Grundzuege einer allgemeinen Theorie der linearen by David Hilbert

By David Hilbert

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The profit-maximising and utility-maximising behaviour of firms and consumers, together with the working of supply and demand, will then ensure that the equilibrium position achieved is a Pareto optimum. To see how the conditions for Pareto optimality are fulfilled in a competitive equilibrium, first consider the equilibrium position of individual consumers. Since they are unable to affect prices their budgetary constraint is linear – a straight line in the case of two goods. Assuming non-satiation of wants, they will maximise their utility by consuming at a point P, where the budget constraint touches the highest indifference curve.

This is the first theorem in welfare economics (on the proof of this theorem see Ng, 1979/83, app. 2B. For a collected volume on general equilibrium see Debreu, 1996). The second theorem states that every Pareto-optimal state can be sustained as a competitive equilibrium by means of an appropriate distribution of resource endowments. This second welfare theorem has to be interpreted carefully to avoid misunderstanding. Suppose that the existing competitive equilibrium, A, gives a Pareto-optimal state such that the rich have high utility levels but the poor have low utility levels.

If we were to lump all three under ‘value judgments’ and refrain, as scientists, from making such judgments, we would have to leave all but undisputable scientific facts to the politicians. Another source of disagreement about policy matters is differences of interest, rather than differences in values or judgments of fact. For example the leaders of a strong trade union may press for a large wage rise despite knowing that it will lead to inflation and/or unemployment. On the ethical level they may admit (although not in public) that their action is not good for society as a whole.

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