Budget constraint ($S_0^T\theta +x=w_0$)

A restriction that the initial cost of all allocated resources equals the available budget. In a portfolio with risky holdings $\theta$ and bank holdings $x$, the displayed equation is the budget identity. Barring borrowing makes $x\ge 0$, equivalently $S_0^T\theta \le w_0$; banning the bank account instead imposes equality.

When a productive stock pays $d_t$ before trading at ex-dividend price $S_t$, an investor entering with ${\theta }_t$ shares receives $d_t{\theta }_t$ and sells or buys shares to leave with ${\theta }_{t+1}$. The displayed budget constraint accounts for the available consumption and the cost of the new holding. Here ${\theta }_t$ is known before date $t$, while $C_t$ and ${\theta }_{t+1}$ may depend on the information revealed at that date.