Recap:
We considered the constrained optimization of of finding which consumption bundles \(x\) maximise \(u(x)\) subject to \(p\ x\leq y\). We intuitively motivated a procedure for computing expressions for \(x_{1},\ldots,x_{n}\) purely in terms of disposable money y and prices. For our illustrative example with \(u\left(x_{1},x_{2}\right)=\log{\left(x_{1}\right)+log(x_{2})}\) we showed that the expressions for \((x_{1},x_{2})\) found following this procedure are a solution to the problem of maximizing \(u(x)\) subject to \(p\ x\leq y\). Actually, this turns out to be the unique solution to the problem. This typically turns out to be true in economic applications. In practice we can therefore simply carry out the procedure we defined here. In the other capsules we will often apply this procedure.