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Unformatted text preview: Mathematical Preliminaries Econ 104B The Mathematics of Optimization • Economic theories assume that an economic agent is seeking to find the optimal value of some function – consumers seek to maximize utility – firms seek to maximize profit • This section introduces the mathematics common to these problems Functions with One Variable • Simple example: Manager of a firm wants to maximize profits ) ( q f = π π = f(q) π Quantity π * q* Maximum profits of π * occur at q * Functions with One Variable • Vary q to see where maximum profit occurs – an increase from q 1 to q 2 leads to a rise in π π = f(q) π Quantity π * q* π 1 q 1 π 2 q 2 ∆ π ∆ q Functions with One Variable • If output is increased beyond q *, profit will decline – an increase from q * to q 3 leads to a drop in π π = f(q) π Quantity π * q* < ∆ π ∆ q π 3 q 3 Derivatives • The derivative of π = f(q) is the limit of ∆π / ∆ q for very small changes in q h q f h q f dq df dq d h ) ( ) ( lim 1 1 + = = π → • The value depends on the value of q 1 Value of a Derivative at a Point • The evaluation of the derivative at the point q = q 1 can be denoted 1 q q dq d = π • In our previous example, 1 π = q q dq d 3 < π = q q dq d = π = * q q dq d First Order Condition for a Maximum • For a function of one variable to attain its maximum value at some point, the derivative at that point must be zero = = * q q dq df Second Order Conditions • The first order condition ( d π / dq ) is a necessary condition for a maximum, but it is not a sufficient condition π Quantity π * q* If the profit function was ushaped, the first order condition would result in q * being chosen and π would be minimized Second Order Conditions • This must mean that, in order for q * to be the optimum, * q q dq d < π for and * q q dq d < π for • At q *, d π / dq must be decreasing – the derivative of d π / dq must be negative at q * Second Derivatives • The derivative of a derivative is called a second derivative • The second derivative can be denoted by ) ( " or or 2 2 2 2 q f dq f d dq d π Second Order Condition • The second order condition to represent a (local) maximum is ) ( " * * 2 2 < = π = = q q q q q f dq d Second Order Conditions  Functions of One Variable • Let y = f ( x ) • A necessary condition for a maximum is that d y /d x = f ’( x ) = 0 – to ensure that the point is a maximum, y must be decreasing for movements away from it Second Order Conditions  Functions of One Variable • The total differential measures the change in y dy = f ’( x ) dx – to be at a maximum, dy must be decreasing for small increases in x – to see the changes in dy , we must use the second derivative of y Second Order Conditions  Functions of One Variable – since d 2 y < 0 , f ’’( x ) dx 2 < 0 – since dx 2 must be > 0, f ’’( x ) < 0 • This means that the function f must have a concave shape at the critical point 2 2 ) ( " ) ( " ] ) ( ' [ dx x f dx dx x f dx dx dx x f d y d = ⋅ = ⋅ = Rules for Finding Derivatives...
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This note was uploaded on 10/15/2010 for the course ECONOMICS Econ 105A taught by Professor Someone... during the Spring '09 term at UC Riverside.
 Spring '09
 Someone...
 Macroeconomics, Utility

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