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Constrained Maximization. What if all values for the x ’s are not feasible? The values of x may all have to be positive A consumer’s choices are limited by the amount of purchasing power available

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constrained maximization
Constrained Maximization
  • What if all values for the x’s are not feasible?
    • The values of x may all have to be positive
    • A consumer’s choices are limited by the amount of purchasing power available
  • One method used to solve constrained maximization problems is the Lagrangian multiplier method
lagrangian multiplier method
Lagrangian Multiplier Method
  • Suppose that we wish to find the values of x1, x2,…, xn that maximize

y = f(x1, x2,…, xn)

subject to a constraint that permits only certain values of the x’s to be used

g(x1, x2,…, xn) = 0

lagrangian multiplier method1
Lagrangian Multiplier Method
  • The Lagrangian multiplier method starts with setting up the expression

L = f(x1, x2,…, xn ) + g(x1, x2,…, xn)

where  is an additional variable called a Lagrangian multiplier

  • When the constraint holds, L = f because g(x1, x2,…, xn) = 0
lagrangian multiplier method2
.

.

.

L/xn = fn + gn = 0

Lagrangian Multiplier Method
  • First-Order Conditions

L/x1 = f1 + g1 = 0

L/x2 = f2 + g2 = 0

L/ = g(x1, x2,…, xn) = 0

lagrangian multiplier method3
Lagrangian Multiplier Method
  • The first-order conditions can be solved for x1, x2,…, xn and 
  • The solution will have two properties:
    • The x’s will obey the constraint
    • These x’s will make the value of L (and therefore f) as large as possible
lagrangian multiplier method4
Lagrangian Multiplier Method
  • The Lagrangian multiplier () has an important economic interpretation
  • The first-order conditions imply that

f1/-g1 = f2/-g2 =…= fn/-gn = 

    • The numerators above measure the marginal benefit that one more unit of xi will have for the function f
    • The denominators reflect the added burden on the constraint of using more xi
lagrangian multiplier method5
Lagrangian Multiplier Method
  • At the optimal choices for the x’s, the ratio of the marginal benefit of increasing xi to the marginal cost of increasing xi should be the same for every x
  •  is the common cost-benefit ratio for all of the x’s
lagrangian multiplier method6
Lagrangian Multiplier Method
  • If the constraint was relaxed slightly, it would not matter which x is changed
  • The Lagrangian multiplier provides a measure of how the relaxation in the constraint will affect the value of y
  •  provides a “shadow price” to the constraint
lagrangian multiplier method7
Lagrangian Multiplier Method
  • A high value of  indicates that y could be increased substantially by relaxing the constraint
    • each x has a high cost-benefit ratio
  • A low value of  indicates that there is not much to be gained by relaxing the constraint
  • =0 implies that the constraint is not binding
duality
Duality
  • Any constrained maximization problem has associated with it a dual problem in constrained minimization that focuses attention on the constraints in the original problem
duality1
Duality
  • Individuals maximize utility subject to a budget constraint
    • Dual problem: individuals minimize the expenditure needed to achieve a given level of utility
  • Firms minimize cost of inputs to produce a given level of output
    • Dual problem: firms maximize output for a given cost of inputs purchased
constrained maximization1
Constrained Maximization
  • Suppose a farmer had a certain length of fence (P) and wished to enclose the largest possible rectangular shape
  • Let x be the length of one side
  • Let y be the length of the other side
  • Problem: choose x and y so as to maximize the area (A = x·y) subject to the constraint that the perimeter is fixed at P = 2x + 2y
constrained maximization2
Constrained Maximization
  • Setting up the Lagrangian multiplier

L = x·y + (P - 2x - 2y)

  • The first-order conditions for a maximum are

L/x = y - 2 = 0

L/y = x - 2 = 0

L/ = P - 2x - 2y = 0

constrained maximization3
Constrained Maximization
  • Since y/2 = x/2 = , x must be equal to y
    • The field should be square
    • x and y should be chosen so that the ratio of marginal benefits to marginal costs should be the same
  • Since x = y and y = 2, we can use the constraint to show that

x = y = P/4

 = P/8

constrained maximization4
Constrained Maximization
  • Interpretation of the Lagrangian multiplier:
    • If the farmer was interested in knowing how much more field could be fenced by adding an extra yard of fence,  suggests that he could find out by dividing the present perimeter (P) by 8
    • The Lagrangian multiplier provides information about the implicit value of the constraint
constrained maximization5
Constrained Maximization
  • Dual problem: choose x and y to minimize the amount of fence required to surround a field of a given size

minimize P = 2x + 2y subject to A = x·y

  • Setting up the Lagrangian:

LD = 2x + 2y + D(A - x - y)

constrained maximization6
Constrained Maximization
  • First-order conditions:

LD/x = 2 - D·y = 0

LD/y = 2 - D·x = 0

LD/D = A - x ·y = 0

  • Solving, we get

x = y = A1/2

  • The Lagrangian multiplier (D) = 2A-1/2
envelope theorem constrained maximization
Envelope Theorem & Constrained Maximization
  • Suppose that we want to maximize

y = f(x1, x2,…, xn)

subject to the constraint

g(x1, x2,…, xn; a) = 0

  • One way to solve would be to set up the Lagrangian expression and solve the first-order conditions
envelope theorem constrained maximization1
Envelope Theorem & Constrained Maximization
  • Alternatively, it can be shown that

dy*/da = L/a(x1*, x2*,…, xn*;a)

  • The change in the maximal value of y that results when a changes can be found by partially differentiating L and evaluating the partial derivative at the optimal point
constrained maximization7
Constrained Maximization
  • Suppose we want to choose x1 and x2 to maximize

y = f(x1, x2)

  • subject to the linear constraint

c - b1x1 - b2x2 = 0

  • We can set up the Lagrangian

L = f(x1, x2) - (c - b1x1 - b2x2)

constrained maximization8
Constrained Maximization
  • The first-order conditions are

f1 - b1 = 0

f2 - b2 = 0

c - b1x1 - b2x2 = 0

  • To ensure we have a maximum, we must use the “second” total differential

d 2y = f11dx12 + 2f12dx2dx1 + f22dx22

constrained maximization9
Constrained Maximization
  • Only the values of x1 and x2 that satisfy the constraint can be considered valid alternatives to the critical point
  • Thus, we must calculate the total differential of the constraint

-b1dx1 - b2dx2 = 0

dx2 = -(b1/b2)dx1

  • These are the allowable relative changes in x1 and x2
constrained maximization10
Constrained Maximization
  • Because the first-order conditions imply that f1/f2 = b1/b2, we can substitute and get

dx2 = -(f1/f2) dx1

  • Since

d 2y = f11dx12 + 2f12dx2dx1 + f22dx22

we can substitute for dx2 and get

d 2y = f11dx12 - 2f12(f1/f2)dx1 + f22(f12/f22)dx12

constrained maximization11
Constrained Maximization
  • Combining terms and rearranging

d 2y = f11 f22- 2f12f1f2 + f22f12 [dx12/ f22]

  • Therefore, for d 2y < 0, it must be true that

f11 f22- 2f12f1f2 + f22f12 < 0

  • This equation characterizes a set of functions termed quasi-concave functions
    • Any two points within the set can be joined by a line contained completely in the set
constrained maximization12
Constrained Maximization
  • Recall the fence problem: Maximize A = f(x,y) = xy subject to the constraint P - 2x - 2y = 0
  • Setting up the Lagrangian [L = x·y + (P - 2x - 2y)] yields the following first-order conditions:

L/x = y - 2 = 0

L/y = x - 2 = 0

L/ = P - 2x - 2y = 0

constrained maximization13
Constrained Maximization
  • Solving for the optimal values of x, y, and  yields

x = y = P/4 and  = P/8

  • To examine the second-order conditions, we compute

f1 = fx= y f2 = fy = x

f11 = fxx = 0 f12 = fxy = 1

f22 = fyy = 0

constrained maximization14
Constrained Maximization
  • Substituting into

f11 f22- 2f12f1f2 + f22f12

we get

0 ·x2 - 2 ·1 ·y ·x + 0 ·y2 = -2xy

  • Since x and y are both positive in this problem, the second-order conditions are satisfied
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