Mathematical Economics  Part I  Optimization. Filomena Garcia. Fall Optimization. Filomena Garcia. Optimization. Existence of Solutions


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1 of Mathematical Economics  Part I  Fall 2009
2 of 1 in R n Problems in Problems: Some of in 8 Continuity: The Maximum Theorem 9 Supermodularity and Monotonicity
3 of An optimization problem in R n is one where the values of a given function f : R n R are to be maximized or minimized over a given set D R n. The function f is called objective function and the set D is called the constraint set. We denote the optimization problem as: max {f (x) x D} A solution to the problem is a point x D such that f (x) f (y) for all y D. We call f (D) the set of attainable values of f in D.
4 of It is worth noting the following: 1 A solution to an optimization problem may not exist Example: Let D = R + and f (x) = x, then f (D) = R + and supf (D) = +, so the problem max {f (x) x D} has no solution. 2 There may be multiple solutions to the optimization problem Example: Let D = [ 1, 1] and f (x) = x 2, then the maximization problem max {f (x) x D} has two solutions: x = 1 and x = 1.
5 of We will be concerned about the set of solutions to the optimization problem, knowing that this set could be empty. argmax [f (x) x D] = {x D f (x) f (y), y D } Two important results to bear in mind: 1 x is a maximum of f on D if and only if it is a minimum of f on D. 2 Let ϕ : R R be a strictly increasing function. Then x is a maximum of f on D if and only if x is also a maximum of the composition ϕ f.
6 Problems in of problems are often presented in parametric form, i.e. both the objective function and/or the feasible set depend on some parameter θ from a set of feasible parameter values Θ. In this case, we denote the optimization problem as: max {f (x, θ) x D(θ)} In general, this problem has a solution which also depends on θ, i.e. x(θ) = argmax {f (x, θ) x D(θ)}
7 Problems: Utility Maximization of 1 Utility Maximization In consumer theory the agent maximizes his utility from consuming x i units of commodity i. The utility is given by u(x 1,..., x n ). The constraint set is the set of feasible bundles given that the agent has an income I and the commodities are priced p = (p 1,...p n ), i.e. B(p, I ) = { x R n + p x I } Notice that this is a parametrized optimization problem, the parameters being (p, I ) Other parameters may exist within the utility function, namely Cobb Douglas weights
8 Problems: Expenditure Minimization of 2 Expenditure Minimization It is the dual problem to the utility maximization. Consists on minimizing the expenditure constrained on attaining a certain level of utility ū. The problem is given by: min {p x u(x) ū}
9 Problems: Profit Maximization of 3 Profit Maximization It is the problem of a firm which produces a single output using n inputs through the production relationship y = g(x 1,..., x n ). Given the production of y units of good, the firm may charge the price p(y) 2. w denotes the vector of input prices. The firm chooses the input mix which maximizes her profits, i.e., max { p(g(x))g(x) w x x R n } + 2 p(y) is the inverse demand function
10 Problems: Cost Minimization of 4 Cost Minimization It is the dual problem to the profit maximization. Consists on minimizing the cost of producing at least ȳ units of output. The problem is given by: min {w x g(x) ȳ}
11 Problems: ConsumptionLeisure Choice of 5 ConsumptionLeisure Choice It is a model of the laborsupply decision process of households in which the income becomes also an object of choice. Let H be the amount of hours that the agent has available. The wage rate of labor time is given by w. If working for L units of time, the agent earns wl. The agent obtains utility from consumption of x and from leisure time, l. Then, the agent s problem is: max {u(x, l) p x wl, L + l H}
12 of of of are: 1 Identify the set of conditions on f and D under which the existence of solutios to optimization problems is guaranteed 2 Obtain a characterization of the set of optimal points, namely: The identification of necessary conditions for an optimum, i.e. conditions that every solution must verify. The identification of sufficient conditions for an optimum, i.e. conditions such that any point that meets them is a solution.
13 of of The identification of conditions ensuring the uniqueness of the solution. The identification of a general theory of parametric variation in a parametrized family of optimization problems. For example: The identification of conditions under which the solution set varies continuously with the parameter θ. In problems where the parameters and actions have a natural ordering, the identification of conditions in which parametric monotonicity is verified, i.e., increasing the value of the parameter, increases the value of the action.
14 How we will proceed of Section 2: Studies the question of existence of solutions. The main result is the Weierstrass Theorem, which provides a general set of conditions under which optimization problems are guaranteed to have both maxima and minima. From Section 3 to Section 5 we will examine the necessary conditions for optima using differentiability assumptions on the undelying problem. We focus on local optima, given that differentiability is only a local property. Section 3 focuses on the situation in which the optimum occurs in the interior the feasible set D. The main result are the necessary conditions that must be met by the objective function at the optimum. Also sufficient conditions that identify optima are provided.
15 How we will proceed of In Sections 4 and 5 some or all of the constraints determine the optima. : Section 4 focuses on equality constraints and the main result is the Theorem of Lagrange (both necessary and sufficient conditions for optima are presented. Section 5 focuses on inequality constraints and the main result is the Theorem of Kuhn and Tucker which describes necessary conditions for optima. Sections 6 and 7 turn to the study of sufficient conditions, i.e., conditions that, when met, will identify points as being global optima. These are mainly related to the convexity and quasiconvexity of the feasible sets and objective functions.
16 How we will proceed of Section 8 and 9 address respectivey the issue of parametric continuity and monotonicity of solutions. The main result of Section 8 is the Maximum Theorem, which states that the continuity assumptions of the primitives of the optimization problem are inherited, although not entirely, by the solutions. Section 9 relies on the concept of supermodularity of the primitives to conclude about the parametric monotonicity of solutions. The main result is the Theorem of Tarski.
17 of Main Result: The Weierstrass Theorem Let D R n be a compact and let f : D R be a continuous function on D. Then f attains a max and a min on D, i.e., z 1, z 2 D: x D f (z 1 ) f (x) f (z 2 )
18 of Some Definitions: Compact set: closed and bounded bounded set: is countained within a finite ball. closed: contains its frontier. (valid even if the frontier is empty like in R) : closed interval in R. Continuous function: for any point a in the domain, we have that: lim x a f (x) = f (a)
19 of Notice: The Weierstrass Theorem gives us sufficient conditions for maxima or minima to exist, however, we don t know what happens when some or even all of the conditions of the Theorem are violated.
20 of Failure of the Weierstrass Theorem: 1 Let D = R and f (x) = x 3. f is continuous but D is not compact (it is not bounded). Since f (D) = R, f does not attain a maximum or a minimum on D. 2 Let D = (0, 1) and f (x) = x. f is continuous but D is not compact (it is not closed). The set f (D) = (0, 1), so f does not attain a maximum or a minimum on D. 3 Let D = [ 1, 1] and let f be given by f (x) = { 0, x = 1, x = 1 1, 1 < x < 1 D is compact, but f fails to be continuous. f (D) is the open interval ( 1, 1)
21 of Intuition of the Proof of the Weierstrass Theorem: 1 First we show that under the stated conditions f (D) is a compact set, hence, closed and bounded. 2 Second, it is shown that if A is a compact set in R, then max A and min A are always well defined, never beeing the case that A fails to contain the supremum and the infimum. Hence max f (D) and min f (D) must exist on D.
22 The Weierstrass Theorem in Applications of Consider the applications mentioned in section 1. The Utility Maximization Problem: Maximize u(x) subject to x B(p, I ) = { x R n + p x I } We assume u(x) to be continuous. By the Weierstrass Theorem, if B is compact, we must have a solution. p >> 0
23 The Weierstrass Theorem in Applications of The Cost Minimization Problem: Minimize wx over x F (y) = { x R n + g(x) y } The objective function is continuous. But the feasible action set is not compact because it is not bounded. The Weierstrass Theorem does not guarantee or preclude a solution. What can we do to guarantee a solution? Bound the set.
24 of optima The constraints do not affect the optimal point; in other words, the optimum is interior. Local Maximum: A point x is a local maximum of f on if there is r > 0 such that f (x) f (y) for all y B(x, r). Global Maximum: A point x is a global maximum of f if for all r > 0, and all y D, f (x) f (y) for all y B (x, r).
25 First Order Conditions of Necessary Conditions If x is a local maximum (or minimum) of the function f on D and f is differentiable, then Df (x ) = 0 Notice, the reverse is not necessarily true. Example: x = 0 for f (x) = x 3 Any point that satisfies the first order conditions (FOC) is a critical point of f on D.
26 Second Order Conditions of Sufficient Conditions Suppose f is a C 2 function on D, and x is a point in the interior of D. Then: If Df (x ) = 0 and D 2 f (x ) is negative definite at x, then x is a local strict local maximum of f on D. If Df (x ) = 0 and D 2 f (x ) is positive definite at x, then x is a local strict local minimum of f on D. Notice: This conditions only identify local optima (not global).
27 Second Order Conditions of Example: Let f (x) = 2x 3 3x 2 and D = R. The first order conditions are: f (x) = 0 and identify as critical points x = 0 and x = 1. At these points, the second order conditions are: f (0) = 6 < 0 and f (1) = 6 > 0. Hence, 0 is a local maximum and 1 is a local minimum. However, these are not global max or min. In fact, this function has no global max nor min (why?). The existence of a global maximum (resp. minimum) in D is guaranteed only if the function is concave (resp. convex) in D
28 Constrained Problems  of 1 It is not often that optimization problems have unconstrained solutions. 2 Typically some, or all of the constraints will matter. 3 In this chapter we will look into the constraints of the form: g(x) = 0 where g : R n R k The problem: Maximize f (x) s.t. g(x)=0
29 Lagrange Theorem  2 variables, one restriction We can solve graphically of The solution is: f x 1 f x 2 = g x 1 g g(x 1, x 2 ) = 0 (1) x 2
30 Lagrange Theorem  2 variables, one restriction of We can solve by substitution Since g(x 1, x 2 ) = 0 in the solution, and g x 2 implicit funtion (what is this?) x 2 (x 1 ). So, the problem becomes: max x1 f (x 1, x 2 (x 1 )) 0, we have an
31 Lagrange Theorem  2 variables, one restriction of The FOC is: f + f x 2 = 0 x 1 x 2 x 1 By the IMPLICIT FUNCTION THEOREM: The solution is: f x 1 f x 2 = x 2 = x 1 g x 1 g x 2 g x 1 g g(x 1, x 2 ) = 0 (2) x 2
32 Lagrange Theorem  2 variables, one restriction of We can rewrite the FOC of the previous problem as: or: where f f x 1 x 2 g x 1 g x 2 = 0 f + λ g = 0 x 1 x 1 λ = f x 2 g x 2 which is the FOC of the Lagrangean.
33 Lagrange Theorem of Lagrange Theorem Let f, g : R 2 R be C 1 functions. Suppose that x = (x 1, x 2 ) is a local maximizer or minimizer of f subject to g(x 1 ; x 2 ) = 0. Suppose also that Dg(x ) 0. Then, there exists a scalar λ R such that (x 1, x 2, λ ) is a critical point of the lagrangean function: L(x 1 ; x 2 ; λ) = f (x 1, x 2 ) + λg(x 1, x 2 ) In other words, at (x 1, x 2, λ ): L x 1 = 0, L x 2 = 0 and L λ = 0
34 Lagrange Theorem of The constraint qualification We must assume Dg(x ) 0. (If there are many variables and restrictions, this amounts to say that the Jacobian matrix of g is invertible or that the rank is equal to the number of constraints). This condition allows us to identify g as an implicit function. The inverse of Dg is used in computing λ.
35 Lagrange Theorem  Necessary conditions of Notice: The Lagrange Theorem provides necessary but not sufficient conditions for local optima. It is not said that if we find x and λ such that g(x) = 0 and the FOC of the Lagrangean are equal to zero, that x, λ are indeed an optimum, even if Dg(x) 0.
36 Lagrange Theorem  Necessary conditions of Then why can we use the Lagrange method so often? Proposition Suppose that the following two conditions hold: 1 A global optimum x exists in the given problem. 2 The constraint qualification is met at x. Then, there exists a λ such that (x, λ ) is a critical point of L. So, under the conditions of Proposition 27, the Lagrangean method does identify the optimum.
37 Lagrange Theorem  Sufficient Conditions of Sufficient Conditions Suppose that there is x and λ which satisfy the constraint qualification and solve the FOC of the Lagrangean. Then: If the Bordered Hessean has a positive determinant, we have a local maximum If the Bordered Hessean has a negative determinant, we have a local minimum
38 of An Example: Max x x 2 y 2 s.t. x 2 + y 2 = 1 First we verify that the conditions of Proposition work: 1 The function f is continuous and the constraint set is compact. 2 The Constraint qualification, namely Dg(x, y) = ( 2x, 2y) 0, is met. Then, we can rely on the Theorem of Lagrange to pinpoint the optima: L(x; y; λ) = f (x, y) + λg(x, y) = x 2 y 2 + λ(1 x 2 y 2 )
39 of An Example (cont.): The first order condtitions are: L x L y = 0 2x λ2x = 0 = 0 2y λ2y = 0 L λ = 0 x 2 + y 2 = 1 The critical points are: (1,0,1);(1,0,1);(0,1,1);(0,1,1). The first two are global maximizers, and the second are global minimizers.
40 constraints and Kuhn Tucker of Most economic problems include nonnegativity restrictions and inequalities in the optimization. For instance, it is natural to impose that consumption be non negative, or work hours, or capital. In this chapter we will study problems of the form: max x s.t.g(x) = 0andh(x) 0 Our tool will be the KuhnTucker Theorem.
41 KuhnTucker Theorem of KuhnTucker Theorem  Maximization Let f : R n R and h j : R n R be C 1 functions, i = 1,..., l. Suppose x is a local maximum of f on D = {x R n h i (x) i = 1,..., l} Let E denote the set of effective (binding) constraints at x, and let h E = (h i ) i E. Suppose that the rank of Dh E = E. Then, there exists a vector λ = (λ 1, λ 2..., λ l ) such that the following constraints are met: 1 λ i 0 and λ i h i(x ) = 0, for i = 1,..., l. 2 Df (x ) + l i=1 λ i Dh i(x ) = 0
42 KuhnTucker Theorem of KuhnTucker Theorem  Minimization Let f : R n R and h j : R n R be C 1 functions, i = 1,..., l. Suppose x is a local minimum of f on D. Let E denote the set of effective (binding) constraints at x, and let h E = (h i ) i E. Suppose that the rank of Dh E = E. Then, there exists a vector λ = (λ 1, λ 2..., λ zl) such that the following constraints are met: 1 λ i 0 and λ i h i(x ) = 0, for i = 1,..., l. 2 Df (x ) l i=1 λ i Dh i(x ) = 0
43 KuhnTucker Theorem  Some Observations of 1 A pair x, λ meets the FOC condition for a maximum in an inequality constrained maximization problem if it satisfies h(x ) 0 and conditions 1 and 2 of the theorem. 2 Condition 1 is called the Complementary slackness condition. 3 KT theorem provides only necessary conditions for local optima which meet the constraint qualification. 4 Df (x ) l i=1 λ i Dh i(x ) = 0
44 KuhnTucker Theorem  Interpreting the conditions of Complementary Slackness condition The Lagrange multipliers give us the effect in the maximum of relaxing the restriction. Hence we can have one of three situations: 1 the restriction is verified with strict inequality, and in that case, relaxing the restricting will not change the maximum value attained: λ = 0 and h > 0. 2 the restriction is verified with equality, and in that case, relaxing the restricting will at least increase the value attained: λ > 0 and h = 0. 3 the restriction is verified with equality, and in that case, relaxing the restricting will maintain the value attained: λ = 0 and h = 0.
45 KuhnTucker Theorem  Constraint Qualification of Constraint Qualification r(dh E (x )) = E An example where the constraint qualification fails: Let: f (x, y) = (x 2 + y 2 ) and h(x, y) = (x 1) 3 y 2. Consider the problem of maximizing f subject to h 0. By inspection we see that f attains its max when (x 2 + y 2 ) attains its min. We also must have x 1 and y 0. So the maximum is obtained at (x, y ) = (1, 0), a point where the constraint is binding. Dh(x, y ) = (3(x 1) 2, 2y ) = (0, 0). The KT theorem fails. In fact: Df (x, y ) + λ(x, y ) = (2, 0).
46 Why the procedure usually works of In general we can rely on the KT theorem due to the following proposition: Proposition Suppose the following conditions hold: 1 A global maximum x exists to the given inequalityconstrained problem. 2 The constraint qualification is met at x. Then, there exists λ such that (x, λ ) is a critical point of L.
47 An Example of Let g(x, y) = 1 x 2 y 2. Consider the problem of maximizing f (x, y) = x 2 y over the set D = (x, y) g(x, y) 0. 1 The conditions of proposition 2 are met: by weierstrass and the fact that if the constraint is met, x and y cannot be simmultaneously zero, and hence r(dg) = 1. 2 We can use the KT theorem. L x L y L(x, y, λ) = x 2 y + λ(1 x 2 y 2 ) = 0 2x 2λx = 0 = 0 1 2λy = 0 λ 0,1 x 2 y 2 0, λ(1 x 2 y 2 ) = 0
48 An Example (cont.) of From (1) either x = 0 or λ = 1. If λ = 1, (2) implies that y = 1/2 and x 2 + y 2 = 1 x = ± 3 2. If x = 0, we may have λ = 0 or λ > 0. Take the first case: λ > 0, then from the CSC we must have y = ±1. y = 1 is inconsistent from (2). so y = 1 and λ = 1 2. λ = 0 is inconsistent with (2). So, the critical points are: (x, y, λ) = ( ± 3 2, 1 2, 1) and (x, y, λ) = (0, 1, 1 2 ) the value of the function is respectively 5/4 and 1, hence we obtain the maxima. (why not the minimum)?
49 Interpreting the Lagrange Multipliers of The Lagrange multipliers of the Lagrange Theorem and of the KuhnTucker Theorem can be interpreted as the sensitivity of the value of the function in the optimum to relaxing the restriction. Case 1: restrictions Suppose that we have the following optimization problem: max x f (x)s.t.g(x) + c = 0 We would solve the problem by writing the FOC of the Lagrangean w.r.t. x and λ, namely: L x = f (x) x + λ g(x) x ; g(x) + c = 0 The optimum is x (c) and the value of the function in the optimum is f (x (c)).
50 of We can compute the effect of c on the optimum of the function. Also: g(x (c)) x f c = f (x (c)) x f (x (c)) x x (c) c = λ g(x (c)) x = 1 (because the constraint must be observed in the optimum), hence: f c = λ
51 Interpreting the Lagrange Multipliers of Case 2: restrictions Suppose that we have the following optimization problem: max x f (x)s.t.h(x) + c 0 We would solve the problem by writing the FOC of the Lagrangean w.r.t. x and λ, namely: L x = f (x) x + λ g(x) x λ 0, h(x) + c 0, λ(h(x) + c) = 0 The optimum is x (c) and the value of the function in the optimum is f (x (c)).
52 of We can compute the effect of c on the optimum of the function. f c = f (x (c)) x f (x (c)) x x (c) c = λ h(x (c)) x If the restriction is not binding, λ = 0 and, hence, f (x (c)) x = 0 If the restriction is binding, h(x) + c = 0 h(x (c)) x So: f c = λ. x (c) c = 1
53 ity and of The notion of convexity occupies a central position in the study of optimization theory. It encompasses not only the idea of convex sets, but also of concave and convex functions. The attractiveness of convexity for optimization theory arises from the fact that when an optimization problem meets suitable convexity conditions, the same firstorder conditions that we have shown in previous to be necessary for local optima, also become sufficient for global optima. Moreover, if the convexity conditions are tightened to what are called strict convexity conditions, we get the additional bonus of uniqueness of the solution.
54 ity Defined of We say that an optimization problem is convex if the objective function is concave (max) or convex (min) and the feasible set is convex. set A set D is said to be convex if given x 1, x 2 D we have λx 1 + (1 λ)x 2 D. concave function A function defined on a convex set D is concave if: given x 1, x 2 D f (λx 1 + (1 λ)x 2 ) λf (x 1 ) + (1 λ)f (x 2 ) convex function A function defined on a convex set D is concave if: given x 1, x 2 D f (λx 1 + (1 λ)x 2 ) λf (x 1 ) + (1 λ)f (x 2 )
55 ity Defined (cont.) of Strictly concave function A function defined on a convex set D is concave if: given x 1, x 2 D f (λx 1 + (1 λ)x 2 ) > λf (x 1 ) + (1 λ)f (x 2 ) Strictly convex function A function defined on a convex set D is concave if: given x 1, x 2 D f (λx 1 + (1 λ)x 2 ) < λf (x 1 ) + (1 λ)f (x 2 )
56 Implications of convexity of Every concave or convex function must also be continuous on the interior of its domain. Every concave or convex function must possess minimal differentiability properties. The concavity or convexity of an everywhere differentiable function f can be completely characterized in terms of the behavior of its second derivative D 2 f
57 ity and of Some General Observations Theorem 1 Suppose D R n is convex and f : D R is concave. Then 1 Any local maximum of f is a global maximum of f. 2 The set argmax {f (x) x D} of maximizers of f on D is either empty or convex.
58 ity and of Some General Observations Theorem 2 Suppose D R n is convex and f : D R is strictly concave. Then 1 Any local maximum of f is a global maximum of f. 2 The set argmax {f (x) x D} of maximizers of f on D is either empty or or contains a single point (uniqueness).
59 ity and of ity and Theorem 3 Suppose D R n is convex and f : D R is concave and differentiable function on D. Then, x is an unconstrained maximum of f on D if and only if Df (x) = 0.
60 ity and of ity and the Theorem of KuhnTucker Theorem 4 Suppose D R n is open and convex and f : D R is concave and differentiable function on D. For i = 1, 2...l, let h i : D R also concave. Suppose that there is some x D such that h i ( x) > 0, i = 1,...l. Then, x maximizes f over the constraint set, if and only if there is λ Rk such that the Kuhn Tucker first order conditions hold.
61 ity and of The Slater Condition The condition that there is some x D such that h i ( x) > 0 is called the Slater Condition 1 The SC is used only in the proof that the KT conditions are necessary at an optimum. It is not related to sufficiency. 2 The rank condition established in the KT theorem can be replaced by the SC and concavity of the constraints.
62 in of ity carries important implications for optimization. However it is a quite restrictive assumption. For instance, such a commonly used utility function, such as the Cobb Douglas function is not concave unless n i=1 α i 1. u(x 1,..., x n = x α 1 αn 1...xn. In this chapter we examine optimization under a weakening of the condition of convexity, which is called quasiconvexity.
63 and Quasiconcavity defined of quasiconcave function A function f defined on a convex set D is quasiconcave if and only if: for all x 1, x 2 D and for all λ (0, 1) it is the case that f (λx 1 + (1 λ)x 2 ) minf (x 1 ), f (x 2 ) quasiconvex function A function defined on a convex set D is concave if: given x 1, x 2 D f (λx 1 + (1 λ)x 2 ) maxf (x 1 ), f (x 2 )
64 and Quasiconcavity defined of strict quasiconcave function A function f defined on a convex set D is quasiconcave if and only if: for all x 1, x 2 D and for all λ (0, 1) it is the case that f (λx 1 + (1 λ)x 2 ) > minf (x 1 ), f (x 2 ) strict quasiconvex function A function defined on a convex set D is concave if: given x 1, x 2 D f (λx 1 + (1 λ)x 2 ) < maxf (x 1 ), f (x 2 )
65 as a generalization of convexity of If a function f is concave on a convex set, it is also quasiconcave on the convex set. Same for concavity. The reverse does not hold. Let f : R R.If f is a nondecreasing function on R, then, f is both quasi concave and quasiconvex.
66 Implications of quasiconvexity of Quasiconcave and quasiconvex functions are not necessarily continuous on the interior of their domains. Quasiconcave and quasiconvex functions may have local optima that are not global optima. First order conditions are not necessary to identify every local optima under quasiconvexity.
67 and of Theorem Suppose that f : D R and h i are strictly quasiconcave and D is convex. Suppose that there are x and λ such that the KT conditions are satisfied. Then, x maximizes f over D provided at least one of the following conditions holds: 1 Df (x ) 0 2 f is concave
68 and of Theorem Suppose that f : D R is strictly quasiconcave and D is convex. Then, any local maximum of f on D is also a global maximum of f on D. Moreover, the set or maximizers of f is either empty or a singleton. Same holds for a minimum and strictly quasiconvex functions.
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