## PS1_Solutions

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## PS1_Solutions

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- Pages:
- 3
- School:
- University of California, Los Angeles
- Course:
- Psych 10 - Introductory Psychology

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ECON 11 Problem Set 1 Solutions 1 Unconstrained Optimization Single Variable Consider the following function f x 3x3 5x2 x where x 0 2 a Find the x s that are critical value points of f x values of x that are a potential minimum or maximum Are those critical values maxima or minima Hint Check the second order derivative Check also the value of the function at the endpoints 0 and 2 to make sure you have the correct answer Solution Take FOC to get critical points f 0 x 9x2 10x 1 0 x 1 Check SOC 1 9 1 f 00 x 18x 10 f 00 0 f 00 1 0 9 Thus there are two critical points x need to check for the END POINTS 1 9 is a local maximum x 1 is a local minimum But we also f 0 0 f 2 6 Compare with 1 13 f 1 1 f 9 243 Therefore f 1 1 f 2 6 are the global minimum and maximum respectively 2 Unconstrained Optimization Two Variables Consider the function f x1 x2 x13 3x23 9x1 x2 1 a Find a minimum given that x1 x2 1 Solution Take FOC f1 3x12 9x2 0 f2 9x22 9x1 0 We get x1 x2 0 and x1 3 9 x2 3 3 But because x1 x2 1 we only have to consider x1 3 9 x2 3 3 Check SOC f11 6x1 6 3 9 0 f12 9 f12 9 f22 18x2 18 3 3 0 3 3 2 f11 f22 f12 6 9 18 3 9 9 182 92 0 Therefore x1 3 9 x2 3 3 is a minimum Note the followings is not required and will not be tested 2 0 when x x 0 So 0 0 is a saddle point But you First if you are interested f11 f22 f12 1 2 are not required to know what a saddle point is Also this function does not have a global maximum or global minimum For example x1 x2 0 then f x1 x2 x1 x2 0 then f x1 x2 3 Constrained Optimization Julia maximizes the following utility function 2 7 5 7 u x1 x2 x1 x2 subject to the budget constraint p1 x1 p2 x2 I where p1 p2 x1 x2 I 0 a Find the x1 x2 that maximizes u x1 x2 i Step 1 Identify the type of problem This is a CONSTRAINED optimization problem with two variables Therefore we need to apply the Lagrangian multiplier method ii Step 2 Setup the Lagrangian 2 7 5 7 L x1 x2 x1 x2 I p1 x1 p2 x2 iii Step 3 Derive the FOCs and solve them for the critical points 2 5 7 5 7 2 5 7 5 7 L1 x1 x2 p1 0 x1 x2 p1 7 7 5 2 7 2 7 5 2 7 2 7 L2 x1 x2 p2 0 x1 x2 p2 7 7 L3

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