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

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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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    C7PARDER.DOC

    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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    C7PARDER.DOC

    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’

    + *E Q E-*QE

    b d db**p P pPb+ +

    -c -c

    Figure 3 Figure 2

    ac;;p(,0 3. 2bbd;;

    which shows that if b (the slope of the demand function, which shows the increase in demand

    when price increases by $1) increases, the equilibrium price will decrease.

    ?,,~1cbdadbcdac;;;;;;Q((,04. 22bbdbd;;;;

    which shows that if b increases, the equilibrium quantity decreases. The magnitude of the

    change depends on a, b, c, and d.

     These two CSD are shown in figures 2 and 3.

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     In figure 2 the original equilibrium point is at E. When a increases, that is, the demand curve shifts upward, holding b, c, and d constant the equilibrium price and quantity increase.

    D,S D,S

    a S aD SD

    E

    +*E EQ- *Q E’

    b d bdd**p P pP+ -

    -c -c

     -c

    Figure 5 Figure 4

     On the other hand, when b increases, holding other parameters constant, then the demand curve shifts downward, holding the intercept a constant. In this case figure 3 shows clearly that, when E changes to E’, both equilibrium price and quantity decrease. These results conform with

    the results obtained from the CSD.

     Similarly we may show that

    Qbp1(0 (;0cbd;;cbd;;

    bacac;;;;pQ(0(,0 22ddbdbd;;;;

    The results are illustrated in figures 4 and 5. Note that in each case only one parameter changes

    while the others are held constant.

    2. A National Income Model From chapter 3, a national income model may be given as

    Y = C + I + G a > 0, 0 < b < 1 00

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    C7PARDER.DOC

    C = a + b(Y - T) c > 0, 0 < t < 1

    T = c + tY

    There are three equations in three endogenous variables (Y, C, T). The exogenous variables are I and G, and the parameters are a, b, c, and t, a total of 6 constants. 00

     The equilibrium values of the model are

    ,??abcIG00(Y1,?bbt

    abcbtIG,?,?1;;;;00 C(1,?bbt

    cbtaIG1,???;;;;00T(1,?bbt

    ***Take the derivatives with respect to the parameters in Y, C, T, each with a, b, c, t, I and G, we 00

    have a total of 3x6 = 18 comparative statics derivatives.

     Interpretation of the comparative statics derivatives: For some CSD there are some

    familiar names

    Y = Investment multiplierI0

    Y=Government multiplier G0

    Y=MPC multiplier b

    Y=Income tax multiplier t

    Example

    What are the effects of changes in the subsistence level of consumption a, on equilibrium national income, equilibrium consumption, and equilibrium tax revenue?

    *** To answer this question we need only take the partial derivatives of Y, C, and T with

    respect to a. They are

    ??::YC= 0k(; aa::

    T = 0tk a

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    C7PARDER.DOC where we denote k = 1/(1-b+bt) = 1/(1-b(1-t)), which is the multiplier with the tax rate.

     Thus when the subsistence level of consumption increases, all equilibrium values increase.

    The magnitudes of the increases depend on the multiplier k.

3. Input-output models

    aad??11121A(,d( ((aad21222)?)?

    where d and d are non-negative numbers. The fundamental equation of the Input-Output 12

    model is

    (I - A)x = d

    Using the inverse matrix method, we have

    -1x = (I - A)d

    or,

    xaad1,,???1122121 ((((x,,aa1dIA221112)?)?)?

    Hence,

    1,,adad;;221122x(1IA 1,?,adad;;122112x(2IA

     Taking d = 1 and d = 0, we see that the first column of the inverse matrix shows the total 12

    *requirement of equilibrium output x when d changes one unit and holding d constant. 112

    TY,C

    Y=+0+G0YCIT

    C=+CabY

    slope = ta

    ca

    45YASY

    Figure 6

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    Similarly, taking d = 0 and d = 1, we see that the second column of the inverse matrix shows the 12

    *total requirement of equilibrium output x when d changes one unit and holding d constant. 221

    ** The effects of changes in d on x and x may be derived separately by using the 112

    comparative statics derivatives.

    x1axa122112(, ( IIdAdA12

    xax1a221211(, (. IIdAdA12

    Thus the above derivatives show how much equilibrium gross output must change, when the final

    demand for commodity 1 changes by one unit, in order to maintain the equilibrium condition in

    the economy.

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