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!



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.