  Asset price bubbles are situations where asset prices exceed the fundamental
values defined by the present value of dividends. This paper presents a
conceptually new perspective: the necessity of bubbles. We establish the Bubble
Necessity Theorem in a plausible general class of economic models: with faster
long-run economic growth ($G$) than dividend growth ($G_d$) and counterfactual
long-run autarky interest rate ($R$) below dividend growth, all equilibria are
bubbly with non-negligible bubble sizes relative to the economy. This bubble
necessity condition naturally arises in economies with sufficiently strong
savings motives and multiple factors or sectors with uneven productivity
growth.
