Lagrange multipliers - Notes on Lagrange Multipliers Susan...

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Notes on Lagrange Multipliers Susan Stratton September 5, 2007 1 Preliminary Concepts/Review Section 1 is here for reference only. We will NOT be going over this material in section. 1.1 Unconstrained Optimization of One Variable To find the maximum value of the function f ( x ) = - x 2 + 10 x + 3 (1) we want to find the point where its first derivative is zero. Remember that the first derivative tells us the slope of the function. Here’s a plot of the function. 0 2 4 6 8 10 0 5 10 15 20 25 30 Unconstrained Problem with 1 Variable x f(x) 1
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Clearly, the maximum point occurs at the ’x’ in the figure, which is also where the slope is 0. Mathematically, we find the first derivative of our function and set it equal to zero. df dx = - 2 x + 10 = 0 (2) or x = 5 . (3) Note that the ’x’ in the graph does indeed occur at x = 5 . 1.2 Unconstrained Optimization of Two Variables Suppose instead we wanted to find the maximum value of f ( x, y ) = - 2 x 2 - 2 y 2 - xy - 5 x (4) The graph of this function looks like this. -10 -5 0 5 10 -10 -5 0 5 10 -600 -400 -200 0 200 Notice that at the maximum point, the graph is “flat” in both directions. This is equiv- alent to saying that the rate of change as we change either x or y is zero at the maximum. Mathematically, this means the partial derivatives of x and y are zero at the maximum. Question What’s a partial derivative ? 2
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Answer A partial derivative is just like a regular derivative only we treat any other variables as fixed constants when taking the derivative. We use a different symbol, , to indicate that we’re only changing one variable and holding everything else fixed. Frequently, we replace the symbol ∂f ∂x with the shorthand f x which refers to the partial derivative of f with respect to x . For our example, we have ∂f ∂x = f x = - 4 x - y - 5 (5) and ∂f ∂y = f y = - 4 y - x (6) To find the maximum, we set both of the above expressions equal to zero. Thus to find our maximum we solve the system of equations - 4 x - y - 5 = 0 (7) - x - 4 y = 0 (8) From (8), we see that x = - 4 y . Substituting into (7), we have 16 y - y - 5 = 15 y - 5 = 0 (9) which implies that y = 1 3 . Since we know that x = 4 y , we have x = - 4 3 . 2 Utility maximization using Lagrange multipliers In your microeconomics class, you probably learned that consumers choose where to consume by setting the ratio of marginal utility equal to the price ratio. If you took EEP 100 or ECON 101A, you’ve probably seen how to derive that rule mathematically. We’re going to go through a simple example here. 2.1 Example 1 Sarah derives utility from the consumption of apples and oranges. We’ll use x to denote her consumption of apples and y to denote her consumption of oranges. She has a utility function U ( · ) which tells us how much utility she gets from consuming apples and oranges.
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