PARTIAL DERIVATIVE

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.

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,

            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

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”.

Example 3

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_L) as the slope of the curve.  The 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_L at L₀ becomes steeper as K increases.  But 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.

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)

Q* = (ad - bc)/(b + d)                                                                   (2)

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

1.

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

2.

which shows that if a increases, the equilibrium quantity also increases.  The magnitude of the change depends on b and d

       3.

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.

4.

which shows that if b increases, the equilibrium quantity decreases.  The magnitude of the change depends on a, b, c, and d.

Figure 2

            These two CSD are shown in figures 2 and 3.

Figure 3

           

            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.

            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

The results are illustrated in figures 4 and 5.  Note that in each case only one parameter changes while the others are held constant.

Figure 4

Figure 5

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

Y = C + I₀ + G₀                             a > 0, 0 < b < 1

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 I₀ and G₀, and the parameters are a, b, c, and t, a total of 6 constants.

            The equilibrium values of the model are

 

Take the derivatives with respect to the parameters in Y^*, C^*, T^*, each with a, b, c, t, I₀ and G₀, we have a total of 3x6 = 18 comparative statics derivatives.

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

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

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

Figure 6

            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

where d₁ and d₂ are non-negative numbers.  The fundamental equation of the Input-Output model is                   

(I - A)x = d

Using the inverse matrix method, we have

x = (I - A)^-1d

or,

Hence,

            Taking d₁ = 1 and d₂ = 0, we see that the first column of the inverse matrix shows the total requirement of equilibrium output x₁^* when d₁ changes one unit and holding d₂ constant.  Similarly, taking d₁ = 0 and d₂ = 1, we see that the second column of the inverse matrix shows the total requirement of equilibrium output x₂^* when d₂ changes one unit and holding d₁ constant. 

            The effects of changes in d₁ on x₁^* and x₂^* may be derived separately by using the comparative statics derivatives.

,   

,     .

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.