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The partial derivative of a function of two or more variables is

By Justin Gonzalez,2014-11-26 12:41
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The partial derivative of a function of two or more variables is

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    C7PARDER.DOC The partial derivative of a function of two or more variables is the ordinary derivative of the

    function with respect to only one of the variables (or “partials”), considering the others as

    constants.

     Let z = f(x, y). Holding y constant, we take the derivative with respect to x, the partial derivative may be denoted in several ways.

    dzf (((((f,f,ffxyxy;;;;xx11dxxy

    Here the subscript 1 denotes the 1st argument of the function, 2 the second, etc.

    Example 1

    Let the budget line be M = px + qy. Then,

    MM ((p and.qxy

     That is, when the consumer purchases one extra unit of x, expenditure must increase by the amount of its price.

    Example 2

    Let the utility function be

    1213//Uxyxy((??U,3223 ;;;;;;

    Then the marginal utility of x and y are

    1??1213//U,xyMUxy;(32323??;;;;;;?1x??2

    3??1213//(3223xy??;;;;???2 1??1223//U,xyMUxy;(?32232;;;;;;?2y??3

    2??1223//(3223xy??;;;;???3

    At x = 1 and y = 3 we may calculate the marginal utility as

    3??1213//U,()()13(312233??;;;;;;?1??2 2??1223//U,()()13(312233??;;;;;;?2??3

    Here the units of utility are given as “utils”.

    Example 3

    Let the production function be given as a function of labor L and capital K.

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    22QKLKLKL((,,F,253 ;;

    Then the marginal productivities of capital and labor are

    Q((,256MPKLLL Q((,252MPLKKK

    Illustration of Q = F(L, K).

     Figure 1 shows the marginal productivity of labor (F) as the slope of the curve. The L

    curve is called the labor productivity curve. The position of this curve depends on the amount of

    capital associated with labor. In general, the more capital associated with labor, the more

    productive the labor becomes. In this case the MP at L becomes steeper as K increases. But L0

    in other cases this may not be so.

    Y??YFLK(,???

    YFLK(,;;

    YFLK(,;;

    FL

    LL00

    Figure 1 The Productivity Curve of Labor

Applications of Partial Derivatives

    Definition: The problem of Comparative Statics Analysis is to find the effects of the

    change in a parameter on the equilibrium value, quantitatively and qualitatively.

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    The problem can be solved by finding the partial derivatives of an equilibrium variable with

    respect to each parameter.

     Two questions may be answered from these partial derivatives. (i) Qualitative changes (direction of changes)

     Increase, decrease, or no changes

    (ii) Quantitative changes (magnitude of changes)

     value of partial derivatives.

Examples.

    1. A Market Model In chapter 3 we have seen that a market model is given as

    D = a - bp

    S = -c + dp

    D = S

    where a, b, c, and d are positive parameters. From Chapter 3 on static analysis, we know that

    the equilibrium values are

    p* = (a + c)/(b + d) (1)

     - bc)/(b + d) (2) Q* = (ad

    The comparative static analysis asks what are the effects of changes in parameters a, b, c, d on the

    equilibrium values p* and Q*?

    (a) the effects on p* are given by

     p*/a, p*/b, p*/c, p*/d,

    (b) the effects on Q* are given by

     Q*/a, Q*/b, Q*/c, Q*/d,

    There are two equilibrium values and four parameters. Here there are a total of 2x4=8 partial

    derivatives. These partial derivatives are called the comparative static derivatives (CSD).

    The CSD are derived as follows

    1p(01. abd;;

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    which shows that an increase in a (the intercept of the demand function) will increase the

    equilibrium price p, and its magnitude depends only on b and d, the slopes of the supply and demand curves

    Qd2. (0 abd;;

    which shows that if a increases, the equilibrium quantity also increases. The magnitude of the

    change depends on b and d

    D,S D,S

    a

    aa S D DS

    E’

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