Download Chapter 5: Univariable calculus

UNIVERSITY OF BRISTOL : DEPARTMENT OF ECONOMICS
MATHEMATICS FOR ECONOMISTS - COURSE 11122
CHAPTER 4 : UNIVARIATE CALCULUS
1.
Concept of Differentiability
Suppose f = f (x) with x and x any two points on the x-axis. The straight line through
( x , f  x  ) and (x, f  x  ) has slope b where:
b 
f  x  f  x 
.
x  x
If x is fixed then b is a function of x ( x  x ) so as we change x , b changes.
f(x)
f(x)
tangent to graph at x-
slope = b
-
f(x)
f(x)
x-
x
x
For the function f graphed above, as x gets closer and closer to x , the straight line gets
closer and closer to the tangent to the graph of f at x , and the slope b gets closer and
closer to the slope of the tangent at x (which is f ( x )). We write it mathematically as
follows:
As x converges to x i.e as x  x , b converges to f   x 
i.e.
lim
x x
b
 lim
x x
f ( x)  f ( x )
 f ( x )
(x  x)
f  x  f  x 
exists we say that f is differentiable at x .
xx
We call the limit the derivative of f at x . If f is differentiable at all x then we say f that
is differentiable and we denote the derivative by f (x).
If lim
x x
Alternatively
lim
x x
let  x  x    x and f  x   f  x    f , then b 
b  lim
 x 0
f
df

at x  x or f ( x )
x
dx
f
.
x
2
Derivatives of standard functions
d
( x n ) = n x n1 for all values of n
dx
d g x 
e
 g   x  e g x 
(ii)
dx
(i)
d
(ax) = ax ln a
dx
f   x
d
(iv)
(ln f  x  )) =
f  x
dx
(iii)
The derivative of a function f is itself a function of x. The derivative may or may not be
differentiable itself but if it is then we can differentiate to obtain the second derivative of the
d2 f
original function. We denote the second derivative of f by f (x) or
. Similarly,
dx 2
d n f  x
the nth derivative of f is denoted f (n)  x  or
.
dx n
2.
Properties of differentiable functions
Let f = f (x), g =g (x), u= u (x) and v = v (x) where  and  are constants; then
(i)
d
du
dv
(u + v) = 
+ 
dx
dx
dx
(ii)
d
dv
du
(uv)= u
+v
dx
dx
dx
(iii)
d  u
 
dx  v 
(iv)
If h(x) = g ( f (x)) then h  (x) = g'( f ( x )). f  (x)

v
du
dv
u
dx
dx
2
v
(function of a function rule).
Result 4.1 If f (x) is differentiable at x , then it is continuous at x .
3


Example 4.1 Differentiate (i) 3 ln 1  x 3  2 1  2x
3.
4
(ii)
3x 2
1  2e 
3x
Applications to economic functions
Example 4.2 Price elasticity of demand
The price elasticity of demand (denoted by  d ) for a demand function Qd  f  p , is defined
  Qd 
as the ratio of the percentage change in the quantity demanded i.e. 
 x 100%
 Qd 
  p
to the percentage change in price i.e. 
 x 100% .
 p 
 Qd
 Qd
Qd
p
Hence  d =
=
.
p
 p Qd
p
When the changes in p are infinitesimal, the above can be written as
d =
dQd
p
.
d p Qd
Similarly the price elasticity of supply is defined as
s =
d Qs p
.
d p Qs
4
Notes
(1) Since  d and  s have been defined in terms of percentage changes, they are therefore
independent of units of measurement.
(2)  d and  s are evaluated at a point on the demand function (curve) and
supply function (curve) respectively so they are known as point elasticities.
Find the price elasticity of demand and the price elasticity of supply for the demand and supply
functions given below at the point p = 4 and comment on them:
p2
(a) Qd  98 
(b) Qs  12  4 p  p 2
2
dQd
16
 90
  p  4 and Qd  98 
2
dp
dQd
p
4
16
  = -0.178
d =
.
= 4x
d p Qd
90
90
Solution (a)
so inelastic demand since demand is weakly responsive to changes in price i.e the percentage
change in demand is less than the percentage change in price.
dQs
 4  2 p = 4 + 8 = 12 and Qs  12  16  16 = 20.
dp
dQs p
4 12
 = 2.4
s =
.
= 12 x
20 5
dp Qs
(b)
so elastic supply since supply is strongly responsive to changes in price i.e the percentage
change in supply is greater then the percentage change in price.
Example 4.3 Marginal utility
Given a utility function u  u x where x is the quantity of a good that a consumer consumes,
the change in total utility resulting from some (infinitesimally small) change in the quantity x is
du
given by the derivative
or u  x which is known as the marginal utility (MU).
dx
In turn the change in MU resulting from some (infinitesimally small) change in x is given by
d  MU  d 2 u
the derivative
=
or u  x  . Clearly if MU declines as x increases infinitesimally,
dx
dx 2
then u  x  0 , indicating the operation of the law of diminishing utility.
A consumer has a utility function u x   x  where   0, 0    1, and x  0 . Does this
utility function possess the property of diminishing marginal utility?
5
Example 4.4 Marginal Cost
In the theory of cost, the firm is assumed to have a total cost function which relates total cost
(C) to the level of output (x) produced by a firm. So given a total cost function C  C x  ,
the change in total cost given some (infinitesimally small) increase in the firm’s output is given
dC
by the derivative
or C  x  , which is known as the marginal cost.
dx
The marginal cost can be interpreted as the extra cost incurred by the firm as an extra unit
of output is produced.
Given the total cost function C  C x   4 x 3  240 x 2  800 x  50 , state whether the marginal
cost is an increasing or decreasing function of the quantity produced.
Example 4.5 Marginal propensity to consume and to save
In macroeconomics we frequently encounter the concept of a consumption function, which
relates aggregate consumption expenditure (C) to the level of national income (Y). Thus given
a consumption function C  CY  , the change in consumption for some (infinitesimally small)
dC
change in income is called the marginal propensity to consume, MPC, so MPC =
or C Y  .
dY
Moreover if saving (S) is defined as the amount of national income not devoted to consumption,
then a saving function can be written as S Y   Y  CY  . This permits us to define the marginal
dS
propensity to save (MPS) as MPS =
or S Y  .
dY
6
(a) Sketch the general shape of the consumption function C  CY   1200 
7200
for Y  0 .
9  Y 
(b) Find the marginal propensity to consume (MPC) when Y = 91.
(c) Find the marginal propensity to save (MPS) when Y = 91.
(d) Determine whether or not MPC and MPS change in the same direction as Y changes.
( for the solution see Worked Example 4.5) .
Example 4.6 Marginal product
In the short-run analysis of production , the production process is assumed to involve
fixed inputs and only one variable input, labour services (l). In this case the production function,
which relates the level of output Q or y to the quantity of labour input, can be written as
dQ
dy
or
is known as the marginal product of labour (MP).
Q or y  f l  . The derivative
dl
dl
If MP declines as l increases, then
d 2Q
 0 or f l   0 , indicating the operation of the law of
dl 2
diminishing marginal product.
Example 4.7 Marginal revenue and marginal revenue product
If a firm has a total revenue function R  R(Q) where Q denotes output, then the
dR
marginal revenue (MR) is the derivative
or R (Q) . It can be interpreted as the extra
dQ
revenue earned by a firm from producing and selling an extra unit of output.
Suppose the firm has a production function Q  f l  , then we can express R as a function of l,
namely R  R( f l ) . The change in total revenue for some (infinitesimally small) change in
dR
quantity of labour input (l) is given by the derivative
or R (l ) . This derivative is known as
dl
the marginal revenue product (MRP) of labour.
To obtain R (l ) , we can use the function of a function rule or chain rule.
i.e. MRP =
dR dR dQ
= R (Q) . f l  = MR . MP where

.
dl dQ dl
A firm’s production and revenue functions are respectively, Q  f l   4l 0.5 and
Rq   120Q  2Q 2 . Find :
(a) the marginal revenue product of labour when l = 16,
(b) the price at which marginal revenue is zero
(c) the price elasticity of demand when marginal revenue is zero.
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4.
The differential of f at x
Suppose we want to approximate a non-linear function f (x) by a linear function at a
particular point x , graphically we are approximating the curve f (x) by the tangent to the curve
at the point x .
The tangent to the curve f (x) at x has slope f ( x ). The equation of a tangent to the curve
f (x) is given by
Hence
f  x   f  x   f  x  x  x 
The differential of f at x is denoted by df (x) and is defined as:
df (x) =
f ( x )  x  x 
(1)
8
Rearranging (1) gives
f  x   f  x   f  x  x  x   df  x 
i.e. the change in f as we move from x to x is approximately equal to df (x).
Hence the differential is a linear approximation to the change in the value of f in going from
x to x . It will be a good approximation when x is close to x .
Alternative (more general) form of the differential.
Let
x  x
= dx then
 df 
df (x) = f  ( x ) dx or df (x) =   dx
 dx 
 df 
More briefly still df =   dx
 dx 
Important note
Example 4.8
2
The profit function of a firm is given by   x    x 3  10 x 2  32 x  3 where x is the
3
output. Find the differential of  at the point x = 3 and use it to estimate the change in
profit when the output increases from 3 units to 3.5 units. Compare this estimated change in profit
with the actual change in profit.
9
A derivative as the ratio of two differentials
 df 
From above df =   dx .
 dx 
Divide both sides of the equation by dx , then
df    df 
 
dx   dx 
i.e.
Differential of f
.
Differential of x
df
 f  x  which has previously been regarded as a single entity can
So that the derivative
dx
the derivative of f 
now be reinterpreted as the ratio of the two differentials df and dx .
Application to elasticities
df
 df 

From above df =   dx so suppose that f(x) = ln x, then
dx
 dx 
So the differential of ln x which is d(ln x) =
From Example 4.2,  d =
dQd
dQd  p
p
.
=
.
d p Qd
dp  Qd
=
i.e the price elasticity of demand is the ratio of the differentials of ln Qd and ln p .
Similarly the price elasticity of supply is  s is the ratio of the differentials of ln Qs and
ln p so that  s =
d ln Qs 
.
d ln p 
Example 4.9 Given Qd   p   , find the price elasticity of demand using logs.
10
5.
Unconstrained Optimisation
Let (a , b) be an open interval and choose any x   (a , b) and x  (a , b).
We say that f has a local maximum at x  if f ( x  )  f (x) for all x  x  in the interval (a , b).
If f ( x  ) > f (x) for all x  x  , we say f has a strict local maximum at x  .
We can illustrate these definitions as follows;
Notice that these definitions are local i.e. they refer only to some interval around x  .
A stronger condition is:
f has a global maximum at x  if f ( x  )  f (x) for all x
f has a strict global max. at x  if f ( x  ) > f (x) for all x  x  .
NOTES
(a) Note that any global maximum is also a local maximum.
(b) If we consider the tangent to a graph at a local maximum we observe that the tangent is
horizontal. An equivalent (and important) way of restating this is contained in the next result
(for which no proof is given).
Result 4.2 If f (x) is a differentiable function and has a local maximum at x  , then
f  x  0 .
 
(c) Sometimes we refer to an x  such that f '( x  ) = 0 as a stationary point or turning point.
We also say that x  satisfies the first order condition for a maximum.
11
Example 4.10 Suppose f (x) = 2 x  x 2 . It has a maximum at x = 1 and f 1  0 .
(d)
 
Now f  x   0 is a necessary condition for a local maximum but it is not
a sufficient condition.
Example 4.11 Suppose f (x) = 3x 3 . The point x = 0 is a stationary point but it is not a
maximum but a point of inflection.
(e) Having derived a necessary condition for a maximum we wish to know if we can find a
(simple) condition that is also sufficient.
It can be shown that a sufficient condition for a function f (x) to
have a local strict maximum at x  is that f ( x  ) < 0.
We normally refer to this as the second order condition for a maximum so we can say that
x  satisfies the second order for a maximum if f ( x  ) < 0.
6. Points of inflection.
Given a function f (x) , whose second derivative f  (x) at the point x = x  is f  ( x  ) = 0,
then the point x  , will be
 
 
if f  x   0 and f  x   0 .
(i) a stationary point of inflection if f  x   0 and f  x   0 ;
(ii) a non-stationary point of inflection


Example 4.12
Show that the cost function C x  
1 3
x  3x 2  8 x  5 where x is the output, has a
3
non-stationary point of inflection when x = 3.
12
7.
More economic examples
Example 4.13
A firm has a total revenue function and a total cost function given by
Rq   20 q  q 2
Cq 
q3
 6q 2  29q  15
3
where q is the level of output.
Find : (a) the firm’s profit-maximising output level ,and the corresponding values of profit, price
and total revenue at this output level.
(b) the revenue-maximising output level, and the corresponding values of profit, price and
total revenue at this output level.
(c) whether or not the minimum profit constraint of   50 will prevent the attainment of
the revenue-maximising output level.
13
Example 4.14
(for the solution see Worked Example 4.14 )
The demand curve for a product is given by q  p  b , where b > 0. A monopolist can supply q at
cost cq where c is a constant. Determine the monopolist’s optimal policies for b < 1, b = 1, and
b > 1 if he wishes to maximise his profit. Interpret b in terms of the elasticity of demand.
14
7.
Further economic applications of differentiation
The contents of Chapter 4 are useful in a great deal of economics. You will hopefully have seen
that there is a direct connection between the derivative and the economic concept of
dy
marginality. The derivative
can be interpreted as the marginal change in the economic
dx
variable y given some (infinitesimally small) change in x and hence is directly related to
the marginal concepts so useful in economics.
Economic analysis is full of such marginal changes : marginal propensity to consume, marginal
cost, marginal propensity to import, marginal rate of substitution, marginal product, marginal
utility and so on.
Below are further economic applications of calculus.
Example 4.15
Consider an economic variable which grows from an initial value of y 0 according to the
exponential function :
y t   y 0 e bt
The proportional rate of growth at time t of y is defined as
y  t 
y t 
Since y  t   b y 0 e bt  b y t  , the proportional rate of growth of y is
d  ln y t 
b y t 
y t 
 b.
 b , the slope of the graph of the natural log of a variable with respect to time
dt
indicates the rate of growth of the variable over time.
The constancy of the slope indicates how close the growth process is to exponential growth
at a constant rate.
Since
Example 4.16
A classic application of the optimisation technique involves a firm that has output q.
This is produced at a cost C(q) and sold with resulting revenue R(q). The firm's profit from
producing output q is given by:
 q   Rq   Cq .
A profit maximising firm seeks a (global) maximum of this function. We have
 q   R q   C q 
 
 
so that at a maximum q  , R  q   C  q  .
15
Thus at a profit maximising point marginal revenue
C q  .
 Rq   is set equal to marginal cost


To ensure that we have a maximum we need to check our second order condition:
 q   0  R q    C q   .
Example 4.17
Total differentials are widely used in macroeconomics.
Suppose, for example, that we have a (consumption) function f(Yt) that gives consumption in
period Ct , as a function of income in period t, Yt . A natural question to ask is: how much
does consumption change if income changes from Yt1 to Yt ? If the change in income is small
and the consumption function is differentiable the (approximate) answer is the total differential
of f at Yt1
i.e.
Ct  Ct 1  f Yt   f Yt 1   df Yt   f Yt 1 Yt  Yt 1  .
The expression f Yt  is the marginal propensity to consume.
READING FOR CHAPTER 4 : CHAPTER 6,
C.Osborne November 2001
BRADLEY AND PATTON