Lecture four

1
Lecture four

• Unconstrained optimization of multivariate
  functions

2
Recapitulation
3
Recapitulation cntnd.

• Example one
• Utility function
• Marginal utility
• MRS
• Diminishing marginal utility

4
Recapitulation cntnd.

• Example two
• Production function
• Marginal productivities
• MRTS
• Diminishing marginal productivity

5
Recapitulation cntnd.

• Example three

Q
Qf(K0,L)
L
MPL,APL
MPL
APL
L
6
Optimization of multivariate functions

• Profit function
• The first order conditions are
• These can be solved simultaneously to give us

7
Optimization of multivariate functions cntnd.

• Example

8
Conditions for optimum

• Given zf(x,y) the total differential of z is
  given by

9
Conditions for optimum

• Analogically to the univariate case, the first
  order necessary condition for optimum is
• dzfxdxfydy0
• It amounts to fx0, fy0, for arbitrary
  values of dx and dy, not both zero. Solving the
  resulting system of simultaneous equations (fx0,
  fy0) gives the stationary point (x0, y0).

10
Conditions for optimum cntnd.

• And once again the second-order sufficient
  condition for optimality amounts to d2zgt0 for a
  minimum and d2zlt0 for a maximum. In the
  multivariate case

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Conditions for optimum cntnd.

• To derive the sufficient conditions for maximum,
  let us make sure that wherever we go d2f stays
  negative.
• (i) For dy0 d2ffxxdx2lt0 iff fxxlt0
• (ii) For dx0 d2ffyydy2lt0 iff fyylt0
• (iii) dx?0, dy ?0

12
Conditions for optimum cntnd.
13
Conditions for optimum cntnd.

• In other words

14
The logic of the optimization conditions
C
y
If the slope of the line segment AC is smaller
than the slope of the tangent AB, the function is
concave
f(x2)
B
f(x1)
A
f
x
x1
x2
15
The logic of the optimization conditions cntnd.
z
f(?(x)(1- ?)y
?f(x)(1- ?)f(y)
f(y)
f(x)
y
(x1,x2)
(y1,y2)
x
When ?f(x)(1- ?)f(y)? f(?(x)(1- ?)y we have a
maximum
?(x)(1- ?)y
16
The logic of the optimization conditions cntnd.

• The opposite is true in the case of convex
  univariate and multivariate functions

y
C
x
B
A
y
x
z
17
The logic of the optimization conditions cntnd.
z
y
z
?z/ ?xlt0
?z/ ?xgt0
x
y0
?z/ ?x
The graph of ?z/ ?x is negatively sloped, hence
?2z/ ?x2lt0
x0
x
x
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The logic of the optimization conditions cntnd.
The logic of the optimization conditions cntnd.
z
y
z
?z/ ?ylt0
?z/ ?ygt0
y
y0
?z/ ?y
The graph of ?z/ ?y is negatively sloped, hence
?2z/ ?y2lt0
x0
x
x
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The logic of optimization conditions contnd.

• In sum, at the maximum of any function zf(x,y)
• Analogically, at the minumum of any function
  zf(x,y)

20
The logic of the optimization
conditions cntnd.

• An additional condition

z
y
S
yo
BB
D
A
F
E
B
C
xo
x
21
Examples

• Find any local minima or maxima of the functions

22
Economic applications

• Perfect competition and profit maximization A
  firm produces two goods and sells them in
  perfectly competitive markets at prices P124 and
  P236. Its total cost function is TC2Q122 Q22.
  (i) Write out the firms profit function. (ii)
  Find the level of output that satisfies the
  first-order conditions for profit maximization.
  (ii) check the second-order sufficient
  conditions.
• Monopoly and profit maximization A monopolist
  produces two goods and faces the following demand
  functions Q110-P1 Q220-P2. The total cost
  function is TCQ12Q22 . (i) Write up the profit
  maximization function, (ii) Find the output
  levels satisfying the first-order conditions,
  (iii) Check the second-order conditions.

23
Economic applications cntnd.

• The demand functions for a firms domestic and
  foreign markets are p150-5q1, p230-4q2 and the
  total cost function is TC1010q, where qq1q2.
  Determine the prices needed to maximize profits
  (a) with price discrimination, (b) without price
  discrimination.
• A monopolist produces the same product at two
  factories. The cost functions of each factory are
  as follows. TC18q1, TC2q22. The demand function
  is P100-2q, where qq1q2. Find the values of q1
  and q2 that maximize profits.

24
Economic applications

• Example 1 The demand functions for a firms
  domestic and foreign markets are p150-5q1,
  p230-4q2 and the total cost function is
  TC1010q, where qq1q2. Determine the prices
  needed to maximize profits (a) with price
  discrimination, (b) without price discrimination.

25
Economic applications cntnd.

• Case 1 No price discrimination

26
Economic applications cntnd.

• Case two price discrimination

27
Economic applications cntnd.

• Example 2 A monopolist produces the same product
  at two factories. The cost functions of each
  factory are as follows. TC18q1, TC2q22. The
  demand function is P100-2q, where qq1q2. Find
  the values of q1 and q2 that maximize profits.

28
Economic applications cntnd.

• Solution