Faculty of EconomicsOptimizationLecture 2Marco HaanFebruary 21, 2005Last weekOptimizing a function of more than 1 variable.Determining local minima and local maxima.First and second-order conditions.Determining global extrema with direct restrictions on variables.This weekConstrained problems.The Lagrange Method.Interpretation of the Lagrange multiplier.Second-order conditions.Existence, uniqueness, and characterization of solutions.2Suppose that we want to maximize some function f(x1,x2) subject to some constraint g(x1,x2) = 0.Example: A consumer wants to maximize utility U(x1,x2) = x1 x2subject to budget constraint2x1 + 3x2 = 10.In this case: f(x1,x2) = x1 x2 and g(x1,x2) = 10 2x1 3x2.3Suppose that, from g(x1,x2) = 0 we can write x2 = (x1).Take the total differential: dx2 = (x1 ) dx1Also: g1(x1, x2) dx1 + g2(x1, x2) dx2 = 0We want to maximize f(x1,x2) subject to g(x1,x2) = 0.Hence:We can now write the objective function as:Weve seen this in Micro 1! 4Theorem 13.1If (x1*, x2*) is a tangency solution to the constrained maximization problemthen we have that x1* and x2* satisfy5Back to the examplef(x1,x2) = x1 x2 and g(x1,x2) = 10 2x1 3x2.We needSoWithHenceThis yieldsNote: this only says that this is a local optimum.6Lagrange MethodAgain, we want toConsider the functionThe first two equalities implyLets maximize this:Hence, we get exactly the conditions we need!7Definition 13.2The Lagrange method of finding a solution (x1*, x2*) to the problem consists of deriving the following first-order conditions to find the critical point(s) of the Lagrange functionwhich are8Back to the examplef(x1,x2) = x1 x2 and g(x1,x2) = 10 2x1 3x2.Again, this only says that this is a local optimum.9The method also works for finding minima. (Definition 13.2)The Lagrange method of finding a solution (x1*, x2*) to the problem consists of deriving the following first-order conditions to find the critical point(s) of the Lagrange functionwhich are10The interpretation of * is the shadow price of the constraint.It tells you by how much your objective function will increase at the margin as the the constraint is relaxed by 1 unit.Later, we go into more details as to why this is the case.In the consumption example, we had income 10 and * = 0.204.This tells us that as income increases by 1 unit, utility increases by 0.204 units.In this example, this is not very informative, as the “amount of utility” is not a very informative number.Yet, in the case of e.g. a firm maximizing its profits, this yields information that is much more useful.11The Lagrange method of finding a solution (x1*,., xm*) to the problem consists of deriving the following first-order conditions to find the critical point(s) of the Lagrange functionwhich areIt also works with more variables and more constraints. (Definition 13.3)12Second-Order ConditionsWith regular optimization in more dimensions, we needed some conditions on the Hessian.We now need the same conditions but on the Hessian of the Lagrange function.This is the Bordered Hessian.13Theorem 13.3A stationary value of the Lagrange function yields amaximum if the determinant of the bordered Hessian is positive,minimum if the determinant of the bordered Hessian is negative.14Again back to the earlier examplef(x1,x2) = x1 x2 and g(x1,x2) = 10 2x1 3x2.Evaluate in Thus, we now know that this is a local maximum.15With more than two dimensions.(Theorem 13.4)If a Lagrange function has a stationary value, then that stationary value is a maximum if the successive principal minor of |H*| alternate in sign in the following way:It is a maximum if all the principal minors of |H*| are strictly negative.Note: Both theorems only give sufficient conditions.16Theorem 13.6The Lagrange method works (in finding a local extremum) if and only if it is possible to solve the first-order conditions for the Lagrange multipliers.17Weierstrasss Theorem:If f is a continuous function, and X is a nonempty, closed, and bounded set, then f has both a minimum and a maximum on X.But when can we be sure that a minimum and a maximum really exist!?18Weierstrasss Theorem:If f is a continuous function, and X is a nonempty, closed, and bounded set, then f has both a minimum and a maximum on X.But when can we be sure that a minimum and a maximum really exist!?f is continuous if it does not contain any holes, jumps, etc.You cannot maximize the function f(x) = 1/x on the interval -1,1.But you can maximize the function f(x) = 1/x on the interval 1,2. 19Weierstrasss Theorem:If f is a continuous function, and X is a nonempty, closed, and bounded set, then f has both a minimum and a maximum on X.But when can we be sure that a minimum and a maximum really exist!?X is nonempty if it contains at least one element. Otherwise the problem does not make sense. If there is no value, there is also no maximum value.20Weierstrasss Theorem:If f is a continuous function, and X is a nonempty, closed, and bounded set, then f has both a minimum and a maximum on X.But when can we be sure that a minimum and a maximum really exist!?X is closed if the endpoints of the interval are also included in X.0 x 1 is an open set. It is not a closed set.0 x 1 is a closed set.You cannot maximize the function f(x) = x on the interval 0 x 0.22But when can we be sure that a local extremum is also a global one!?Not always.g(x)f increasesnot a global maximumglobal maximum23To give a formal derivation, we need some more mathematics.Convex setConsider some set X.Take any two points in X.Draw a line between these points.If the entire line is within X, and this is true for any two points in the set, then the set is convex.Convex setNot a convex set24NoteA “convex set” is something entirely different than a “convex function”.There is no such thing is a “concave set”.25To give a formal derivation, we need some more mathematics.Quasi-concavityConsider some function f(x).Take some point x1.Consider the set X0 consisting of all points x0 that have f(x0) f(x1).If this set is convex, and this is true for all possible x1, then the function is quasi-concave.x1This function is quasi-concave, but not concave!26To give a formal derivation, we need some more mathematics.x1This function is not quasi-concave.Quasi-concavityConsider some function f(x).Take some point x1.Consider the set X0 consisting of all points x0 that have f(x0) f(x1).If this set is convex, and this is true for all possible x1, then the function is quasi-concave.27To give a formal derivation, we need some more mathematics.Quasi-convexityConsider some function f(x).Take some point x1.Consider the set X0 consisting of all points x0 that have f(x0) f(x1).If this set is convex, and this is true for all possible x1, then the function is quasi-convex.x1This function is quasi-convex, but not convex!28Quasi-convexityConsider some function f(x).Take some point x1.Consider the set X0 consisting of all points x0 that have f(x0) f(x1).If this set is convex, and this is true for all possible x1, then the function is quasi-convex.x1This function is not quasi-convex.To give a formal derivation, we need some more mathematics.29Important to note.A function that is concave, is also quasi-concave.A function that is convex, is also quasi-convex.In almost all of the cases we run into, well have convex and concave functions.Note also that a utility function that is strict quasi-concave if and only if it yields indifference curves that are strictly convex.30Theorem 13.7In a constrained maximization problemIf f is quasiconcave, all gs are quasiconvex,then any locally optimal solution to the problem is also globally optimal.Thus, if these conditions are satisfied, solving the Lagrange yields the global optimum!31Theorem 13.8: UniquenessIn a constrained maximization problemwhere f and all the gs are increasing, then if f is strictly quasiconcave and the gs are convex, or f is quasiconcave and the gs are strict convex,then a locally optimal solution is unique and also globally optimal.Example: the consumer problem! Utility function is increasing and strictly quasi-concave, Budget constraint is increasing and convex.The theorem says that solving the FOCs yields a unique and global optimum.32Another exampleA firm produces some output using the following production functionThis is a CES production function (constant elasticity of supply).Its general form is:33Another exampleA firm produces some output using the following production functionPrices of inputs: Question: What are the minimal costs of producing 1 unit of output? 34Of course, second order conditions also have to be checked.35ExampleA representative student spends 60 hours per week studying.She takes two subjects.Her objective: allocate time between the two subjects such that the average grade is maximized.Subjects differ with respect to their production function.Objective function:36ThusWe have that f is concave and g is convex. So this is a maximum. Spending one hour more studying leads to an increase in grade average of 1.5.37This weeks exercisespg. 615: 1, 3, 5, 7.pg. 622: 1.38