Intertemporal budget constraint notion

Question

Let $W$ denote the investor's wealth $B$ is the the investment in the risk free asset (suppose that the risk free rate is taken to be zero) and $X$ is the investment on the risky security which has price $P$. Se the investors total wealth is: $$W=B+PX$$ The capital gain that is generated in some some time interval $[t-1,t]$ is about $X_{t-1}(P_t-P_{t-1})$ and the portfolio rebalancing then occurring to the intertemporal budget constrained is: $$B_t+P_tX_t=B_{t-1}+P_tX_{t-1}$$ I wonder why does the last equation hold? Does this come from the intertemporal budget constraint and how?

In the sequel it refers to the wealth change that is: $$W_{t}-W_{t-1}=X_{t-1}(P_{t}-P_{t-1})$$

$\underline{Note:}$ It is from Kerry Back 1992 paper

Answer 1

At time $(t-1)$, the investor buys some risk free bond, $B_{t-1}$ and some risky asset $X_{t-1}$ at price $P_{t-1}$, such that the budget constraint holds, i.e. $$ W_{t-1} = B_{t-1} + P_{t-1}X_{t-1} $$ At period $t$, one unit of risk free bond pays off one unit, so $B_{t-1}$ units of risk free bonds pays off $B_{t-1}$ units of wealth at $t$.

For a risky asset, one unit of $X$ is now worth $P_t$. Since the agent holds $X_{t-1}$ units of the risky asset, she effectively has wealth $P_tX_{t-1}$ from risky assets.

Adding the two gives us the budget at period $t$, i.e. $$ W_{t} = B_{t-1} + P_t X_{t-1} $$ And period $t$ portfolio choice satisfies $$ W_t = B_t + P_t X_t $$ Combining the two gives us the desired equation.