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I am reading this paper and got confused by Equation (6) in said paper. Suppose there is an investor that can trade in riskless bonds that pay interest rate $r_t$, and holds a market portfolio with value $P_t$ that pays instantaneous dividends $D_t^A$ per unit of time. Furthermore, this investor earns flow labour income $w_t$. There is also a market for annuities ala Blanchard (1985), but for the purposes of this question, we can forget it. I will also ignore the overlapping generations structure in that paper. Then, according to Equation (6), an investor with initial financial wealth $W_0=0$ is subject to

$$ \mathrm{d}W_t = r_tW_t\mathrm{d}t+ (w_t - c_t)\mathrm{d}t + \theta_t(\mathrm{d}P_t +D_t^A\mathrm{d}t - r_tP_t\mathrm{d}t), $$ where $c_t$ is flow consumption. Now the instantaneous return $\mathrm{d}R_t$ to the market portfolio should be

$$ \mathrm{d}R_t = \frac{\mathrm{d}P_t + D_t^A\mathrm{d}t}{P_t}. $$ Thus, the flow budget constraint becomes

$$ \mathrm{d}W_t = r_tW_t\mathrm{d}t+ (w_t - c_t)\mathrm{d}t + \theta_t(\mathrm{d}R_t - r_t\mathrm{d}t)P_t, $$

which looks very similar to what I would expect, except instead of $P_t$, I would expect to see $W_t$ on the right hand side. So implicitly, it seems to be saying that the value of my financial wealth is such that $W_t = P_t$. Given that the risk free asset is in zero net supply, I could see this being true in equilibrium, but not otherwise. Any clarification here would be very useful. Thanks!

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