🚀 go-pugleaf

\documentstyle[12pt]{article} 
\topmargin -0.2in 
\textheight 8.0in 
\textwidth 5.8in 
\oddsidemargin 0.20in 
\def\baselinestretch{1.55} 
\begin{document} 
\underline{\bf Probelm 1 \rm} \\
\\The agent maximizes the expected discounted utility 
$$E_{0}\sum_{t=0}^{\tilde{T}}    \beta^{t}U(c_{t}),$$
( where $ \tilde{T}\sim[T,
\sigma_{T}^{2}]$,is normaly distributed.)
\\
\\subject to budget constraints:
$$	c_{t} + p^{s}_{t} s_{t}  + 
    p^{b}_{t} b_{t}
   =  s_{t-1} ( p^{s}_{t}  +  d_{t}  ) +  
    b_{t-1} + l_{t}  ;  $$  
where $ s_{t} ,   p^{s}_{t} ,  b_{t} ,    p^{b}_{t},  $ represent, 
respectively, holding and price of one share of stock, and 
holding and price of a riskless real government
 bond, $ d_{t} $ is dividend of the stock, and $ l_{t}$  is the wage
 received by the agent at period t. $d_{t}$ and $l_{t}$ are stochastic processes to
be specified, possibly, as Markov processes. $ E_{0}$  denotes the conditional expectation 
operator given all information available to the agent in time 0.\\

\underline{\bf Problem 2\rm}
\\
\\The agent maximizes the expected discounted utility 
$$E_{0}\sum_{t=0}^{\infty}    \beta^{t}U(c_{t})     $$
subject to budget constraints:
$$	c_{t} + p^{s}_{t} s_{t}  +  
    p^{b}_{t} b_{t}
   =  s_{t-1} ( p^{s}_{t}  +  d_{t}  ) +  
    b_{t-1} + l_{t} - \xi p^{s}_{t} \mid s_{t}-s_{t-1}\mid  ; $$ 
where $\xi$ is a constant, and  all other notations have the same meanings 
as in the previous model. The term $ \xi  p^{s}_{t} \mid s_{t}-s_{t-1}\mid $ represents
transaction cost incurred in trading stock.
\end{document}