chap. 8 optimization (maximization or minimization) of single … · chap. 8 optimization...
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one independent variable
constraint is closed interval [a0,b0]
inner solution of maximization problem or
minimization problem
Chap. 8 Optimization
(Maximization or Minimization)
of Single Variable Functions
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First order conditions of
optimization
f ’(a)>0 ⇒ If x increases infinitesimally from
a, then f (x) goes up.
f ’(a)<0 ⇒ If x decreases infinitesimally
from a, then f (x) goes down.
f ’(a)=0 ⇒ slope of tangency line to f (x) at
x=a is zero ↓
x=x* is an inner solution of the problem
⇒ f ’(x*)=0 (first order condition)
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y=f(x)
a1 a2 a3 a4
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concave function and maximum
differentiable and concave function y=f(x)
the slope of tangency line of f (x) at x=a is f ’(a)
equation ))(()( axafafy
xaxafafxf ))(()()(
xafxfaf )()(0)(
f (x) is maximized at x=a，
the maximum is f (a)
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graph and tangency line of a
concave function
a x2x1
f(a)
f(x2)
f(x1)
f(a)+(x1-a)f ’(a)
f(a)+(x2-a)f ’(a)
f(x)
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maximum and graph of a
concave function
a
f(a)
f(x)
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convex function and minimum
differentiabel and convex function y=f(x)
the slope of tangency line f (x) at x=a is f ’(a)
equation ))(()( axafafy
xaxafafxf ))(()()(
xafxfaf )()(0)(
f (x) is minimized at x=a，
the minimum is f (a)
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a x2x1
f(a)
f(x2)
f(x1)
f(a)+(x1-a)f ’(a)
f(a)+(x2-a)f ’(a)f(x)
graph and tangency line of a
convex function
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a
f(a)
f(x)
minimum and graph of a convex
function
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second order derivative
If the function y=f ’(x) is differentiable at x=a
y=f (x) is second order differentiable
(f ’(a))’= f ’ ’(a) second order derivative
axaxdx
xfd
dx
ydaf
2
2
2
2 )()(
second order derived function(f ’(x))’= f ’ ’(x)
n order derived function
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Conditions for concave
functions and convex functions
f '(x)≦0⇒ slope of tangency line is negative
⇒y=f(x) is monotone decreasing
second order derivative f ’’(x)≦0 (for any x)
⇒ function y=f '(x) is monotone decreasing
⇒ slope of tangency line is decreasing
⇒ concave function
⇒ f (x) is minimized at a such that f '(a) =0
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Conditions for concave functions
and convex functions
2nd derivative f ''(x)≦0 (for all x)
⇒ concave function
⇒ f (x) is maximal at a such that f '(a)=0
2nd derivative f ''(x)≧0 (for all x)
⇒ convex function
⇒ f (x) is minimal at a such that f ' (a)=0
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Conditions for concave functions
and convex functions
2nd derivative f '' (x)<0 (for all x)
⇒ strictly concave function
⇒ a such that f ' (a)=0 is unique if it exists
( f (x) has maximum)
2nd derivative f '' (x)>0 (for all x)
⇒ strictly convex function
⇒ a such that f ' (a)=0 is unique if it exists ( f (x) has minimum)
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Exercises Judge concavity or convexity with the
following functions.
baxxf 2)(
cbxaxxf 2)()1(
axf 2)(
convex. then ,0 if concave; then ,0 If aa
axexf )()2(axaexf )( axeaxf 2)(
convex 0and02 axea
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Exercises Judge concavity or convexity with the
following functions.
2)(
x
axf
x
axf )()3(
3
2)(
x
axf
concave 0 convex,0 xaxa
xaxf log)()4(
x
axf )(
2)(
x
axf
convex 0 concave, 0 ,0 Since 2 aax
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Local maximum and minimum maximal in the neighborhood→local maximum
minimum in the neighborhood→local minimum
function y=f (x) is maximum in x=x0
⇒ f (x0) is maximum in open-interval (a,b)
⇒ f (x0) is local maximum
function y=f (x) is minimum in x=x0
⇒ f (x0) is minimum in open-interval (a,b)
⇒ f (x0) is local minimum
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Maximum and local maximum
(Minimum and local minimum)
maximum among the local maximums ⇒maximum
minimum among the local minimums ⇒minimum
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Graph and fluctuation of function
0)( af 0)( bf
Indep. v. x ・・・ a ・・・ b ・・・
derivative f ’(x) ＋ 0 ー 0 ＋
fluctuationlocal
max
Local
min
function f(x) f(a) f(b)
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Fluctuation of the functiony=f(x)
a b
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1st order conditions for local
maximum and minimum
f '(a)>0 ⇒ f (x) increases when x increases
from a infinitesimally
f '(a)<0 ⇒ f (x) decreases when x increases
from a infinitesimally
f '(a)=0 ⇒zero slope of tangent of f (x) in x=a
maximum (minimum) in x=x * ⇒ f '(x*)=0
1st order conditions for maximum (minimum)
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Conditions to be maximum and
minimum
2nd order derivative f ’ ’(a)≦0
⇒ f (x) is maximum in a such that f ’ (a) =0
2nd order derivative f ’ ’(b)≧0
⇒ f (x) is minimum in b such that f ’ (b) =0
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Marginal concepts in economics
The marginal productivity
labor imput production
production function
Increment of the production by adding one
unit of labor
)(fq
q
)()()()( ffffq
product marginal)(
fq
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marginal concepts in economics
Diminishing marginal productivity
function concave strictly()( ） xf
decreasing is)(ty productivi marginal f
0)( f
total differentiation dfdf )()(
total differential
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Exercises Show the marginal productivity of f(x). Judge
whether or not the marginal productivity
diminishes. Moreover, compute a total differential.
2
1)( f
)()1( f
decreasing
04
1)(
3
f
ddf2
1)(
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Exercises
)0()1log()()2( aaf
1)(
a
f
decreasing
0)1(
)(2
af
da
df1
)(
Show the marginal productivity of f(x). Judge
whether or not the marginal productivity
diminishes. Moreover, compute a total differential.
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Exercises
aef 1)()3(
alaef )(
decreasing
0)( 2 aeaf
daedf a)(
Show the marginal productivity of f(x). Judge
whether or not the marginal productivity
diminishes. Moreover, compute a total differential.
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Marginal concepts in economics
marginal cost
cost function
increment of the cost by producing one
additional unit of the product
)(qCC
q
CqCMC
)(cost marginal
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Marginal concepts in economics
The law of diminishing marginal productivity
f’(l) is decreasing → f’’(l)<0 (strictly concave)
The law of increasing marginal cost
C’(q) is increasing → C’’(q)>0 (strictly convex)
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The solution of the profit
maximization problem total revenue R(q) total cost C(q)
profit function π(q)=R(q)C(q)
MCMRq 0)(
MCMRqCqRq )()()(
CMRMqCqRq )()()(
CMRMq 0)(maxprofit
always holds if decreasing marginal revenue,
increasing marginal cost
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Example of profit maximization
Profit maximization problem of a firm
which inputs labor and produces some
good
Firm behaves as the price taker
labor input product (variable)
wage rate product price (fixed)
production function
q
w p
)(fq
profit → maximizationwpq
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Profit maximization problem
steps of solution
step 1: find necessary minimum of labor
step 2: find necessary minimum total cost
step 3: Find production maximizing profit
Assumptions of the production function
q=f(l) is twice differentiable and f(0)=0
f’(l)>0 strictly monotone increasing
f’’(l)<0 strictly concave
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Step 1 : efficient production
Find a minimum labor input function
)()(1 qhqf
function h(q) is a strictly convex h''(q)>0
)0()(:Example 2 qqqhq
0202
1,0
2
13
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Step 2 : Cost minimization
Find the total cost function
)()( qhwqC
h(q) is strictly convex C''(q)=wh''(q)>0
)0()(:Example 2 qwqqCq
02)(2)()( wqCwqqCqMC
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Step 3 : Profit maximization
Find profit function and use 1st order
conditions)()( qhwpqqCpq
Profit function π is strictly concave
π''= C''(q).<0
)0(:Example 2 qwqpqq
022
02 ww
pqwqp
maxprofit 0)( qhwp
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The derivation of the cost
function
total variable cost
function)()( qwhqTVC
)()( 1 qfqh
TFCqwh
TFCqTVCqTCC
)(
)()(
necessary minimum
labor inputs
total cost
function
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A variety of average costs
Average total cost ATC, AC
Average variable cost AVC
Average fixed cost AFC
q
qTVCqAVC
)()(
)()()( qAFCqAVCqATC
q
qTCqATC
)()(
q
qTFCqAFC
)()(
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Relation between average variable
cost and marginal cost
)()(0)( qAVCqMCqCAV
)()(0)( qAVCqMCqCAV
)()(0)( qAVCqMCqCAV
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)()(0)( qATCqMCqCAT
)()(0)( qATCqMCqCAT
)()(0)( qATCqMCqCAT
Relation between average variable
cost and marginal cost
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C
c0
0qq2q1q0
A
B
D
E
C(q)
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0 q0 q1 q2q
AFC
AVC
ATCMC
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Maximization of short-run profit
max Π(q)=pq-C(q)
By the increase of the additional one unit of
production, the revenue increases in price p.
On the other hand, the cost increases only
by the marginal cost MC to enhance
production of the unit. The profit maximum
attains where both make a balance.
p=MC (price=marginal cost)
marginal cost is increasing MC'>0
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C,R
π
0
c0
-c0
q**
q* q
π (q)
R=pqC(q)
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Short-run supply curve price=marginal cost, increasing marginal cost
→ optimal provision → supply curve
MC increasing → upward-sloping part of MC
curve
Supply is q3 for price p3. Minimal profit in q3'.
Supply is q4 for price p4. Deficit is fewer than FC.
Candidacy for p5 is q5. It isn't supplied because
deficit is larger than FC.
The supply curve is a part of MC curve above
the shutdown point
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0 q0 q1 q2q
AVC
ATCMC
q3' q5 q3
q4
p
p3
p2
p4
p1
p5
A
BA : break-even point
B : shutdown point
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The shape with average total
cost function
2)()(
q
TFCqCAVqCAT
32)()(
q
TFCqCAVqCAT
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Relation of the arithmetic mean
and the geometrical average
for any positive numbers x, y
xyyx
2equality holds only for x=y
yxyxxyyx 2)()(2)( 22
0)( 2 yx
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The minimum of the average
total cost
)()()( qAFCqAVCqATC
equality holds only when
AVC(q)=AFC(q)
)()(2 qAFCqAVC
ATC(q) is minimal when AVC(q)=AFC(q)