In a certain economy, time is discrete with periods $t=0,1,2,...$. The economy is populated by many households and identical firms. The utility of a household is:
$\displaystyle\sum^{\infty}_{t=0}\beta^t\Big(\log c_t - \gamma L_t^{\frac{1}{\gamma}}\Big)$
where $c_t$ is consumption and $L_t$ is labor in hours worked. There are two assets: capital, $K_t$, and a one-period risk-free bond $b_t$.
Capital is rented to the firms at a rate $v_t$. The law of motion for capital is $K_{t+1}=(1-\delta)K_t+I_T$. $I_t$ is gross investment and $I_t-\delta K_t$ is net investment.
The bond, is in zero net supply and yields $r_t$ per period.
Firms are owned by households and produce output $Y_t^p=K_t^{\alpha}L_t^{1-\alpha}$ where $0<\alpha<1$. Wages are $w_t$.
The government has a sequence of public spending $\{G_t\}_{t=0}^{\infty}$ that will be paid with a combination of taxed bonds at a rate $\tau_t^b$, taxed net investment at a rate $\tau_t^I$, taxed labor income at a rate $\tau_t^w$, and taxed consumption at a rate $\tau_t^c$. The government cannot borrow or save, $Y_t=Y_t^p$.
I need to set up the firm's problem, the household's problem, and the social planner's problem but have no idea how to incorporate the tax structure.
For example, here is what I have for the household's problem, but I'm pretty sure it's not right:
$\begin{aligned} \max_{c_t, L_t, K_{t+1}} \quad & \displaystyle\sum_{t=0}^{\infty} \beta^t (c_t, L_t)\\ \textrm{s.t.} \quad & c_t + I_t + \tau_t = Y_t = K_t^{\alpha} L_t^{1-\alpha} + G_t\\ \end{aligned}$
I'd appreciate any hints for setting up these problems.
