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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.
dz
dx

y
f
 fx x, y   f1 x, y   fx  f1
x
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,
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.
Example 2
Let the utility function be
U  Ux, y   3x  2
1/ 2
2y
 3
1 /3
Then the marginal utility of x and y are
 1
1/ 2
1/3
U 1 x, y   MU x    3x  2
32y  3
 2
 3
1 / 2
   3x  2
2y  3 1/3
 2
U 2 x, y   MU y  3x  2
1/ 2  1 
  2y  3
 3
2/3
2
 2
1/ 2
2/3
   3x  2 2y  3
 3
At x = 1 and y = 3 we may calculate the marginal utility as
 3
1/ 2
1 /3
U 1 1,3    31
( )  2
( )  3
23
 2
 2
1/ 2
2/3
U 2 1,3    31
( )  2 23
( )  3
 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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Q  FK, L  25KL  K 2  3L2
Then the marginal productivities of capital and labor are
Q
 MPL  25K  6L
L
Q
 MPK  25L  2K
K
Illustration of Q = F(L, K).
Figure 1 shows the marginal productivity of labor (FL) 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 MPL at L0 becomes steeper as K increases. But in
other cases this may not be so.
Y


Y  F  L, K 


 
Y  F L, K

Y  F L, K

FL
0
L0
L
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)
Q* = (ad - bc)/(b + d)
(1)
(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.
p 

a
1
 0
b  d
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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
2.
Q 

a
d
 0
b
  d
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’
a
a
S
D
D
S
E’
+
E
Q*
-
E
Q
*
b
d
p*
E’
d
P
+
-c
b
p
*
b’
P
+
-c
Figure 2
3.
Figure 3
a  c
p 
 
 0
b
b  d2
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.
c b  d   ad  bc
d a  c 
   1
Q 

 
 0
2
b
b  d
b  d2
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
a
D,S
S
D
a
S
D
E’
+
E
- Q*
Q
*
E
E’
b
d
p*
b
d’ d
P
p
+
-c
*
-
P
-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
 p
1

0
 c b  d 
Q 

c
a  c
p 
 
 0
d
b  d2
ba  c 
Q 

 0
d
b  d2
b
 0
b  d
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 + I0 + G0
a > 0, 0 < b < 1
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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
Y 
C 
T 
a  bc  I 0  G0
1  b  bt
a  bc  b 1  t  I 0  G0 
1  b  bt
c 1  b   t  a  I 0  G0 
1  b  bt
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
Y
= Investment multiplier
 I0
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
 Y
 C
=k
0
a
a
 T
= tk  0
a
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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
a11 a12 
d1 
A  
, d   

a21 a22 
d2 
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,
a12  d1 
x1 
1  a22
1
x  
 a
1
 a11  d2 
I

A
21
 2

Hence,
x1 
x2 
1  a22 d1
 a12d2
I A
a12d2  1  a11 d2
I A
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.
Y,C
Y
T

Y=C+I0+G0
T
C


C=a+bY
slope = t
a’
c
a
45
AS
Figure 6
Y


Y
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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.
x1
1  a22
,

d1
I A
x1
a12

d2
I A
x2
a21
,

d1
I A
x2
1  a11
.

d2
I A
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.