If economic agents have to determine in advance their supply or demand in reaction to different market prices we may assume that their strategic instruments are supply or demand functions. The best examples for such markets are the spot markets for electricity in England and Wales, in Chile, in New Zealand, in Scandinavia and perhaps elsewhere. A further example is computerized trading in stock markets, financial markets, or commodity exchanges. The functional form of equilibria is explicitly determined in this paper. Under a certain condition, equilibria exist for every finite spread of (stochastic) autonomous demand, i.e. demand from small, non-strategically acting consumers. Contrary to competition with supply functions alone, however, there is no tendency for market prices to converge to 0 if the spread of autonomous demand increases infinitely. Lower bounds of market prices can be computed instead.