# Flow budget constraint in a paper by Garleanu and Panageas

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!