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Intro Optimization

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).

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Intro Optimization

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  1. Intro Optimization Constraints

  2. 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.

  3. 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)

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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)

  9. 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.

  10. 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

  11. 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

  12. Example • Area = XY • Constraint 100 = 2X + 2Y • L = XY + λ(100 – 2X – 2Y) • Lx = Y - 2λ • Ly = X - 2λ • Lλ = 100 – 2X – 2Y

  13. 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

  14. 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?

  15. 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.

  16. 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.

  17. 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

  18. 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.

  19. 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!

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