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Transcript
Intro Optimization Constraints Optimization Why this is important: In microeconomics we want to represent the ways that consumers and producers behave in a way that allows us to forecast or predict how they might respond to economic changes (for example – taxes). The basis of the model of Demand & Supply is linked to the mathematical issue of constrained optimization. It might be helpful for you to remember that consumers & producers really don’t make calculus decisions when they buy or sell. But the models we develop do work and can accurately model behavior. The models are well suited to econometric estimation. Classic Problem A farmer has 100 feet of fencing and wants to build a rectangular enclosure to maximize area. What are the optimal dimensions? Area = XY (Objective Function) 100 = 2X + 2Y (Contraint) Classical Via the process of embedding we can collapse a 2 variable decision problem into a one variable decision problem. This eliminates one degree of freedom. For example, X = 50 – Y. Area = XY = (50-Y)Y=50Y-Y2 Fence A = 50Y – Y2 A’ = 50 – 2Y FOC Y* = 25 A’’ = -2 < 0 Evaluated at Y* Thus X* = 50 – Y* = 50 – 25 = 25 So the optimal design is to utilize the fencing and make a square enclosure Fencing Continued Now let’s assume the farmer can build the enclosure along the side of a stream or existing building. So the constraint is now something like 100 = 2X + Y Or Y = 100 – 2X Or A = XY = X(100-2X) = 100X – 2X2 Fence Continued A = 100X – 2X2 A’ = 100 – 4X X* = 25 A’’ = - 4 < 0 So Y* = 100 – 2X* = 50 Now the optimal enclosed region is not square Embedding It may not always be convenient to embed the constraints into the objective function to collapse the set of choice variables. The Lagrangian method is frequently used in economics and statistics to accommodate constrained optimization problems. (I.E. RLS instead of OLS) Lagrangian Multiplier This method is algorithmic which makes it outstanding for many problems. Step by step we take our objective function (eg, f(x,y)) and rewrite any constraints in the form gi(x,y) = 0 where gi represents the ith constraint. Lagrangian We form a new function L(x,y,λ) where λ is a new variable called the Lagrangian multiplier. Note we would have as many λ’s as we would constraints. L = f + λg Now we take derivatives L(x,y,λ) We will arrive at a system of equations which we simultaneously solve, generating our FOC’s Lx = 0 Ly = 0 Lλ = 0 Example Area = XY Constraint 100 = 2X + 2Y L = XY + λ(100 – 2X – 2Y) Lx = Y - 2λ Ly = X - 2λ Lλ = 100 – 2X – 2Y 3 Equations 3 Unknowns Lx = Y - 2λ = 0 Ly = X - 2λ = 0 Lλ = 100 – 2X – 2Y = 0 Solution? From Eqns 1 and 2 we see X = Y Substitute in Eqn 3 to get X = Y = 25 What about SOC’s? This involves finding the principal minors of a matrix called the bordered Hessian. What are principal minors? Why are they important? Hessians Hessians are square matrices comprised of 2nd order partial derivatives. When evaluated at critical values we might be able to determine if a Hessian matrix has what is called sign definiteness. For example, the function x2 + y2 is always ≥ 0. x2 + y2 fx = 2x, fy = 2y, fxx = 2, fxy = 0, and fyy = 2. Ultimately we want to find the determinants of the minors of the Hessian and the sign patterning associated with them. 2 0 0 2 For our class We will deal with easy to solve utility and production functions which yield solutions having anticipated SOC’s. The probability of your having to calculate the signs of the principal minors for a problem on the final exam is P(m) with 0 ≤ P(m) ≤ 0.1 Maple Maple is a mathematical calculator that is useful in learning calculus. It can also solve systems of equations and has built-in optimizing algorithms. Although you won’t be able to use Maple on the final exam, it is a very good tool to facilitate learning. Maple Let’s take a look at some problems from our textbook and translate them into the Maple language. Maple is available for you to use in the computer lab. I think Maple (or Mathematica) is also available at Pantip Plaza!