# Budget constraint in Radner Sequential Trade Equilibria

Suppose that $$q$$ is a k-tuple vector of prices for the k assets whose quantities are given by the k-tuple $$\theta$$. I have just read that in the Radner Sequential Trade Equilibrium (not sure if this goes by another name), where the trading of assets occurs in period t=0 (as opposed to t=1), the period-zero budget can be expressed as $$q*\theta$$, where $$*$$ denotes the inner product, and that this budget is non-positive.

I feel like I must be misunderstanding something about this particular setting, because I have no idea how a budget could possibly be non-positive, as opposed to non-negative.

Can someone give me some intuition for what's happening here?

## 1 Answer

In Radner's model each actor starts with an initial endowment, an asset vector $$\omega$$. You can trade this to any asset portfolio $$x$$ that has lower or equal value, so $$x * q \leq \omega * q.$$ Denoting asset trades by $$\theta = x - \omega$$, you can also express this as $$\theta * q \leq 0.$$

• So in the initial stage t=0, I've been assigned some quantity of each asset, and I can trade with some second party to possibly change the quantities I possess and these new quantities I end up with are the vector $x$, correct? Also, am I right to say that the only reason trade occurs is because there is uncertainty about which state will be realized in t=1? – Dion Apr 16 '19 at 21:08
• @David 1. Yes. 2. This depends on the exact modell you are using. In case the assets are not just financial but can also represent goods, trade is also possible without uncertainty, though this is not the main point of Radner. Also if you have new questions please post them as new questions. – Giskard Apr 17 '19 at 4:50