The time series of consumption is explained as a consequence of expenditure. The quantity theory hypothesis relates the level of consumption in money terms to the nominal quantity of money. This is, of course, a variant of the normal quantity theory where the level of money income is determined by the amount of money. By subtracting investment from the dependent variable, one makes the quantity theory formulation directly comparable to its Keynesian rival. Money exerts its influence on consumption directly or via elements of expenditure such as investment. With a different definition of autonomous expenditure, we get rather different results for United Kingdom data. Specifically, the monetary hypothesis is more successful for our early period up to the First World War, while the inter-war years are a strongly Keynesian period. After the Second World War, neither model has very high explanatory power, while for the overall period, there is a slightly better fit with expenditure. Exogeneity of Money

THIS PAPER DISCUSSES the problem of aggregating probability judgements and shows that Arrow-type paradoxes arise in this context, just as in the context of aggregating preferences. The need to aggregate probability judgements may arise in several different sorts of situation. Two examples which concern decision making under risk are: (i) when an individual, prior to making a decision, consults a number of experts who differ in their assessments of the probabilities of alternative states of nature; (ii) when the individuals constituting a society have to make a joint decision on the basis of identical utility functions but, again, differing assessments of the probabilities of alternative states of nature.2 We illustrate (i) above by means of a simple example: An individual consults two legal experts, who give him their probability judgements concerning the success or failure (there are just two possible outcomes) of proposed litigation. Scenario 1: The individual knows that litigation will succeed if judge J presides and barrister B defends, but will not succeed otherwise. From the experts' probability judgements the individual is able to infer that one expert has received a message that judge J will preside, and the other a message that barrister B will defend. (Either message, or both, may be false.) Given these messages, the individual updates in Bayesian fashion the prior probabilities he assigns to success and failure. Scenario 2: Success or failure depends on interpretation of the law. Pooling the information on which the experts' probability judgements are based is, we suppose, impracticable. (Perhaps they meet to discuss the case and cannot agree.) Aggregation of probability judgements in situations such as Scenario 1 (and according to axioms which we discuss in the next section) is clearly a wrong procedure which may lead to totally erroneous results. In situations such as Scenario 2 such aggregation may however have a useful role. Formulae for aggregating probability judgements have been discussed by, among others, Wagner (1982), Bordley (1982), Genest, Weerahandi, and Zidek (1984), and Fishburn and Rubinstein (1984). In this paper we generalize some results contained in the latter two papers, before going on to consider paradoxes. An early, Arrow-type impossibility result, based on rather strong assumptions, is due to Dalkey (1972). In Section 2 notation and axioms are introduced, and some results given. In Section 3 three propositions are stated and proved.