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\begin{document}
\section{Financial Sector Weakness and Macroeconomic Volatlity}
\bigskip Assaf Razin and Hui Tong
\bigskip \bigskip\ \bigskip
\section{\protect\bigskip Moral Hazard and Crises}
September , 2005
Assaf Razinand Hui Tong
Notes on Investment, Moral Hazard, Bubble and Crash, by Paul Krugman
\bigskip
Model's Features:
(1) One sector model
(2) Investment convex cost of adjustmen
(3)growth rate of labor is a random variable
\bigskip
Notation: small letters are logs of cap letters
\bigskip Equations:
(1) l$_{t+1}=l_{t}+\alpha _{\substack{ t+1}}$
$\alpha ^{e}=$expected $\alpha $; $\alpha ^{H}=$ Maximum value of $\alpha $.
\bigskip
(2) $Cost$ of adjustment = $[1+\frac{1}{2}\frac{1}{\upsilon }\frac{I}{K}]$
(3) \ $k_{t+1}=k_{t}+\upsilon (q_{t}-1)-l_{t}$
$q_{t}$ = Tobin q
\bigskip
The capital rental rate:
(4) $R_{t}=R[k_{t}-l_{t}]$
r = The world rate of interest
\bigskip
I. A Two-Period Model
\bigskip
A. No Moral Hazard
\bigskip
(5) $(1+r)q_{1}=E_{1}[R[k_{2}]]$
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = $E_{1}\left\{ R[k_{1}+\upsilon
(q_{1}-1)-\alpha _{2}]\right\} $
$(1+r)dq_{1}=E_{1}\left\{ R^{\prime }[k_{1}+\upsilon (q_{1}-1)-\alpha
_{2}]\right\} [dk_{1}-\alpha _{2}]$
$(1+r-\upsilon E_{1}\left\{ R^{\prime }[k_{1}+\upsilon (q_{1}-1)-\alpha
_{2}]\right\} )dq_{1}=E_{1}\left\{ R^{\prime }[k_{1}+\upsilon
(q_{1}-1)-\alpha _{2}]\right\} [dk_{1}-\alpha _{2}]$
$dq_{1}=\frac{E_{1}\left\{ R^{\prime }[k_{1}+\upsilon (q_{1}-1)-\alpha
_{2}]\right\} [dk_{1}-\alpha _{2}]}{(1+r-\upsilon E_{1}\left\{ R^{\prime
}[k_{1}+\upsilon (q_{1}-1)-\alpha _{2}]\right\} )}.$
B. Moral Hazard
We introduce a financial intermediary which borrows and uses the proceeds to
buy capital. The intermediary borrows at the rate r to buy capital at the
price $q_{1}$. The realized return on the intermediary's investment \ is
random $R_{2}=R[k_{1}-\alpha _{2}].$
(1 )Competitive
(2)no capital adequacy requirement
(3) the intermediary's liabilities are guaranteed:
If $R>(1+r)q_{1}$, \ the intermediary makes profits;
If $R<(1+r)q_{1}$, bail out and zero profits.
\bigskip
Competition among intermediaries drives up the price $q_{1}$ to:
$(1+r)q_{1}$\ \ \ \ = $E_{1}\left\{ R[k_{1}+\upsilon (q_{1}-1)-\alpha
^{H}]\right\} $
$q_{1}^{H}$\ \ \ \ = $\frac{R[k_{1}+\upsilon (q_{1}^{H}-1)-\alpha ^{H}]}{%
(1+r)}.$
\bigskip
II. The Infinite Horizon Model
\bigskip
\bigskip
II(1). No Moral Hazard
Linearizing R(k-l) yields:
(6) R$_{t}=\bar{A}-\rho (k_{t}-l_{t})$
(7) (1+r)q$_{t}=E_{t}$\ \ $\left( R_{t+1}\right) +E_{t}q_{t+1}$
(8) $\ \ \ \ \ E_{t}\ \ \left( R_{t+1}\right) =E_{t}\left( \bar{A}-\rho
(k_{t+1}-l_{t+1})\right) $
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = $E_{t}\left( \bar{A}-\rho
(k_{t}+\upsilon (q_{t}-1)-\alpha _{t+1})\right) $
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =$\bar{A}+\rho \upsilon +\rho
\alpha ^{e}-\rho \upsilon q_{t}-\rho k_{t}$
(7) and (8) imply:
(9) \ \ q$_{t}=\frac{1}{1+r+\rho \upsilon }[\bar{A}+\rho \upsilon +\rho
\alpha ^{e}-\rho k_{t}+E_{t}q_{t+1}]$
Solution by a "guess":
(10) q$_{t}=A+B(k_{t}-\alpha _{t})$
(10') E$_{t}$q$_{t+1}=A+BE_{t}(k_{t+1}-\alpha _{t+1})$
but
$E_{t}(k_{t+1})=k_{t}+\upsilon (q_{t}-1)-\alpha _{t}^{e}$
$\ \ \ \ \ \ \ \ \ \ \ \ E_{t}q_{t+1}=A+B(k_{t}+\upsilon (q_{t}-1)-\alpha
_{t}^{e}-\alpha _{t+1})$
\ \ \ \ \ \ \ \ \ \ substitute into \ (9) yields
q$_{t}=\frac{1}{1+r+\rho \upsilon }[\bar{A}+\rho \upsilon +\rho \alpha
^{e}-\rho k_{t}+A+B\left( (k_{t}+\upsilon (A+Bk_{t})-\alpha _{t}^{e})\right)
]=A+Bk_{t}$
rearranging terms yields:
$\left( B+\frac{\rho -B-\upsilon B^{2}}{1+r+\rho \upsilon }\right) k_{t}=%
\frac{1}{1+r+\rho \upsilon }[\bar{A}+\rho \upsilon +\rho \alpha
^{e}+A+B\upsilon A-\alpha ^{e})]-A$
Implying:
\bigskip (11) $\left( B+\frac{\rho -B-\upsilon B^{2}}{1+r+\rho \upsilon }%
\right) =0$
(12) $\frac{1}{1+r+\rho \upsilon }[\bar{A}+\rho \upsilon +\rho \alpha
^{e}+A+B\upsilon A-\alpha ^{e})]-A=0$
Equations (11) and (12) can be solved for A and B as follows:
B = $\frac{r+\rho \upsilon -(r+\rho \upsilon )^{2}+4\rho \upsilon }{%
2\upsilon }<0$
A = (1 -$\frac{1+B\upsilon }{1+r+\rho \upsilon })^{-1}\frac{1}{1+r+\rho
\upsilon }[\bar{A}+\rho \upsilon +\rho \alpha ^{e}+A+B\upsilon A-\alpha
^{e})].$
\bigskip
Proposition 1: The jumping variable q$_{t}$ is negatively related to the
state variable k$_{t}.$
\bigskip
II(2). Moral Hazard
\bigskip
Payoffs:
If $R_{t+1}^{\ast }=\bar{A}-\rho (k_{t+1}-l_{t+1})>(1+r)q_{t}$, \ the
intermediary makes profits;
If $R_{t+1}^{\ast }=\bar{A}-\rho (k_{t+1}-l_{t+1})<(1+r)q_{t}$, bail out and
zero profits.
Competition among the intermediaries bid up the asset price so that asset
price so that the return is given by R$_{t}^{\ast }=\bar{A}-\rho
(k_{t}-\alpha ^{H})$
(7') (1+r)q$_{t}^{\ast }=R_{t+1}^{\ast }+q_{t+1}^{\ast }$
(8') $\ \ \ \ \ R_{t+1}^{\ast }=\bar{A}-\rho (k_{t+1}-\alpha ^{H}$
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = $\left( \bar{A}-\rho
(k_{t}+\upsilon (q_{t}-1)-\alpha ^{H})\right) $
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =$\bar{A}+\rho \upsilon +\rho
\alpha ^{H}-\rho \upsilon q_{t}^{\ast }-\rho k_{t}$
(7') and (8') imply:
(9') \ \ q$_{t}^{\ast }=\frac{1}{1+r+\rho \upsilon }[\bar{A}+\rho \upsilon
+\rho \alpha ^{H}-\rho k_{t}+q_{t+1}^{\ast }]$
Solution by a "guess":
(10') q$_{t}^{\ast }=A+B(k_{t}-\alpha _{t})$
(10') q$_{t+1}^{\ast }=A+B(k_{t+1}-\alpha ^{H})$
but
$k_{t+1}=k_{t}+\upsilon (q_{t}-1)-\alpha _{t}^{H}$
$\ \ \ \ \ \ \ \ \ \ \ \ q_{t+1}=A+B(k_{t}+\upsilon (q_{t}-1)-\alpha
_{t}^{H}-\alpha _{t+1})$
\ \ \ \ \ \ \ \ \ \ substitute into \ (9') yields
q$_{t}=\frac{1}{1+r+\rho \upsilon }[\bar{A}+\rho \upsilon +\rho \alpha
^{H}-\rho k_{t}+A+B\left( (k_{t}+\upsilon (A+Bk_{t})-\alpha ^{H})\right)
]=A+Bk_{t}$
rearranging terms yields:
$\left( B^{\ast }+\frac{\rho -B^{\ast }-\upsilon B^{\ast 2}}{1+r+\rho
\upsilon }\right) k_{t}=\frac{1}{1+r+\rho \upsilon }[\bar{A}+\rho \upsilon
+\rho \alpha ^{H}+A^{\ast }+B\upsilon A^{\ast }-\alpha ^{H})]-A^{\ast }$
Implying:
\bigskip (11) $\left( B^{\ast }+\frac{\rho -B^{\ast }-\upsilon B^{\ast 2}}{%
1+r+\rho \upsilon }\right) =0$
(12) $\frac{1}{1+r+\rho \upsilon }[\bar{A}+\rho \upsilon +\rho \alpha
^{H}+A^{\ast }+B^{\ast }\upsilon A-\alpha ^{H})]-A^{\ast }=0$
\bigskip
Proposition 2: A$^{\ast }>A;B^{\ast }=B.$
\bigskip
III. Financial Crises and Moral Hazard
\bigskip
The government deficits\ \ from bailouts:
\bigskip
D$_{t}=\left( (1+r)q_{t}^{\ast }-R_{t+1}^{\ast }+q_{t+1}^{\ast }\right)
k_{t} $
D$_{t}>0$, except for $\alpha _{t}=\alpha ^{H}.$
\bigskip
Note that stronger creditor protection would require less frequent bailouts.
This can be captured by having the coefficient $\alpha ^{H}$ shifting down;
hence smaller amounts of bailouts and deficits.
Assume that the asset market bubble is sustained until
\bigskip D$_{t}=D^{\max }.$
Now consider a bad draw of $\alpha ,$ such that
The government bails out creditors who lent money in period t-1\ (or t-x, x
is some lag) but not those who lent money later. Then the asset price drops
from the asset price guaranteed level, $q^{\ast }(k_{t})\ ,\ $to the
coressponding unprotected asset price level, $q(k_{t}).$
Features:
(1) The actual loss of the intermediary is magnified by the drop in the
current price of its assets.
(2) There are self fulfilling multiple equilibrium possibilities because the
actual loss of the intermediary depens on market expectations for $q(k_{t})$%
, which in turn depend on the loss of the intermediary.
If
$\left( (1+r)q_{t-1}-R_{t+1}+q_{t}\right) k_{t}>D^{\max }.>\left(
(1+r)q_{t-1}-R_{t}+q_{t}^{\ast }\right) k_{t}$
\bigskip then
In one equilibrium: potential lenders expect that the regime with guarantees
will last one more period.
In the second equilibrium potential lenders expect the collapse of the
regime.Notes and questions:
1. In your original note, labour growth is the random variable. Here I
switched to technology shock. There is no real difference. It is just that
labour may introduce a wage issue which is absent form the model.
2. One change is the equation (5b) in this updated version (or equation 3 in
the first version). In the first version, we have $k_{t+1}=k_{t}+v\left(
q_{t}-1\right) +l_{t\text{. }}$I am not sure that we need $l_{t\text{ }}$
after we already specify $k_{t+1}=k_{t}+v\left( q_{t}-1\right) .$ Indeed, if
we add $l_{t}$, then we will get equation 12 in the first version:%
\[
A=(1-\frac{1+B\upsilon }{1+r+\rho \upsilon })^{-1}\frac{1}{1+r+\rho \upsilon
}[\bar{A}+\rho \upsilon +\rho \alpha ^{e}+A+B\upsilon A-\alpha ^{e})].
\]%
From this $A$, one would infer that as $\alpha ^{e}$ increases, $\left( \rho
-1\right) \alpha ^{e}$ would decrease, then $A$ and $q_{t}$ will decrease
too. However, if we exclude $l_{t}$, then we get equation 14 in this updated
version: \
\[
A_{1}=\frac{\bar{A}+v\rho +\alpha ^{e}-B_{1}v}{-B_{1}v+r+v\rho }
\]%
From $A_{1}$, as $\alpha ^{e}$ increases, $q_{t}$ will increase rather than
decrease. I think $A_{1}$ is more reasonable than $A$ in that higher
productivity should increase capital price. Our updated equation 5b is also
consistent with David Romer (page 372, equ 8.11). Please let me know if I
misunderstood or missed anything in the first version.
3. I am not sure about the best way to connect creditor protection with the
volatility of $q_{t\text{. }}$I gave it some try in the last section of this
draft. Could you give any suggestions? Thanks!
\section{Literature Review}
\bigskip 1. Assaf Razin: Trade Balance Dynamics. http://www.tau.ac.il/%
\symbol{126}razin/fry7.pdf
2. Paul krugman:Bubble, Boom, Crash: Theoretical notes on Asia's crisis.
Mimeo. MIT 1998 and also http://web.mit.edu/krugman/www/DISINTER.html.
3. David Romer: Advanced Macroeconomics, chapter 8.
4. Hayashi (1982).
\section{\protect\bigskip Model}
\bigskip
\subsection{Assumptions}
Model's Features:
(1) One sector model:
\[
Y_{t}=C_{t}K_{t}^{1-\rho }
\]
note that $C$ stands for technology.
(2) convex adjustment cost, then gross investment $Z_{t}$:
\[
Z_{t}=I_{t}\left( 1+\frac{1}{2}\frac{1}{v}\frac{I_{t}}{K_{t}}\right)
\]
(3) Technology is a random variable
\bigskip
\[
\ln (C_{t+1})=\ln (C)+\alpha _{t+1}
\]
Notation: small letters are logs of cap letters:
\[
c_{t+1}=c+\alpha _{t+1}
\]
Assume $\alpha _{\substack{ t+1}}$ follows a uniform distribution over the
region $[\alpha ^{L},\alpha ^{H}]$, with $\alpha ^{e}$ being the mean. \
\bigskip
\subsection{First Order Conditions}
(4) Denote $r$ as the world interest rate, a representative firm will
maximize the following Lagrangian:
\bigskip
\[
\acute{L}=\Sigma _{t=0}^{\infty }\frac{1}{\left( 1+r\right) ^{t}}\left[
CK_{t}^{1-\rho }-Z_{t}+q_{t}\left( K_{t}+I_{t}-K_{t+1}\right) \right]
\]
where $q_{t}$ is Tobin $q$.
From the Lagrangian, we can obtain a first order condition:
\bigskip (5)
\[
Z_{t}^{^{\prime }}=q_{t}
\]
That is
(5a)
\begin{eqnarray*}
Z_{t}^{^{\prime }} &=&q_{t}\Longrightarrow \\
1+\frac{1}{v}\frac{I_{t}}{K_{t}} &=&q_{t}\Rightarrow \\
\ \frac{K_{t+1}}{K_{t}} &=&v\left( q_{t}-1\right) +1
\end{eqnarray*}
Denote $\ln \left( K_{t}\right) $ as $k_{t}$, then
\[
k_{t+1}-k_{t}=\ln \left( v\left( q_{t}-1\right) +1\right)
\]
Linearizing $\ln \left( \upsilon \left( q_{t}-1\right) +1\right) $ gives:
\bigskip (5b)
\[
k_{t+1}=k_{t}+v\left( q_{t}-1\right)
\]
\bigskip Another first order condition from equation (4) is
(6)
\[
(1+r)q_{t}=E_{t}\left( R_{t+1}\right) +E_{t}q_{t+1}
\]
where $R_{t+1}$ is the capital rental rate. Note that
\begin{eqnarray*}
R_{t+1} &=&\left( 1-\rho \right) C_{t+1}K_{t+1}^{-\rho }\Longrightarrow \\
\ln \left( R_{t+1}\right) &=&\ln \left( 1-\rho \right) +c_{t+1}-\rho k_{t+1}
\end{eqnarray*}
\bigskip Linearizing $\ln \left( R_{t+1}\right) $ gives
\begin{eqnarray*}
R_{t+1}-1 &=&\ln \left( 1-\rho \right) +c_{t+1}-\rho k_{t+1} \\
&=&\ln \left( 1-\rho \right) +c+\alpha _{t+1}-\rho k_{t+1}
\end{eqnarray*}
Denote $\bar{A}\equiv 1+\ln \left( 1-\rho \right) +c$, then
\[
R_{t+1}=\bar{A}-\rho k_{t+1}+\alpha _{t+1}
\]
Therefore equation (6) becomes
(7)
\[
(1+r)q_{t}=E_{t}\left( \bar{A}-\rho k_{t+1}+\alpha _{t+1}\right)
+E_{t}q_{t+1}
\]
\bigskip
\subsection{ No Moral Hazard}
\bigskip To recap, we have
(5b)
\[
k_{t+1}=k_{t}+v\left( q_{t}-1\right)
\]
and (6)
\[
(1+r)q_{t}=E_{t}\left( R_{t+1}\right) +E_{t}q_{t+1}
\]
\bigskip
We introduce a financial intermediary which borrows and uses the proceeds to
buy capital. The intermediary borrows at the rate $r$ to buy capital at the
price $q_{t}$. The realized return on the intermediary's investment \ is
random $R_{t+1}$. Assume perfect competition among financial intermediaries
and no capital adequacy requirement.
If there is no government bailout guarantee, then
(8)
\begin{eqnarray*}
E_{t}\left( R_{t+1}\right) &=&E_{t}\left( \bar{A}-\rho k_{t+1}+\alpha
_{t+1}\right) \\
&=&E_{t}\left( \bar{A}-\rho \left( k_{t}+v\left( q_{t}-1\right) \right)
+\alpha _{t}\right) \\
&=&\bar{A}+\rho v-\rho k_{t}-\rho vq_{t}+\alpha ^{e}
\end{eqnarray*}
(6) and (8) imply:
(9)
\[
q_{t}=\frac{1}{1+r+\rho v}[\bar{A}+\rho v-\rho k_{t}+\alpha
^{e}+E_{t}q_{t+1}]
\]
\ \
Solution by a ''guess'':
(10)
\[
q_{t}=A+Bk_{t}
\]
(10a)
\[
E_{t}q_{t+1}=A+BE_{t}(k_{t+1})
\]
Since
\[
E_{t}(k_{t+1})=k_{t}+v\left( q_{t}-1\right)
\]
then (10b)
\[
E_{t}q_{t+1}=A+B\left( k_{t}+v\left( q_{t}-1\right) \right)
\]
Substituting (10b) into \ (9) yields
\[
A+Bk_{t}=\frac{1}{1+r+\rho v}[\bar{A}+\rho v-\rho k_{t}+\alpha
^{e}+A+B\left( k_{t}+v\left( A+Bk_{t}-1\right) \right) ]
\]
Rearranging terms yields:
\bigskip
\[
\left( B+\frac{\rho -B-B^{2}v}{1+r+\rho v}\right) k_{t}=\frac{1}{1+r+\rho v}[%
\bar{A}+\rho v\ +\alpha ^{e}+A+Bv\left( A-1\right) ]-A
\]
This implies equation (11):
\[
B+\frac{\rho -B-B^{2}v}{1+r+\rho v}=0
\]
and equation (12):
\[
\frac{1}{1+r+\rho v}[\bar{A}+\rho v\ +\alpha ^{e}+A+Bv\left( A-1\right) ]=A
\]
Equations (11) and (12) can be solved for $A$ and $B$ as follows:
\bigskip
$%
\begin{array}{c}
B+\frac{\rho -B-B^{2}v}{1+r+\rho v}=0%
\end{array}
$, Solution is: $\left\{ B_{1}=\frac{1}{2v}\left( r+v\rho -\sqrt{\left(
r^{2}+2v\rho r+v^{2}\rho ^{2}+4v\rho \right) }\right) \right\} ,\allowbreak
\left\{ B_{2}=\frac{1}{2v}\left( r+v\rho +\sqrt{\left( r^{2}+2v\rho
r+v^{2}\rho ^{2}+4v\rho \right) }\right) \right\} \allowbreak $
As the jumping variable $q_{t}$ is negatively related to the state variable $%
k_{t}$, we choose
\bigskip (13):
\[
B_{1}=\frac{1}{2v}\left( r+v\rho -\sqrt{\left( r^{2}+2v\rho r+v^{2}\rho
^{2}+4v\rho \right) }\right)
\]
From equation (12)
\[
\frac{1}{1+r+\rho v}[\bar{A}+\rho v\ +\alpha ^{e}+A+Bv\left( A-1\right) ]-A=0
\]
\bigskip we get (14):
\[
A=\frac{\bar{A}+v\rho +\alpha ^{e}-B_{1}v}{-B_{1}v+r+v\rho }
\]
\bigskip
\subsection{Moral Hazard}
We introduce a financial intermediary which borrows and uses the proceeds to
buy capital. The intermediary borrows at the rate $r$ to buy capital at the
price $q_{t}$. The realized return on the intermediary's investment \ is
random $R_{t+1}$. Let us assume competitiveness among financial
intermediaries.
Now suppose that the government will guarantee the investment if $\alpha
_{t}g$, the net payoff to financial intermediary is
\[
E_{t}\left( R_{t+1}\right) +E_{t}q_{t+1}-(1+r)q_{t}.
\]
Therefore (6) becomes (6)':
\[
(1+r)q_{t}=((1+r)q_{t})\ast \left( \Pr \left( a_{t+1}g\right) \right)
\]%
Combing (5b) and (6)', we get
\[
(1+r)\left( 1-\Pr \left( a_{t+1}g\right) \right) ,
\]%
which gives (9)'
\begin{eqnarray*}
\ \ q_{t}^{\ast } &=&\frac{1}{1+r+\rho \upsilon }[\bar{A}+\rho \upsilon +E%
\left[ \alpha _{t+1}|a_{t+1}>g\right] -\rho k_{t}+E_{t}\left( q_{t+1}^{\ast
}\right) ] \\
&=&\frac{1}{1+r+\rho \upsilon }[\bar{A}+\rho \upsilon +\alpha ^{e\ast }-\rho
k_{t}+E_{t}\left( q_{t+1}^{\ast }\right) ]
\end{eqnarray*}
Note that equation (9)' has the similar form as equation (9). Actually in
equation (9), $\alpha ^{e}=\frac{a^{H}+a^{L}}{2}$, but in equation (9)', $%
\alpha ^{e\ast }=\frac{a^{H}+g}{2}$. Therefore, we could use the same guess
method.
Solution by a "guess":
(10)'
\[
q_{t}^{*}=A^{*}+B^{*}k_{t}
\]
and get $\ \ \ \ \ \ \ \ \ \ \ \ $
\bigskip (13)':
\[
B^{\ast }=\frac{1}{2v}\left( r+v\rho -\sqrt{\left( r^{2}+2v\rho r+v^{2}\rho
^{2}+4v\rho \right) }\right)
\]
(14)'
\[
A^{*}=\frac{\bar{A}+v\rho +\alpha ^{e*}-B_{1}^{*}v}{-B_{1}^{*}v+r+v\rho }
\]
Comparing equation (13), (13)', (14) and (14)', we have \textbf{Proposition
1:}
\[
A^{\ast }>A;B^{\ast }=B
\]
\bigskip
\section{Financial Crises and Moral Hazard}
The government deficits\ \ from bailouts at $t$ is (only if $a_{t}g$, then there is no bailout and $D_{t}=0$.
Assume that the asset market bubble is sustained until
\[
D_{t}=D^{\max }.
\]%
Now consider a bad draw of $\alpha _{t}$, such that the government bails out
creditors who lent money in period $t-1\ $(or $t-x$, $x$ is some lag) but
not those who lent money later. Then the asset price drops from the asset
price guaranteed level, $q^{\ast }(k_{t})$,$\ $to the corresponding
unprotected asset price level, $q(k_{t}).$
Features:
(1) The actual loss of the intermediary is magnified by the drop in the
current price of its assets (from $q^{\ast }(k_{t})$ to $q(k_{t})$).
(2) There are self fulfilling multiple equilibrium possibilities because the
actual loss of the intermediary depends on market expectations for $q(k_{t})$%
, which in turn depend on the loss of the intermediary.
Suppose
\[
\left( (1+r)q_{t-1}^{\ast }-R_{t}-q_{t}\right) k_{t}>D^{\max }>\left(
(1+r)q_{t-1}^{\ast }-R_{t}-q_{t}^{\ast }\right) k_{t}
\]%
then there are two equilibriums. In one equilibrium, potential lenders
expect that the regime with guarantees will last one more period. In this
case, price at time $t$ will be $q^{\ast }(k_{t})$ as defined in equations
(10)', (13)' and (14)'. And the bailout (if any) $D_{t}=\left(
(1+r)q_{t-1}^{\ast }-R_{t}-q_{t}^{\ast }\right) k_{t}D^{\max }$, which justifies the expectation of
the collapse of the government's guarantee. In this case, we have equation
(16),
\begin{eqnarray*}
q_{t}-q_{t-1}^{\ast } &=&A+Bk_{t}-A^{\ast }-B^{\ast }k_{t-1} \\
&=&A-A^{\ast }+B^{\ast }\left( k_{t}-k_{t-1}\right) \\
&=&\frac{\frac{a^{H}+a^{L}}{2}-\frac{a^{H}+g}{2}}{-B_{1}^{\ast }v+r+v\rho }%
+B^{\ast }v\left( q_{t-1}^{\ast }-1\right) \\
&=&\frac{\frac{\ a^{L}}{2}-\frac{\ g}{2}}{-B_{1}^{\ast }v+r+v\rho }+B^{\ast
}v\left( q_{t-1}^{\ast }-1\right)
\end{eqnarray*}
The volatility to capital price $q$ comes from the uncertainty of whether
the price at time $t$ will be $q_{t}$ or $q_{t}^{\ast }$ (note that price at
$t-1$ is always $q_{t-1}^{\ast }$). Now let us look at the impact of
creditor protection on\ the volatility of $q$. Note that stronger creditor
protection would require less frequent bailouts. This can be captured by
having the coefficient $g$ shifting down.
The drop of $g$ has two effects:
First, lowering the cost of bailout. This may then lower the likelihood of $%
q_{t}^{\ast }$ becoming $q_{t}$ (Note that since there are two equilibriums,
the exact probability of $q=$ $q_{t}^{\ast }$ (i.e., $q\neq q_{t}$) cannot
be calculated for some $\alpha _{t}$).
Secondly, as $g$ decreases, the absolute difference between $%
q_{t}-q_{t}^{\ast }$, i.e., $\left| \frac{\frac{\ a^{L}}{2}-\frac{\ g}{2}}{%
-B_{1}^{\ast }v+r+v\rho }\right| $, will decrease too. Thus the capital
price volatility could be reduced.
Both effects suggest that higher creditor protection (lower $g$) will reduce
the volatility of capital price.
\section{\protect\bigskip UPDATE }
November 2005
\bigskip
Features: The tobin $q$: $\,q_{t}=B_{0}+B_{1}a_{t}+B_{2}k_{t}$( equation
10). After solving these coefficients, we get (equ 12):
a. \ If the serial correlation $\gamma \,$is $0$, then the coefficient for $%
a_{t}$ in equation (10) is $0$, back to our previous case.
b. I have looked at two cases of government's guarantee:
The first case is that the guarantee is a constant. I find that the marginal
change in the guarantee level $g$ does not affect the volatility of $%
q_{t}^{*}$. Then to get the result that government guarantee increases the
volatility of $q_{t}^{*}$, we may need to refer to the situation where the
government cannot \ honor its commitment.
The second case is that the guarantee is proportional to the technology
shock. In that case, higher guarantee level $g$ will increase the volatility
of $q_{t}^{*}$.
\section{Literature Review}
\bigskip 1. Assaf Razin: Trade Balance Dynamics. http://www.tau.ac.il/%
\symbol{126}razin/fry7.pdf
2. Paul krugman:Bubble, Boom, Crash: Theoretical notes on Asia's crisis.
Mimeo. MIT 1998 and also http://web.mit.edu/krugman/www/DISINTER.html.
3. David Romer: Advanced Macroeconomics, chapter 8.
4. Hayashi (1982).
\section{Model}
\subsection{Assumptions}
Model's Features:
(1) One sector model:
\[
Y_{t}=A_{t}K_{t}^{1-\rho }
\]
note that $A_{t}$ stands for technology.
(2) convex adjustment cost, then gross investment $Z_{t}$:
\[
Z_{t}=I_{t}\left( 1+\frac{1}{2}\frac{1}{v}\frac{I_{t}}{K_{t}}\right)
\]
(3) Technology is a random variable
\bigskip
\[
\ln (A_{t+1})=\gamma \ln (A_{t})+\varepsilon _{t+1}
\]
Notation: small letters are logs of cap letters:
\[
a_{t+1}=\gamma a_{t}+\varepsilon _{t+1}
\]
Assume $\varepsilon _{t+1}$ follows a uniform distribution over the region $%
[-1,1]$. \
\subsection{First Order Conditions}
(4) Denote $r$ as the world interest rate, a representative firm will
maximize the following Lagrangian:
\bigskip
\begin{eqnarray*}
\acute{L} &=&\Sigma _{t=0}^{\infty }\frac{1}{\left( 1+r\right) ^{t}}\left[
A_{t}K_{t}^{1-\rho }-Z_{t}+q_{t}\left( K_{t}+I_{t}-K_{t+1}\right) \right] \\
&=&\Sigma _{t=0}^{\infty }\frac{1}{\left( 1+r\right) ^{t}}\left[
A_{t}K_{t}^{1-\rho }-I_{t}\left( 1+\frac{1}{2}\frac{1}{v}\frac{I_{t}}{K_{t}}%
\right) +q_{t}\left( K_{t}+I_{t}-K_{t+1}\right) \right]
\end{eqnarray*}
where $q_{t}$ is Tobin $q$.
From the Lagrangian, we can obtain one first order conditions w.r.t. $I_{t}$:
\bigskip (5)
\[
Z_{t}^{^{\prime }}=q_{t}
\]
That is
(5a)
\begin{eqnarray*}
Z_{t}^{^{\prime }} &=&q_{t}\Longrightarrow \\
1+\frac{1}{v}\frac{I_{t}}{K_{t}} &=&q_{t}\Rightarrow \\
\ \frac{K_{t+1}}{K_{t}} &=&v\left( q_{t}-1\right) +1
\end{eqnarray*}
Denote $\ln \left( K_{t}\right) $ as $k_{t}$, then
\[
k_{t+1}-k_{t}=\ln \left( v\left( q_{t}-1\right) +1\right)
\]
Linearizing $\ln \left( \upsilon \left( q_{t}-1\right) +1\right) $ gives:
\bigskip (5b)
\[
\mathbf{k}_{t+1}\mathbf{=k}_{t}\mathbf{+v}\left( q_{t}-1\right)
\]
\bigskip Another first order condition from equation (4) w.r.t. $K_{t}$ is
(6)
\[
(1+r)q_{t}+\frac{1}{2}\frac{1}{v}\left( \frac{I_{t}}{K_{t}}\right)
^{2}=E_{t}\left( R_{t+1}\right) +E_{t}q_{t+1}
\]
where $R_{t+1}$ is the capital rental rate. Note that
\begin{eqnarray*}
R_{t+1} &=&\left( 1-\rho \right) A_{t+1}K_{t+1}^{-\rho }\Longrightarrow \\
\ln \left( R_{t+1}\right) &=&\ln \left( 1-\rho \right) +a_{t+1}-\rho k_{t+1}
\end{eqnarray*}
\bigskip Linearizing $\ln \left( R_{t+1}\right) $ gives
\begin{eqnarray*}
R_{t+1}-1 &=&\ln \left( 1-\rho \right) +a_{t+1}-\rho k_{t+1} \\
&=&\ln \left( 1-\rho \right) +a_{t+1}-\rho k_{t+1}
\end{eqnarray*}
Denote $\pi \equiv 1+\ln \left( 1-\rho \right) $, then
\[
R_{t+1}=\pi -\rho k_{t+1}+a_{t+1}
\]
Without loss of generality, assume that $\frac{1}{2v}\left( \ \frac{I_{t}}{%
K_{t}}\right) ^{2}$ is small enough compared with other terms in (6).
Therefore equation (6) becomes (7):
\[
\mathbf{(1+r)q}_{t}\mathbf{=E}_{t}\left( \pi -\rho k_{t+1}+a_{t+1}\right)
\mathbf{+E}_{t}\mathbf{q}_{t+1}
\]
\bigskip
\section{No Moral Hazard}
\bigskip To recap, we have
(5b)
\[
\mathbf{k}_{t+1}\mathbf{=k}_{t}\mathbf{+v}\left( q_{t}-1\right)
\]
and (6)
\[
\mathbf{(1+r)q}_{t}\mathbf{=E}_{t}\left( R_{t+1}\right) \mathbf{+E}_{t}%
\mathbf{q}_{t+1}
\]
\bigskip
We introduce a financial intermediary which borrows and uses the proceeds to
buy capital. The intermediary borrows at the rate $r$ to buy capital at the
price $q_{t}$. The realized return on the intermediary's investment \ is
random $R_{t+1}$. Assume perfect competition among financial intermediaries
and no capital adequacy requirement.
If there is no government bailout guarantee, then
(8)
\begin{eqnarray*}
E_{t}\left( R_{t+1}\right) &=&E_{t}\left( \pi -\rho k_{t+1}+a_{t+1}\right) \\
&=&E_{t}\left( \pi -\rho \left( k_{t}+v\left( q_{t}-1\right) \right)
+a_{t+1}\right) \\
&=&\pi +\rho v-\rho k_{t}-\rho vq_{t}+\gamma a_{t}
\end{eqnarray*}
(9)
\[
q_{t}=\frac{1}{1+r+\rho v}[\pi +\rho v-\rho k_{t}+\gamma a_{t}\
+E_{t}q_{t+1}]
\]
Solution by a ''guess'':
(10)
\[
q_{t}=B_{0}+B_{1}a_{t}+B_{2}k_{t}
\]
(10a)
\[
q_{t+1}=B_{0}+B_{1}a_{t+1}+B_{2}k_{t+1}
\]
Since
\[
E_{t}(k_{t+1})=k_{t}+v\left( q_{t}-1\right)
\]
then (10b)
\[
E_{t}q_{t+1}=B_{0}+B_{1}\left( \gamma a_{t}\right) +B_{2}\left(
k_{t}+v\left( q_{t}-1\right) \right)
\]
Substituting (10b) into \ (9) yields (11):
$\ $%
\[
B_{0}+B_{1}a_{t}+B_{2}k_{t}=\frac{1}{1+r+\rho v}[\pi +\rho v-\rho
k_{t}+\gamma a_{t}+B_{0}+B_{1}\left( \gamma a_{t}\right) +B_{2}\left(
k_{t}+v\left( B_{0}+B_{1}a_{t}+B_{2}k_{t}-1\right) \right) ]
\]
i.e.,
\[
\left( B_{2}-\frac{1}{1+r+\rho v}\left( -\rho +B_{2}\left( 1+vB_{2}\right)
\right) \right) k_{t}+\left( B_{1}-\frac{1}{1+r+\rho v}\left(
vB_{2}B_{1}+\gamma +B_{1}\gamma \right) \right) a_{t}+B_{0}-\allowbreak
\frac{1}{1+r+\rho v}\left( \pi +\rho v+vB_{2}\left( B_{0}-1\right)
+B_{0}\right) \allowbreak =0
\]
\bigskip
We solve $B_{0}$, $B_{1}$, $B_{2}$. As the jumping variable $q_{t}$ is
negatively related to the state variable $k_{t}$, we choose (12)
$%
\begin{array}{c}
\ B_{2}=\frac{1}{2v}\left( r+\rho v-\sqrt{\left( r^{2}+2r\rho v+\rho
^{2}v^{2}+4\rho v\right) }\right) \allowbreak \\
B_{1}=\frac{\gamma }{1+r+\rho v-vB_{2}-\gamma } \\
B_{0}=\frac{-\pi -\rho v+vB_{2}}{-r-\rho v+vB_{2}}%
\end{array}
$
\smallskip $\ $Note that (13):
\begin{eqnarray*}
q_{t+1}-q_{t} &=&\left( B_{0}+B_{1}a_{t+1}+B_{2}k_{t+1}\right) -\left(
B_{0}+B_{1}a_{t}+B_{2}k_{t}\right) \\
&=&B_{1}\left( a_{t+1}-a_{t}\right) +B_{2}(k_{t+1}-k_{t}) \\
&=&B_{1}\left( \gamma a_{t}+\varepsilon _{t+1}-a_{t}\right) +B_{2}(v\left(
q_{t}-1\right) )
\end{eqnarray*}
\bigskip Conditional on information available at time $t$ (i.e., $a_{t}$, $%
k_{t}$, $q_{t}$), the price variance is (14):
\begin{eqnarray*}
Var\left( q_{t+1}-q_{t}\right) &=&Var\left( B_{1}\left( \gamma
a_{t}+\varepsilon _{t+1}-a_{t}\right) +B_{2}(v\left( q_{t}-1\right) )\right)
\\
&=&Var\left( B_{1}\varepsilon _{t+1}\right) \\
&=&\frac{1}{3}\left( B_{1}\right) ^{2}
\end{eqnarray*}
Note that $\varepsilon _{t+1}$ has \ a uniform distribution over $\left[ -1,1%
\right] $.
\section{Moral Hazard}
\subsection{Government Guarantee Is a Constant}
Now suppose that the government will guarantee the investment if $\alpha
_{t}g$, the net payoff to financial intermediary is
\[
E_{t}\left( R_{t+1}\right) +E_{t}q_{t+1}-(1+r)q_{t}.
\]
Then (6) becomes (6)':
\[
(1+r)q_{t}=((1+r)q_{t})\ast \left( \Pr \left( a_{t+1}g\right) \right)
\]
(5b)
\[
k_{t+1}=k_{t}+v\left( q_{t}-1\right)
\]
Combing (5b) and (6)', we get
\[
(1+r)\left( 1-\Pr \left( a_{t+1}g\right) \right)
\]
Which gives (9)'
\begin{eqnarray*}
\ \ q_{t}^{*} &=&\frac{1}{1+r+\rho \upsilon }[\pi +\rho \upsilon +E\left[
\alpha _{t+1}|a_{t+1}>g\right] -\rho k_{t}+E_{t}\left( q_{t+1}^{*}\right) ]
\\
&=&\frac{1}{1+r+\rho \upsilon }[\pi +\rho \upsilon +\alpha ^{e*}-\rho
k_{t}+E_{t}\left( q_{t+1}^{*}\right) ]
\end{eqnarray*}
$E\left[ \alpha _{t+1}|a_{t+1}>g\right] $ stands for the expectation of $%
\alpha _{t+1}$, conditioned on that $\alpha _{t+1}>g$. \ Note that $\alpha
_{t+1}=\gamma a_{t}+\varepsilon _{t+1}$ and has a uniform distribution over $%
\left[ \gamma a_{t}-1,\gamma a_{t}+1\right] $. Assume that $g\in \left[
\gamma a_{t}-1,\gamma a_{t}+1\right] $, then
\begin{eqnarray*}
\alpha ^{e*} &\equiv &E\left[ a_{t+1}|a_{t+1}>g\right] \\
&=&\frac{g+\gamma a_{t}+1}{2}
\end{eqnarray*}
\smallskip
Note that equation (9)' has the similar form as equation (9). Actually in
equation (9), $\alpha ^{e}=\gamma \alpha _{t}$, but in equation (9)', $%
\alpha ^{e\ast }=\frac{g+\gamma \alpha _{t}+1}{2}$. Therefore, we could use
the same guess method.
Solution by a "guess":
(10)'
\[
q_{t}^{\ast }=B_{0}^{\ast }+B_{1}^{\ast }a_{t}+B_{2}^{\ast }k_{t}
\]
and get $\ \ \ \ \ \ \ \ \ \ \ \ $
\bigskip (11)':
\[
B_{0}^{*}+B_{1}^{*}a_{t}+B_{2}^{*}k_{t}=\frac{1}{1+r+\rho v}(\pi +\rho
v-\rho k_{t}+\frac{g+\gamma a_{t}+1}{2}+B_{0}^{*}+B_{1}^{*}\left( \frac{%
g+\gamma a_{t}+1}{2}\right) +B_{2}^{*}\left( k_{t}+v\left(
B_{0}^{*}+B_{1}^{*}a_{t}+B_{2}^{*}k_{t}-1\right) \right) )
\]
\bigskip That is:$\ \allowbreak $
$\left( B_{2}^{\ast }-\frac{1}{1+r+\rho v}\left( -\rho +B_{2}^{\ast }\left(
1+vB_{2}^{\ast }\right) \right) \right) k_{t}+\left( B_{1}^{\ast }-\frac{1}{%
1+r+\rho v}\left( \frac{1}{2}B_{1}^{\ast }\gamma +\frac{1}{2}\gamma
+vB_{2}^{\ast }B_{1}^{\ast }\right) \right) a_{t}+B_{0}^{\ast }-\frac{1}{%
1+r+\rho v}\left( \pi +\rho v+B_{0}^{\ast }+\frac{1}{2}g+vB_{2}^{\ast
}\left( B_{0}^{\ast }-1\right) +\frac{1}{2}+B_{1}^{\ast }\left( \frac{1}{2}g+%
\frac{1}{2}\right) \right) =0$
\bigskip
Solve for $B_{0}^{\ast }$, $B_{1}^{\ast }$, and $B_{2}^{\ast }$ gives (12)':
$%
\begin{array}{c}
B_{0}^{\ast }=\frac{1}{2}\frac{2\pi +2\rho v+g-2vB_{2}^{\ast }+1+B_{1}^{\ast
}g+B_{1}^{\ast }}{r+\rho v-vB_{2}^{\ast }} \\
B_{1}^{\ast }=\frac{\gamma }{2+2r+2\rho v-\gamma -2vB_{2}^{\ast }} \\
B_{2}^{\ast }=\frac{1}{2v}\left( r+\rho v-\sqrt{\left( r^{2}+2r\rho v+\rho
^{2}v^{2}+4\rho v\right) }\right)%
\end{array}
$
$\ $
Note that (13)'
\begin{eqnarray*}
q_{t+1}^{\ast }-q_{t}^{\ast } &=&\left( B_{0}^{\ast }+B_{1}^{\ast
}a_{t+1}+B_{2}^{\ast }k_{t+1}\right) -\left( B_{0}^{\ast }+B_{1}^{\ast
}a_{t}+B_{2}^{\ast }k_{t}\right) \\
&=&B_{1}^{\ast }\left( a_{t+1}-a_{t}\right) +B_{2}^{\ast }(k_{t+1}-k_{t}) \\
&=&B_{1}^{\ast }\left( \gamma a_{t}+\varepsilon _{t+1}-a_{t}\right)
+B_{2}^{\ast }(v\left( q_{t}^{\ast }-1\right) )
\end{eqnarray*}
\bigskip Conditional on information available at time $t$ (i.e., $a_{t}$, $%
k_{t}$, $q_{t}^{*}$), the price variance is (14)'
\begin{eqnarray*}
Var\left( q_{t+1}^{\ast }-q_{t}^{\ast }\right) &=&Var\left( B_{1}^{\ast
}\left( \gamma a_{t}+\varepsilon _{t+1}-a_{t}\right) +B_{2}^{\ast }(v\left(
q_{t}^{\ast }-1\right) )\right) \\
&=&Var\left( B_{1}^{\ast }\varepsilon _{t+1}\right) \\
&=&\frac{1}{3}\left( B_{1}^{\ast }\right) ^{2}
\end{eqnarray*}
Note that $\varepsilon _{t+1}$ has \ a uniform distribution over $\left[ -1,1%
\right] $.
\ Since $B_{1}^{*}$ does not depend on $g$, then $Var\left(
q_{t+1}^{*}-q_{t}^{*}\right) $ is not affected by the marginal change of $g$.
\subsection{Government Guarantee is Proportional to Technology Shock}
Now instead suppose that government's guarantee is proportional to the
highest value of \bigskip $A_{t+1}^{H}$, and $g=sa_{t+1}^{H}$, where $s$ is
a constant between $0$ and $1$. Then we have (9)''\
\begin{eqnarray*}
\ \ q_{t}^{*^{\prime \prime }} &=&\frac{1}{1+r+\rho \upsilon }[\pi +\rho
\upsilon +E\left[ \alpha _{t+1}|a_{t+1}>g\right] -\rho k_{t}+E_{t}\left(
q_{t+1}^{*^{\prime \prime }}\right) ] \\
&=&\frac{1}{1+r+\rho \upsilon }[\pi +\rho \upsilon +\alpha ^{e*^{\prime
\prime }}-\rho k_{t}+E_{t}\left( q_{t+1}^{*^{\prime \prime }}\right) ]
\end{eqnarray*}
Note that $a_{t+1}=\gamma a_{t}+\varepsilon _{t+1}$ and has a uniform
distribution over $\left[ \gamma a_{t}-1,\gamma a_{t}+1\right] $. That is to
say, $A_{t+1}^{H}=\gamma a_{t}+1$. Then,
\begin{eqnarray*}
\alpha ^{e*^{\prime \prime }} &\equiv &E\left[ a_{t+1}|a_{t+1}>g\right] \\
&=&\frac{\left( 1+s\right) \left( \gamma a_{t}+1\right) }{2}
\end{eqnarray*}
Equation (9)''\ has the similar form as equation (9)'. Therefore, we could
use the same guess method.
Solution by a "guess":
(10)''\
\[
q_{t}^{*^{\prime \prime }}=B_{0}^{*^{\prime \prime }}+B_{1}^{*^{\prime
\prime }}a_{t}+B_{2}^{*^{\prime \prime }}k_{t}
\]
and get $\ \ \ \ \ \ \ \ \ \ \ \ $
\bigskip (11)'':
\[
B_{0}^{*^{\prime \prime }}+B_{1}^{*^{\prime \prime }}a_{t}+B_{2}^{*^{\prime
\prime }}k_{t}=\frac{1}{1+r+\rho v}(\pi +\rho v-\rho k_{t}+\frac{\left(
1+s\right) \left( \gamma a_{t}+1\right) }{2}+B_{0}^{*^{\prime \prime
}}+B_{1}^{*^{\prime \prime }}\left( \frac{\left( 1+s\right) \left( \gamma
a_{t}+1\right) }{2}\right) +B_{2}^{*^{\prime \prime }}\left( k_{t}+v\left(
B_{0}^{*^{\prime \prime }}+B_{1}^{*^{\prime \prime }}a_{t}+B_{2}^{*^{\prime
\prime }}k_{t}-1\right) \right) )
\]
Solve for $B_{0}^{*^{\prime \prime }}$, $B_{1}^{*^{\prime \prime }}$, and $%
B_{2}^{*^{\prime \prime }}$ gives (12)'':
$\
\begin{array}{c}
B_{0}^{*^{\prime \prime }}=\frac{1}{2r+2v\rho -2vB_{2}^{*^{\prime \prime }}}%
\left( 2\pi +s+B_{1}^{*^{\prime \prime }}+2v\rho +sB_{1}^{*^{\prime \prime
}}-2vB_{2}^{*^{\prime \prime }}+1\right) \\
B_{1}^{*^{\prime \prime }}=\frac{\gamma \left( 1+s\right) }{2r+2v\rho
-2vB_{2}^{*^{\prime \prime }}+2-\gamma \left( 1+s\right) } \\
B_{2}^{*^{\prime \prime }}=\frac{1}{2v}\left( r+\rho v-\sqrt{\left(
r^{2}+2r\rho v+\rho ^{2}v^{2}+4\rho v\right) }\right)%
\end{array}
$
We get (13)''
\begin{eqnarray*}
q_{t+1}^{*^{\prime \prime }}-q_{t}^{*^{\prime \prime }} &=&\left(
B_{0}^{*^{\prime \prime }}+B_{1}^{*^{\prime \prime
}}a_{t+1}+B_{2}^{*^{\prime \prime }}k_{t+1}\right) -\left( B_{0}^{*^{\prime
\prime }}+B_{1}^{*^{\prime \prime }}a_{t}+B_{2}^{*^{\prime \prime
}}k_{t}\right) \\
&=&B_{1}^{*^{\prime \prime }}\left( a_{t+1}-a_{t}\right) +B_{2}^{*^{\prime
\prime }}(k_{t+1}-k_{t}) \\
&=&B_{1}^{*^{\prime \prime }}\left( \gamma a_{t}+\varepsilon
_{t+1}-a_{t}\right) +B_{2}^{*^{\prime \prime }}(v\left( q_{t}^{*^{\prime
\prime }}-1\right) )
\end{eqnarray*}
Conditional on information available at time $t$ (i.e., $a_{t}$, $k_{t}$, $%
q_{t}^{*^{\prime \prime }}$), the price variance is (14)''
\begin{eqnarray*}
Var\left( q_{t+1}^{*^{\prime \prime }}-q_{t}^{*^{\prime \prime }}\right)
&=&Var\left( B_{1}^{*^{\prime \prime }}\left( \gamma a_{t}+\varepsilon
_{t+1}-a_{t}\right) +B_{2}^{*^{\prime \prime }}(v\left( q_{t}^{*^{\prime
\prime }}-1\right) )\right) \\
&=&Var\left( B_{1}^{*^{\prime \prime }}\varepsilon _{t+1}\right) \\
&=&\frac{1}{3}\left( B_{1}^{*^{\prime \prime }}\right) ^{2}
\end{eqnarray*}
Note that $\varepsilon _{t+1}$ has \ a uniform distribution over $\left[ -1,1%
\right] $.
Since the marginal increase of $s$ (and therefore $g$) will increase $%
B_{1}^{*^{\prime \prime }}$, we conclude that higher government guarantee
will increase $Var\left( q_{t+1}^{*^{\prime \prime }}-q_{t}^{*^{\prime
\prime }}\right) $.
\end{document}