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I have the following budget constraint

Consumption in period 2 = Wealth in period 2 = Wealth in period 1 + Capital gain from risky assets + The endowment

where

Gain of risky assets = # of holding asset in period 1 * (Sell in period 2 – Buy in period 1)

That's, $$= \theta_1 ( S_2 - S_1) $$

and therefore, the budget constraint is

$$C_2 = W_1 +\theta_1 ( S_2 - S_1)+z(D-\bar{D} ) $$

Now, I would like to entegre capital gain tax rate $\tau \in [0, 1]$ and lump-sum transfer payments $T_0$.

I re-write the budget constraint as follow:

$$C_2 = W_1 + (1-\tau)\theta_1 ( S_2 - S_1)+z(D-\bar{D} ) + T_0 $$

Is this budget constraint correct or not?

Please share your ideas with me? Thank you

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