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
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.
Here the subscript 1 denotes the 1st argument of the function, 2 the second, etc.
Let the budget line be M = px + qy. Then,
：M：M ((p and.q：x：y
That is, when the consumer purchases one extra unit of x, expenditure must increase by the amount of its price.
Let the utility function be
Then the marginal utility of x and y are
At x = 1 and y = 3 we may calculate the marginal utility as
Here the units of utility are given as “utils”.
Let the production function be given as a function of labor L and capital K.
Then the marginal productivities of capital and labor are
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.
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.
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.
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
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
aa S D DS