)σkt σ if λ λ where τ is a tax rate imposed on the date t net worth of the non loss-making entrepreneurs (ie H-types in period t ) For technical reasons ie in order to derive entrepreneur s consumption function explicitly we consider the case that the government taxes entrepreneur s net worth T e t increases with λ in 0 λ λ This means that as λ rises aggregate H-investments expand during bubbly periods which increases tax revenues from the non loss-making entrepreneurs when bubbles collapse This increase in tax revenues reduces tax burden for workers When we solve for tax burden per unit of workers Tt u measure) we learn (recall that there are workers with unit (H) T u t = λ X T e t = F (λ)σk σ t

with (H2) F (λ) = λ βϕ(λ) βϕ(λ) τ α H ( θ)p (α L θα H ) ϕ(λ) + (α H α L )p if 0 λ λ λ βϕ(λ) βϕ(λ) τ( θ) if λ λ It follows that T u t is a decreasing function of τ By using (H) and (H2) W (λ) is replaced with (H3) M(λ) = log σ σf (λ) + βσ log + F (λ) βσ + βσ ) ( βσ β log + αh α L α L θα p βα L σ + β log( σ) H β From (30) together with (H3) we see how an increase in τ affects workers welfare We learn that (30) is an increasing function of τ ie workers welfare increases with τ Intuition is very simple If the government imposes higher tax rate on the non loss-making entrepreneurs tax burden per unit of workers decreases which increases workers consumption when bubbles collapse thereby improving their welfare We also learn from (30) how an increase in bailout guarantees affects workers welfare in this case We find that even in this case if (A) holds then partial bailouts are optimal for workers Moreover when we compute (30) with (H3) under the benchmark parameter case then we find that λ increases with τ and approaches λ This means that optimal bailouts for workers approach the bailout level that maximizes ex-ante output efficiency When the government taxes the non lossmaking entrepreneurs tax revenues from those entrepreneurs increase together with an increase in λ since T e t is an increasing function of λ This increase in tax revenues lowers tax burden for workers As a result the welfare-enhancing effect captured by the first term of (3) dominates the welfare-reducing effect captured by the second term of (3) even in greater values of λ < λ We can also compute welfare for entrepreneurs in this case When computing it we need to take into account the fact that entrepreneurs are taxed when bubbles collapse if they are H-types in one period before bubbles collapsing (see the Technical Appendix for derivation of the value function in this case) We find that welfare for entrepreneurs monotonically increases with λ even in this case 2

Appendix I Derive the demand function for bubble assets of a L-entrepreneur Each L-entrepreneur chooses optimal amounts of b i t x i t and z i t so that the expected marginal utility from investing in three assets is equalized The first order conditions with respect to x i t and b i t are (I4) (x i t) : c i t = πβ + c iπ t+ + ( π)λβ d t+ c i( π)λ t+ (I5) (b i t) : c i t = πβ r t c iπ t+ + ( π)λβ r t c i( π)λ t+ + ( π)( λ)β r t c i( π)( λ) t+ where c iπ t+ = ( β)(q t+ α L z i t r t b i t + + x i t) c i( π)λ t+ = ( β)(q t+ α L z i t r t b i t + m i t+) and c i( π)( λ) t+ = ( β)(q t+ α L z i t r t b i t) 24 The RHS of (I4) is the gain in expected discounted utility from holding one additional unit of bubble assets at date t + With probability π bubbles survive in which case the entrepreneur can sell the additional unit at + but with probability π bubbles collapse in which case with probability λ he/she is rescued and receives d t+ units of consumption goods per unit of bubble assets and with probability λ he/she is not rescued and receives nothing The denominators reflect the respective marginal utilities of consumption The RHS of (I5) is the gain in expected discounted utility from lending one additional unit It is similar to the RHS of (I4) except for the fact that lending yields r t at date t + irrespective of whether or not bubbles collapse From (7) (I4) and (I5) we can derive the demand function for bubble assets of a type i L-entrepreneur in the main text Appendix J Aggregation The great merit of the expressions for each entrepreneur s investment and demand for bubble assets z i t and x i t is that they are linear in period-t net worth e i t Hence aggregation is easy: we do not need to keep track of the distributions 24 Since the entrepreneur consumes a fraction β of the current net worth in each period the optimal consumption level at date t + is independent of the entrepreneur s type at date t + It only depends on whether bubbles collapse and whether government rescues the entrepreneur 3

From (6) we learn the aggregate H-investments: (J6) Z H t = βpa t θq t+α H r t where A t q t K t + X is the aggregate wealth of entrepreneurs at date t and i H t e i t = pa t is the aggregate wealth of H-entrepreneurs at date t From this investment function we see that the aggregate H-investments are both historydependent and forward-looking because they depend on asset prices as well as cash flows from the investment projects in the previous period q t K t In this respect this investment function is similar to the one in Kiyotaki and Moore (997) There is a significant difference In the Kiyotaki-Moore model the investment function depends on land prices which reflect fundamentals (cash flows from land) while in our model it depends on bubble prices Aggregate L-investments depend on the level of the interest rate: (J7) Zt L = βa t βpa t θαh α L X if r t = q t+ α L 0 if r t > q t+ α L When r t = q t+ α L L-entrepreneurs may invest positive amount In this case we know from (9) that aggregate L-investments are equal to aggregate savings of the economy minus aggregate H-investments minus aggregate value of bubbles When r t > q t+ α L L-entrepreneurs do not invest The aggregate counterpart to (8) is (J8) X t = δ(λ) + r t + β( p)a t r t where i L t e i t = ( p)a t is the aggregate net worth of L-entrepreneurs at date t (J8) is the aggregate demand function for bubble assets at date t 4

Appendix K Worker s Behavior We verify that workers do not save nor buy asset bubbles in equilibrium First we verify that workers do not save When the borrowing constrained binds workers do not save The condition that the borrowing constraint binds is c u t > πβ r t c uπ t+ + ( π)β r t c u π t+ We know that c u t = w t and c uπ t+ = w t+ if workers do not save nor buy bubble assets Then the above can be written as (K9) > π + ( π) When r t = q t+ α L (K9) can be written as σ σ λ βϕ(λ) βϕ(λ) σ β Kσ t r Kt+ σ t (K0) H(λ) σ > π + ( π) βσαl σ λ βϕ(λ) σ βϕ(λ) Since H(λ)/βσα L > in the bubble regions and the right hand side of (K0) is an increasing function of σ and equals one with σ = 0 (K0) holds if σ is sufficiently small When r t = θq t+ α H βϕ(λ)/ p ϕ(λ) (K0) can be written as (K) H(λ) p ϕ(λ) θβσα H ϕ(λ) σ > π + ( π) σ λ βϕ(λ) σ βϕ(λ) Since H(λ) p ϕ(λ)/θβσα H ϕ(λ) > in the bubble regions and the right hand side of (K0) is an increasing function of σ and equals one with σ = 0 (K) holds if σ is sufficiently small Under the reasonable parameter values in our numerical examples both (K0) and (K) hold Next we verify that workers do not buy bubble assets When the short sale constraint binds workers do not buy bubble assets The condition that the short sale constraint binds is We know c u t c u t > πβ + c uπ t+ = w t and c uπ t+ = w t+ if workers do not save nor buy bubble assets 5

Then the above can be written as which is true > πβ w t w t+ + = πβ Appendix L Behavior of H-types We verify that H-types do not buy bubble assets in equilibrium When the short sale constraint binds H-types do not buy bubble assets In order that the short sale constraint binds the following condition must hold: (L2) c i t > βe t c i t+ + Since the borrowing constraint is binding for H-types we have (L3) c i t = βe t rt c i t+ We also know that c i t+ = ( β) rt α H ( θ) (L2) yields r t θq t+ α H q t+ α H ( θ) r t θq t+ α H if (L2) is true Inserting (L3) into (L4) β r t q t+ α H δ(λ) + + θq t+ α δ(λ) H + r t > 0 c i t+ r t θq t+ α H If (L4) holds then the short sale constraint binds We see that the second term in the numerator is positive as long as ϕ > 0 and we know that ϕ > 0 on the saddle path Thus if the first term is positive (L4) holds The condition that the first term is positive is q t+ α H > δ(λ) + On the saddle path since follows according to (L5) = βϕ(λ) X βϕ(λ) σkσ t 6

Using (L5) the above inequality condition can be written as (L6) σα H K σ t > δ(λ)k t+ First we show that (L6) holds in 0 λ λ In 0 λ λ aggregate capital stock follows (28) Thus (L6) can be written as (L7) which is equivalent to α H > δ(λ) ( + αh α L α L θα H p)βαl βα L ϕ(λ) / βϕ(λ) (L8) α H > δ(λ) ( + αh α L α L θα H p)βαl βα L ( p) / δ(λ)β( p) The right hand side of (L8) is an increasing and convex function of λ in 0 λ λ Thus (L6) holds in 0 λ λ if (L6) is true at λ = λ At λ = λ we know ϕ = α L ( p) θα H /(α L θα H ) Inserting this relation into (L7) yields which is true α H ( β) + αl pβα H α L θα H δ(λ ) > 0 Next we show that (L6) holds in λ λ In λ λ aggregate capital stock follows (28) Thus (L6) can be written as βϕ > δ(λ)β( ϕ) which is true since βϕ > ϕ and δ(λ)β < Appendix M Derivation of taxpayer s value function Suppose that at date t bubbles collapse After the date t the economy is in the bubbleless economy Let Vt BL be the value function of taxpayers at date t when bubbles collapse and the government bails out entrepreneurs First we solve Vt+ BL 7

Given the optimal decision rules the Bellman equation can be written as (M9) with (M20) V BL t+ (K t+ ) = log c t+ + βv BL t+2 (K t+2 ) after date t + K t+2 = c t+ = w t+ after date t + + αh α L p βα L σk σ α L θα H t+ after date t + We guess that the value function is a linear function of logk : (M2) V BL t+ (K t+ ) = f + g log K t+ after date t + From (M9)-(M2) applying the method of undetermined coefficients yields f = ) βσ log( σ) + ( β β βσ log + αh α L α L θα p βα L σ H Thus we have V BL t+ (K t+ ) = (M22) g = σ βσ ) βσ log( σ) + ( β β βσ log + αh α L α L θα p βα L σ H + σ βσ log K t+ after date t + Next we derive the value function of taxpayers at date t when bubbles collapse and the government bails out entrepreneur by taking into account the effects of bailouts on the date t consumption and the date t + aggregate capital stock The value function of taxpayers at date t satisfies (M23) Vt BL (K t ) = log c t + βvt+ BL (K t+ ) 8

with (M24) K t+ = c t = w t λ X = w t λ βϕ(λ) βϕ(λ) σkσ t + αh α L p βα L σ + λ βϕ(λ) K σ α L θα H βϕ(λ) t From (M22) (M23) and (M24) we have (29) in the text Now we are in a position to derive the value function at any date t in the bubble economy Let Vt BB (K t ) be the value function of taxpayers at date t in the bubble economy Given optimal decision rules the Bellman equation can be written as (M25) Vt BB (K t ) = log c t + β πvt+ BB (K t+ ) + ( π)vt+ BL (K t+ ) with the optimal decision rule of aggregate capital stock until bubbles collapse: (M26) K t+ = H(λ)K σ t We guess that the value function is a linear function of logk : (M27) V BB t (K t ) = s + Q log K t From (29) and (M25)-(M26) applying the method of undetermined coefficients yields s = β( π) log( σ) + βπ βπ M(λ) + βσ log H(λ) βπ βσ Thus we have (30) in the text Q = σ βσ 9

Appendix N Appendix N Derivation of entrepreneur s value function the case where the government does not tax entrepreneurs Suppose that at date t bubbles collapse After the date t the economy is in the bubbleless economy Let Wt BL (e t K t ) be the value function of the entrepreneur at date t who holds the net worth e t at the beginning of the period t before knowing his/her type of the period t First we solve W BL t+(e t K t ) Given the optimal decision rules the Bellman equation can be written as (N28) W BL t+(e t+ K t+ ) = log c i t++β pwt+2(r BL t+βe H t+ K t+2 ) +( p)wt+2(r BL t+βe L t+ K t+2 ) after date t+ where R t+βe H t+ and R t+βe L t+ are the date t + 2 net worth of the entrepreneur when he/she was H-type and L-type at date t + respectively R t+ H and R t+ L are realized rate of return per unit of saving from date t + to date t + 2 in the bubbleless economy and they satisfy (N29) R H t Aggregate capital stock follows: = q t+α H ( θ) θαh α L after date t R L t = q t+ α L after date t (N30) K t+2 = ( + αh α L α L θα H p)βα L σk σ t+ after date t + We guess that the value function are linear functions of log K and log e : (N3) W BL t+(e t+ K t+ ) = f + g log K t+ + h log e t+ 0

From (N28)-(N3) applying the method of undetermined coefficients yields (N32) f = β log( β) + β ( β) log( β) + β 2 ( β) log σ 2 β + p log αh ( θ) + ( p) log α L ( β) 2 θαh α L β(σ ) + ( ( β) 2 βσ log + αh α L α L θα H p)βαl σ (N33) g = βσ σ βσ β (N34) h = β Next we derive the value function at date t when bubbles collapse and the government bails out entrepreneurs by taking into account the effects of bailouts on the date t + aggregate capital stock Given the optimal decision rules the value function at date t satisfies (N35) with Wt BL (e t K t ) = log c t + β pw BL (N36) K t+ = From (N3)-(N36) we obtain (N37) Wt BL β(σ ) (e t K t ) = f + β t+(r t H βe t K t+ ) + ( p)w BL t+(r L t βe t K t+ ) + αh α L α L θα p βα L σ + λ βϕ(λ) K H t σ βϕ(λ) βσ log + λ βϕ(λ) + βσ σ βϕ(λ) βσ β log K t+ β log e t Now we are in a position to derive the value function at any date t in the bubble economy Wt BB (e t K t ) is the value function of the entrepreneur at any date t in the bubble economy who holds the net worth e t at the beginning of the period t before knowing his/her type of the period t Given optimal decision rules the Bellman

equation can be written as (N38) W BB t (e t K t ) = log c t + βπ pwt+ BB (Rt H βe t K t+ ) + ( p)wt+ BB (Rt L βe t K t+ ) pwt+(r BL t H βe t K t+ ) + ( p)λwt+(r BL t L βe t K t+ ) + β( π) +( p)( λ)w BL t+(r LL t βe t K t+ ) where Rt H βe t Rt L βe t and Rt LL βe t are the date t + net worth of the entrepreneur in each state Rt H Rt L and Rt LL are realized rate of return per unit of saving from date t to date t + and in 0 λ λ they satisfy (N39) R H t R L t = δ(λ) + = q t+α H ( θ) θαh α L = δ(λ) q t+α L p ϕ(λ) δ(λ)( p) ϕ(λ) R LL t = q t+α L p ϕ(λ) p and in λ λ they satisfy (N40) R H t R L t = δ(λ) + = q t+α H ( θ) ϕ(λ) p = δ(λ) q t+θα H ϕ(λ) δ(λ)( p) ϕ(λ) R LL t = q t+θα H ϕ(λ) p Aggregate capital stock until bubbles collapse follows: (N4) K t+ = H(λ)K σ t We guess that the value function are linear functions of log K and log e : (N42) W BB t (e t K t ) = m + l log K t + n log e t From (N37)-(N42) and (N38) applying the method of undetermined coefficients 2

yields m = βπ log( β) + βπ β(σ ) + log H(λ) βπ βσ β { β( π) β(σ ) + f + βπ β βσ log + β βπ β πj + ( π)j 2 β β log β + βπ β β log σ + λ βϕ(λ) } βϕ(λ) where in 0 λ λ and in λ λ J = p log αh ( θ) θαh α L J 2 = p log αh ( θ) θαh α L l = +( p)( λ) log J = p log αh ( θ) ϕ(λ) p βσ(σ ) βσ n = β β + ( p) log δ(λ) αl p ϕ(λ) δ(λ)( p) ϕ(λ) + ( p)λ log δ(λ) αl p ϕ(λ) δ(λ)( p) ϕ(λ) α L p ϕ(λ) p + ( p) log δ(λ) θα H ϕ(λ) δ(λ)( p) ϕ(λ) + ( p)λ log J 2 = p log αh ( θ) ϕ(λ) p θα H ϕ(λ) +( p)( λ) log p Thus we have (32) in the text θα H ϕ(λ) δ(λ) δ(λ)( p) ϕ(λ) 3

Appendix N2 the case where the government taxes entrepreneurs When the government taxes entrepreneurs who do not suffer losses from bubble investments m and J 2 change as follows: m = βπ log( β) + βπ β(σ ) + log H(λ) βπ βσ β { β( π) β(σ ) + f + βπ β βσ + β βπ β πj + ( π)j 2 β β log β + βπ } log + F (λ) β β log σ in 0 λ λ J 2 = p log ( τ)αh ( θ) θαh α L α L p ϕ(λ) +( p)( λ) log p in λ λ J 2 = p log ( τ)αh ( θ) ϕ(λ) p θα H ϕ(λ) +( p)( λ) log p + ( p)λ log δ(λ) αl p ϕ(λ) δ(λ)( p) ϕ(λ) + ( p)λ log θα H ϕ(λ) δ(λ) δ(λ)( p) ϕ(λ) Appendix O Procedures to derive numerical examples of entrepreneur s welfare When we compute (32) we make the following assumptions: aggregate capital stock in the initial period is set to the steady-state value of the bubbleless economy; population measure of entrepreneurs is assumed to be equal to one; in the initial period each entrepreneur is endowed with the same amount of capital kt i = k t 4

and one unit of bubble assets and owes no debt Under these assumptions all entrepreneurs hold the same amount of net worth in the initial period ie e 0 = q 0 k 0 + P 0 By using determination of equilibrium bubble prices (L5) e 0 can be written as e 0 (λ) = Inserting the above relation into (32) yields βϕ(λ) σkσ 0 W0 BB (K 0 ) = m(λ) + β log σ + βσ log K 0 log βϕ(λ) β Figure 4 describes the relationship between W BB 0 and λ 5