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

<p>For a risky asset, one unit of <span class="math-container">$X$</span> is now worth <span class="math-container">$P_t$</span>. Since the agent holds <span class="math-container">$X_{t-1}$</span> units of the risky asset, she effectively has wealth <span class="math-container">$P_tX_{t-1}$</span> from risky assets.</p>

<p>Adding the two gives us the budget at period <span class="math-container">$t$</span>, i.e.
<span class="math-container">$$
W_{t} = B_{t-1} + P_t X_{t-1}
$$</span>
And period <span class="math-container">$t$</span> portfolio choice satisfies
<span class="math-container">$$
W_t = B_t + P_t X_t
$$</span>
Combining the two gives us the desired equation.</p>

CC BY-SA 4.0
2019-12-05T20:35:19.100Z
Walrasian Auctioneer you are the best!
Nav89
CC BY-SA 4.0
2019-12-05T15:52:42.727Z
2019-12-05T15:52:42.727Z
Walrasian Auctioneer
<p>Let <span class="math-container">$W$</span> denote the investor's wealth <span class="math-container">$B$</span> is the the investment in the risk free asset (suppose that the risk free rate is taken to be zero) and <span class="math-container">$X$</span> is the investment on the risky security which has price <span class="math-container">$P$</span>. Se the investors total wealth is:
<span class="math-container">$$W=B+PX$$</span>
The capital gain that is generated in some some time interval <span class="math-container">$[t-1,t]$</span> is about <span class="math-container">$X_{t-1}(P_t-P_{t-1})$</span> and the portfolio rebalancing then occurring to the intertemporal budget constrained is:
<span class="math-container">$$B_t+P_tX_t=B_{t-1}+P_tX_{t-1}$$</span>
I wonder why does the last equation hold? Does this come from the intertemporal budget constraint and how?</p>

<p>In the sequel it refers to the wealth change that is:
<span class="math-container">$$W_{t}-W_{t-1}=X_{t-1}(P_{t}-P_{t-1})$$</span></p>

<p><span class="math-container">$\underline{Note:}$</span> It is from Kerry Back 1992 paper</p>

<microeconomics><self-study><capital-returns>
Intertemporal budget constraint notion
Nav89
# Intertemporal budget constraint notion

(Asked by: username_0 on 2019-12-04)

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

**Tags:** microeconomics, self-study, capital-returns

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## 1 answers

### Answer by username_1 on 2019-12-05. Score: 3 | Accepted answer

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.

> **Comments:**
> **username_0**: Walrasian Auctioneer you are the best!  

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