# Neoclassical model with proportional taxes

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.

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In your household problem you have a typo, they maximize their utility so I think you meant $$\sum_{t=0}^\infty\beta^tu(c_t, L_t)$$. Now their income comes from the profits of the firm $$\pi$$ their wages $$w_tL_t$$ and what they borrow-lend (not directly the production $$Y$$ or government expenditures $$G$$) and their expenditures are consumption, paying their debts (or receiving money from what they lend yesterday) and what they pay in taxes. Obviously they only pay consumption taxes, bond taxes, and labor income taxes, since consumers don't directly invest. We usually model that investment decisions are made by firms, so investment taxes will enter the firm's problem.