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EC 352: Intermediate Microeconomics, Lecture 2 Economics 352: Intermediate Microeconomics Lots of Fun Practice! Here are a lot of practice problems. They are not required. The intention is to give you sufficient opportunity to practice the techniques presented above. You should do these practice problems until you feel you are proficient with the techniques and concepts described above. Derivatives and the Profit Function Graph, find the first derivatives and calculate maximum values or minimum values for the following functions. 1. y = f(x) = 5 f’(x)=0 No min or max 2. y = f(x) = 5x f’(x)=5 No min or max 3. y = f(x) = 5x + 5 f’(x)=5 No min or max 4. y = f(x) = 5x – 5 f’(x)=5 No min or max 5. y = f(x) = 5x + 5x f’(x)=5+5=10 No min or max 6. y = f(x) = 5x - 5x f’(x)=5-5=0 No min or max 7. y = f(x) = 5x2 f’(x)=10x A minimum at x=0, no max 8. y = f(x) = 5x2 + 5 f’(x)=10x A minimum at x=0, no max 9. y = f(x) = 5x2 – 5 f’(x)=10x A minimum at x=0, no max 10. y = f(x) = 5x2 + 5x f’(x)=10x+5 A minimum at x=-0.5 11. y = f(x) = 5x2 - 5x f’(x)=10x-5 A minimum at x=0.5 12. y = f(x) = -5x2 + 5x f’(x)=-10x+5 A maximum at x=0.5 13. y = f(x) = 5x2 + 5x + 5 f’(x)=10x+5 A minimum at x=-0.5 14. y = f(x) = 5x2 + 5x – 5 f’(x)=10x+5 A minimum at x=-0.5 15. y = f(x) = 5x2 - 5x + 5 f’(x)=10x-5 A minimum at x=0.5 16. y = f(x) = -5x2 + 5x + 5 f’(x)=-10x+5 A maximum at x=0.5 17. y = f(x) = -5x2 - 5x + 5 f’(x)=-10x-5 A maximum at x=-0.5 18. π = 3q π’(q)=3 No maximum or minimum 19. π = 3q – 5 π’(q)=3 No max or min 20. π = 3q – 15 π’(q)= 3 No max or min 21. π = 3q2 – 5 π’(q)=6q Min at q=0 22. π = 3q2 – 15 π’(q)=6q Min at q=0 23. π = 60q - 3q2 – 15 π’(q)=60-6q Max at q=10 24. π = 60q - 3q2 – 25 π’(q)=60-6q Max at q=10 25. π = 60q - 3q2 – 55 π’(q)=60-6q Max at q=10 26. π = 60q - 3q2 – 65 π’(q)=60-6q Max at q=10 27. π = 60q - 5q2 – 5 π’(q)=60-10q Max at q=6 28. π = 60q - 2q2 – 15 π’(q)=60-4q Max at q=15 Second Derivatives and Curvature For each of the following functions, calculate the value of the second derivative at the value of x or q for which the first derivative is zero and then determine whether that value of x or q is a maximum, a minimum or neither. You should check your graphs from the first set of practice problems and confirm that your answer is consistent with the graph. These are largely answered above. Here are the second derivatives. 1. y = f(x) = 5 f”(x)=0 2. y = f(x) = 5x f”(x)=0 3. y = f(x) = 5x + 5 f”(x)=0 4. y = f(x) = 5x – 5 f”(x)=0 5. y = f(x) = 5x + 5x f”(x)=0 6. y = f(x) = 5x - 5x f”(x)=0 7. y = f(x) = 5x2 f”(x)=10 2 8. y = f(x) = 5x2 + 5 f”(x)=10 9. y = f(x) = 5x2 – 5 f”(x)=10 10. y = f(x) = 5x2 + 5x f”(x)=10 11. y = f(x) = 5x2 - 5x f”(x)=10 12. y = f(x) = -5x2 + 5x f”(x)=-10 13. y = f(x) = 5x2 + 5x + 5 f”(x)=10 14. y = f(x) = 5x2 + 5x – 5 f”(x)=10 15. y = f(x) = 5x2 - 5x + 5 f”(x)=10 16. y = f(x) = -5x2 + 5x + 5 f”(x)=-10 17. y = f(x) = -5x2 - 5x + 5 f”(x)=-10 Functions of Several Variables For each of the indicated functions, take the partial derivatives with respect to each explanatory variable. 1. y = f(x,w) = 5w + 6x + 7xw 2. y = f(x,w) = 5w2 + 6x3 + 7x2w 3. y = f(x,w) = 5w2 + 6x2 + 7xw2 4. y = f(x,w) = w/x 5. y = f(x,w) = 5w2 + 6x3/w2 + 7w/x3 6. y = f(x,w) = 5w + 6x - 7xw 7. y = f(x,w) = 5w2 - 6x3 - 7x2w 8. y = f(x,w) = 5w2 + 6x-2 + 7xw2 9. y = f(x,w) = wx 10. y = f(x,w) = 5w2 + 6x3w-2 + 7wx-3 3 For each of the following functions calculate the cross partials fxw and fwx and confirm that they are identical. 1. y = f(x,w) = 5w + 6x + 7xw fx=6+7w fw=5+7x 2. y = f(x,w) = 5w2 + 6x3 + 7x2w fx=18x2+14xw 3. y = f(x,w) = 5w2 + 6x2 + 7xw2 fx=12x+7w2 4. y = f(x,w) = w/x fx=-w/x2 fw=1/x fwx=7 fw=10w+7x2 fwx=14x fw=10w+14xw fwx=14w fwx=-1/x2 5. y = f(x,w) = 5w2 + 6x3/w2 + 7w/x3 fx=18x2/w2-21w/x4 fw=10w-12x3/w3+7/x3 fwx= -36x2/w3-21/x4 6. y = f(x,w) = 5w + 6x - 7xw fx=6-7w fw=5-7x 7. y = f(x,w) = 5w2 - 6x3 - 7x2w fx=-18x2-14xw fwx=-7 fw=10x-7x2 fwx=-14x 8. y = f(x,w) = 5w2 + 6x-2 + 7xw2 fx= fw= fwx= 9. y = f(x,w) = wx fx= fw= fwx= 10. y = f(x,w) = 5w2 + 6x3w-2 + 7wx-3 fx= fw= fwx= For each of the following functions, determine whether there is a maximum by finding the values of x and w for which fx and fw are equal to zero and then examining the second derivatives at those values. 1. y = f(x,w) = 30w2 + 45x2 - 5xw fx=90x-5w fw=60w-5x fxx=90 fww=60 Both of these are positive, so when the first derivatives are zero we’ll find a minimum. 2. y = f(x,w) = 24w2 - 6x3 - 8x2w Constrained Optimization - Lagrangians Solve the following utility maximization problems by using the Lagrangian technique described above. In these questions, the two goods are cleverly named A and B, their prices are P A and PB, and income is M. For each question, calculate the marginal utility of income (the value of λ) at the utility maximizing solution. 4 1. PA=10, PB=20, M=600, U(A,B)=100A2B L=100A2B + λ(600-10A-20B) 200AB - 10 λ=0 -> 200AB = 10 λ 100A2-20 λ=0 -> 100A2 = 20 λ 200AB/100A2 = 10λ/20 λ 2B/A = ½ 4B=A 600 – 10(4B) – 20B = 0 600 – 40B – 20B = 0 B=10 A=4B=40 λ =200AB/10 = 200*40*10/10 = 80000/10=8000 λ =100A2/20 = 100*402/20 = 160000/20=8000 2. PA=10, PB=20, M=600, U(A,B)=10A2B 3. PA=10, PB=20, M=600, U(A,B)=A2B 4. PA=10, PB=20, M=600, U(A,B)=100AB1/2 5. PA=10, PB=20, M=600, U(A,B)=10AB1/2 6. PA=10, PB=20, M=600, U(A,B)=10A2/3B1/3 7. PA=10, PB=20, M=600, U(A,B)=AB L=AB + λ(600-10A-20B) B - 10 λ=0 -> B = 10 λ A - 20 λ=0 -> A = 20 λ B/A = ½ 2B=A 600 – 10(2B) – 20B = 0 600 – 20B – 20B = 0 B=15 A=2B=30 5 λ =B/10 = 15/10 = 1.5 λ =A/20 = 30/20 = 1.5 8. PA=10, PB=20, M=600, U(A,B)=A1/2B1/2 9. PA=10, PB=20, M=600, U(A,B)=A1/3B1/3 10. PA=20, PB=20, M=600, U(A,B)=100A2B 11. PA=30, PB=20, M=600, U(A,B)=100A2B 12. PA=10, PB=30, M=600, U(A,B)=100A2B 13. PA=10, PB=40, M=600, U(A,B)=100A2B 14. PA=10, PB=20, M=600, U(A,B)=100A2B 15. PA=10, PB=20, M=1200, U(A,B)=100A2B 16. PA=10, PB=20, M=2400, U(A,B)=100A2B 17. PA=10, PB=20, M=600, U(A,B)=A1/2B1/3 18. PA=10, PB=20, M=1200, U(A,B)=A1/2B1/3 19. PA=10, PB=20, M=2400, U(A,B)=A1/2B1/3 20. Why are the utility maximizing combinations for questions 1 through 6 all the same? 21. Why are the utility maximizing combinations for questions 7 through 9 all the same? 22. In questions 10 through 13, what happens to the utility maximizing bundle as the price of good A rises and then as the price of good B rises? 23. In questions 14 through 16, what happens to the utility maximizing bundle as income rises? 24. In questions 14 through 16 and 17 through 19, what happens to the value of the marginal utility of income (λ) as income rises? 6