By Donald A. R. George (auth.)

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**Sample text**

The dual variable A. is the shadow price of storage while !! is the shadow price of pollution. This similarity between dual variables and the 'multipliers' of Chapter 3 should now be clear. It should not therefore be a surprise that these dual variables can be thought of as the agent's subjective valuation of inputs. They indicate the maximum amount the agent would be willing to pay in order to have the corresponding constraint relaxed by one unit at the margin. Shadow prices have important roles in planning of various kinds, cost-benefit analysis and other branches of economics.

5 clearly has an interior minimum at the origin. At this point it is clear that the tangent plane has a zero slope in all directions. Thus both its partial derivatives vanish there. This property can be stated as a necessary condition for an interior maximum or minimum of a function of several variables. At such a point all the function's partial derivatives must equal zero. 7 Let f: D-R, where DC Rn. If Xo ED is an interior maximum or an interior minimum of f then = 1, ... , n i NB A point at which f;(x 0 ) = 0 (i = 1, ...

The partial derivative [;(x) can be calculated by holding xi constant where i of= j and differentiating f with respect to X; as if it were a function of X; alone. Example Suppose Then and It is clear that partial derivatives have some immediate economic interpretations. 7) above) relates inputs to outputs. The partial derivative f gives a measure of how much extra output could be produced by increasing the labour input by one unit at the margin, holding the capital input constant. This is, of course, precisely what is meant by the marginal product of labour.