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Unformatted text preview: CHAPTER 7 Partial Differentiation From the previous two chapters we know how to differentiate functions of one variable . But many functions in economics depend on several variables: output depends on both capital and labour, for example. In this chapter we show how to differentiate functions of several vari ables and apply this to find the marginal products of labour and capital, marginal utilities, and elasticities of demand . We use differentials to find the gradients of isoquants, and hence determine marginal rates of substitution . ./ 1. Partial Derivatives The derivative of a function of one variable, such as y ( x ), tells us the gradient of the function: how y changes when x increases. If we have a function of more than one variable, such as: z ( x,y ) = x 3 + 4 xy + 5 y 2 we can ask, for example, how z changes when x increases but y doesnt change. The answer to this question is found by thinking of z as a function of x , and differentiating, treating y as if it were a constant parameter: z x = 3 x 2 + 4 y This process is called partial differentiation. We write z x rather than dz dx , to emphasize that z is a function of another variable as well as x , which is being held constant. z x is called the partial derivative of z with respect to x z x is pronounced partial dee z by dee x. Similarly, if we hold x constant, we can find the partial derivative with respect to y : z y = 4 x + 10 y Remember from Chapter 4 that you can think of z ( x,y ) as a surface in 3 dimensions. Imagine that x and y represent coordinates on a map, and z ( x,y ) represents the height of the land at the point ( x,y ). Then, z x tells you the gradient of the land as you walk in the direction of increasing x , keeping the y coordinate constant. If z x > 0, you are walking uphill; if it is negative you are going down. (Try drawing a picture to illustrate this.) 117 118 7. PARTIAL DIFFERENTIATION Exercises 7.1 : Find the partial derivatives with respect to x and y of the functions: (1) (1) f ( x,y ) = 3 x 2 xy 4 (3) g ( x,y ) = ln x y (2) h ( x,y ) = ( x + 1) 2 ( y + 2) Examples 1.1 : For the function f ( x,y ) = x 3 e y : (i) Show that f is increasing in x for all values of x and y . f x = 3 x 2 e y Since 3 x 2 > 0 for all values of x , and e y > 0 for all values of y , this derivative is always positive: f is increasing in x . (ii) For what values of x and y is the function increasing in y ? f y = x 3 e y When x < 0 this derivative is positive, so f is increasing in y . When x > 0, f is decreasing in y . 1.1. Secondorder Partial Derivatives For the function in the previous section: z ( x,y ) = x 3 + 4 xy + 5 y 2 we found: z x = 3 x 2 + 4 y z y = 4 x + 10 y These are the firstorder partial derivatives. But we can differentiate again to find second order partial derivatives. The second derivative with respect to x tells us how z x changes as x increases, still keeping...
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This note was uploaded on 04/12/2010 for the course ECON DEAM taught by Professor Vines during the Spring '10 term at Oxford University.
 Spring '10
 Vines
 Economics

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