Transcript Economics 214

Economics 214
Lecture 18
Ceteris Paribus


Economic analysis often proceeds by
considering the consequences of a
certain event, ceteris paribus.
The advantage of this approach is that
it identifies the exclusive impact of the
variable under study rather than
confounding its effect with the effect of
other variables.
Ceteris Paribus and
Multivariate Functions
Suppose you have the following demand function :
Q   - P  I  PS where Q  Quantity, P  own price, I  income
and PS  price of substitute good.
The effect of a 1 unit increase in income holding price of the good and
price of substitute constant is
Q QI 1  QI  - P   (I  1)  PS    - P  I  PS 


I
1
1

Graph of income effect
P

D0
D1
Q/T
Demand Example Continued
Now let the demand equation be
Q  100 I
5P  2 PS 
The difference in the quantity demanded given a change in income is now
Q 100( I  1) 5P  2 PS   100 I 5P  2 PS 
100


5P  2 PS 
I
I 1- I
The gain from an increase income decreases as the price of the product
rises and rises as its substitute ' s price rises.
Effect of income change on
Demand for model 2
P
P*
P**
D1
D0
Q/T
Partial Derivatives
A partial derivative of a multivariate function with
respect to any one of its arguments represents the
rate of change of the value of that function due to a
very small change in that argument while all the
other variables that are also arguments of this
function are held constant. The difference quotient
of the previous example introduces the ceteris
paribus characteristic of a partial derivative.
Partial Derivative
The partial derivative of the function y  f(x1,x2 ,...,xn ) with
respect to its argument x j , written as y x j is
f x1 ,, x j  x j ,, xn   f x1 , , x j , , xn 
y
 lim
x j x j 0
x j
provided the limit exists.
We may also denote the partial derivative of y with respect
to x j as f j(x1,x2 ,...,xn ) or f j .
Partial Derivative

The partial derivative of a multivariate
function with respect to one of its
arguments is found by applying the
rules for univariate differentiation and
treating all the other arguments of the
function as constant.
Examples
3x12
Let y 
x2
y
6 x1

x1
x2
Let y  e x1x2
y
 x2 e x1x2
x1
and
and
y
3x12

3/ 2
x2
2 x2 
y
 x1e x1x2
x2
Economic Example
Suppose you have the following utility function :
U  U  x1,x2   200 ln( x1 )  200 ln( x2 )  x1 x2
Marginal Utility of good x1
U 200
 x2

U1 
x1
x1
Marginal Utility of good x2
U2 
U 200
 x1

x2
x2
Geometric Interpretation of
Partial Derivative

As a special type of derivative, a partial
derivative is a measure of the
instantaneous rate of change of some
variable, and in that capacity it again
has a geometric counterpart in the
slope of a particular curve.
A Production Function and Its
“Slices”
3 D graph of Production
Function
Graph of Partial Derivative of Y
with respect to K
Y
Y/K
K
L
L1
K1
L2 K2
Second Partial Derivative &
Cross Partial Derivatives
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
A second partial derivative is a measure of
how a partial derivative with respect to one
argument of a multivariate function changes
with a very small change in that argument.
A cross partial derivative is a measure of how
a partial derivative taken with respect to one
of the arguments of the multivariate function
varies with a very small change in another
argument of that function.
Partial Derivatives
We have the function y  f(x1,x2 )
The two partial derivates are f1(x1,x2 ) and f 2(x1,x2 ) or
y
y
and
x2
x1
Second partial derivative s
  y   2 y
  2

f11(x1,x2 ) 
x1  x1  x1
  y   2 y
  2

f 22(x1,x2 ) 
x2  x2  x2
Cross Partial Derivative
The cross partial derivative representi ng the partial
derivative of f 1(x1 ,x2 ) taken wit h respect to x2 is denote as
  y 
2 y

 
f 21(x1,x2 ) 
x2  x1  x2 x1
and the cross partial derivate of f 2(x1 ,x2 ) taken wit h respect
to x1 is denoted as
  y 
2 y

 
f12(x1,x2 ) 
x1  x2  x1x2
Example Utility Function
U  U ( X , Y )  X  Y 1 0    1 and   0
Marginal Utilities
1
U
Y 
 X  1Y 1    
UX 
X
X
0

U
X
 1   X  Y   1       0
UY 
Y
Y 
Diminish Marginal Utility or second partial derivative s
 2U
   1X   2Y 1  0
U XX 
2
X
 2U
  1


0
Y
X



1



U YY 
2
Y
Example Cross Partial
Marginal Utilities
1
U
Y 
UX 
 X  1Y 1    
X
X
0

U
X
UY 
 1   X  Y   1       0
Y
Y 
The cross partials are


Y
2
 U   U
U YX
 1   X  1Y   0


 X  YX
  U   2U
U XY 
  1   X  1Y   0


X  Y  XY
These goods are compliment s in consumptio n. More of
one increases the marginal utility of the other.