Economics 701 Problem Set 1 1. Let ( ) 4 4 4. Find the critical values of (note: there are three of them).0 B œ B  B  B  B4 3 2 For each one, use the second–order condition to indicate whether the critical value yields a local maximum or minimum. 2. For each multivariate function below, calculate , , , , , . Verify in each case0 0 0 0 0 0" 2 11 22 12 21 that Young's theorem holds. a. ( , ) 20 B B œ B B1 2 1 23 b. ( , ) 14 2 20 B B œ B  B B  B  B1 2 1 1 2 1 22 2 c. ( , ) – 4 – 6 20 B B œ B B  B  B B  B1 2 1 2 1 22 21 2 d. ( , )0 B B œ B 68B1 2 1 2 e. ( , ) e0 B B œ B1 2 1 ( )B  B1 2 2 3. Consider the function in question 2b) above. Find the critical values of and thatB B1 2 maximize/minimize the function. Use the second–order conditions to determine whether or not these values yield a local maximum or minimum. 4. Repeat question 3, but use the function in question 2c) above. 5. Consider the function ( , ) 2 .C œ 0 B B œ )B B  B B1 2 1 22 2 3 31 2 a. Derive the total differential of the function. b. Use the total differentail to estimate the change in if initally, 1, and increasesC B œ B œ B1 2 1 by 0.5 and simultaneously increases by 0.2. Compute the B2 actual change in .C 6. Consider the implicit function: ( ) 3 7 0. Use the implicit function1 Bß C œ B  BC  C  œ2 3 theorem to caluclate and evaluate this derivative at 4, 3..C.B B œ C œ 7. Consider the function ( , ) 2 4 , where is a strictly positiveC œ 0 B + œ  B  +B  + +2 parameter. a. Use the first order condtion for a maximum to derive the critical value of , which will be aB function of , ( )+ B + Þ* In parts b) and c) below, you will calculate the effect of a unit increase in on the maximum+ value of ( , ). In part b), you will calculate the effect by substitution and taking the derivative.0 B + In part c, you will apply the envelope theorem. b. Substitute ( ) from part a) into ( , ) to obtain ( ( ), ). Calculate .B + 0 B + C œ 0 B + +* * * .C.+ * c. Now use the envelope theorem to compute ..C.+ * d. Suppose 0.5 initially. Use differential approximation to estimate the change in the+ œ maximum value of ( , ) if were to increase by 20 B + + !Þ Þ e. Continuing with part d), calculate the actual change in the maximized value of ( , ).0 B +