University of Toronto Department of Economics ECO 204 Summer 2013 Ajaz Hussain TEST 1 SOLUTIONS GOOD LUCK!

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1 University of Toronto Department of Economics ECO 204 Summer 2013 Ajaz Hussain TEST 1 SOLUTIONS TIME: 1 HOUR AND 50 MINUTES DO NOT HAVE A CELL PHONE ON YOUR DESK OR ON YOUR PERSON. ONLY AID ALLOWED: A CALCULATOR YOU CANNOT LEAVE THE EXAM ROOM DURING THE LAST 10 MINUTES OF THE TEST STAY SEATED UNTIL ALL TESTS HAVE BEEN COLLECTED AND THE PROCTORS ANNOUNCE THAT YOU CAN LEAVE THE ROOM IF YOU DETACH ANY PAGES FROM THE TEST, THEN YOU MUST RE-STAPLE THESE LOOSE PAGES TO THE TEST GOOD LUCK! LAST NAME (AS IT APPEARS IN ROSI): FIRST NAME (AS IT APPEARS IN ROSI): MIDDLE NAME (AS IT APPEARS IN ROSI) 9-DIGIT STUDENT ID # (AS IT APPEARS IN ROSI) SIGNATURE: DO NOT WRITE BELOW. FOR GRADER S USE ONLY Question Maximum Possible Points Score Total Points = 100 Page 1 of 26

2 Question 1 [20 POINTS] [ALL PARTS ARE INDEPENDENT OF EACH OTHER] (a) [5 POINTS] Consider the following problem: Under what conditions is a stationary point (i.e. where ) a (or the) solution to this problem? If we ve found a stationary point then we can be assured it is a solution to the problem if the function is concave, i.e., and that it is the solution if the function is strictly concave, i.e.. In such cases, there s no need to check for boundary solutions. Page 2 of 26

3 (b) [5 POINTS] Consider the following problem: At the optimal solution to this problem, why will the optimal value of the Lagrange equation equal calculations and state all assumptions.? Show all This problem is solved as follows: The Lagrange Method Since we see that: [ ] Page 3 of 26

4 (c) [5 POINTS] Consider the following problem: At the optimal solution to this problem, why will the optimal value of the Lagrange equation equal calculations and state all assumptions.? Show all This problem is solved as follows: Since the animal we see that: Page 4 of 26

5 (d) [5 POINTS] Give an example of a single-variable function that is both concave and convex but not strictly concave nor strictly convex. A concave but not strictly concave function is defined as but not while a convex but not strictly convex function is defined as but not. Thus, for a function to be concave and convex we require that everywhere. The only function with this property is a linear function like. Page 5 of 26

6 Question 2 [10 POINTS] [ALL PARTS ARE INDEPENDENT OF EACH OTHER] (a) [5 POINTS] A business analyst has solved the following problem: Here After solving the problem, the analyst finds that the value of the Lagrange multiplier is and, if appropriate, make a recommendation to the analyst. Explain your answer.. Interpret this result To answer this question we need to know what measures. By the envelope theorem the change in due to a small change in is given by: Thus: This means that increasing capacity by 1 unit will decrease revenues by $0.50. You should recommend that capacity be reduced until. Page 6 of 26

7 (b) [5 POINTS] A business analyst has solved the following problem: The analyst tells you that the value of the Lagrange multiplier is Explain your answer.. What do you recommend the analyst do? This is an inequality constrained problem and we know that for such problems. The fact that means that the analyst made a mistake in his calculations. Page 7 of 26

8 Question 3 [15 POINTS] [ALL PARTS ARE INDEPENDENT OF EACH OTHER] (a) [5 POINTS] Graph and express mathematically the consumption set of a consumer for whom: Assume neither good can be consumed in negative amounts. Here burgers and soda cans must be consumed in integer amounts so that: { } Page 8 of 26

9 (b) [5 POINTS] Graph and express mathematically the consumption set of a consumer for whom: Assume neither good can be consumed in negative amounts. Here burgers must be consumed in integer amounts while gallons of sodas can be consumed in any amount: { } Page 9 of 26

10 (c) [5 POINTS] Graph and express mathematically the consumption set of a consumer for whom: Assume neither good can be consumed in negative amounts. Here pounds of burgers and gallons of sodas can be consumed in any amount: { } Page 10 of 26

11 Question 4 [20 POINTS] [ALL PARTS ARE INDEPENDENT OF EACH OTHER] (a) [4 POINTS] A consumer perceives goods 1 and 2 to be good goods as well as perfect substitutes with a marginal rate of substitution. Write down two utility functions representing this consumer s preferences and use one of these utility functions to state the simplest possible UMP. Do not solve the UMP but do explain how you simplified the UMP. The consumer s utility function is: This has: We are told that: Hence one utility function is: Another utility function be obtained by doing any positive monotonic transformation such as: Now the general linear UMP is: Since and everywhere in the consumption set, we see that the consumer can choose a bundle anywhere in the consumption set including the boundaries. As such, we cannot drop the non-negativity constraints. Page 11 of 26

12 (b) [4 POINTS] A consumer perceives goods 1 and 2 to be good goods as well as imperfect substitutes. She tells you that she must consume both goods and that she ll always spend 43% of her income on good 2. Write down two utility functions representing this consumer s preferences and use one of these utility functions to state the simplest possible UMP. Do not solve the UMP but do explain how you simplified the UMP. We know that the Cobb-Douglas UMP should be used to model consumers who perceive all goods to be good goods as well as imperfect substitutes. A property of the Cobb-Douglas model is that the expenditure on any good is always a constant fraction of income. In fact we know that for: That: Now we know that: Thus: This implies that: The utility function is: Another utility function be obtained by doing any positive monotonic transformation such as: The UMP is: Now: Now notice that: Page 12 of 26

13 { To see whether there could be a boundary solution we check: The UMP becomes: Page 13 of 26

14 (c) [4 POINTS] A consumer perceives and to be good goods as well as complements. A unit of consists of a combination of and : the consumer perceives 2 units of to be a perfect substitute for 5 units of, and 2 units of to be a perfect substitute for a unit of. Write down a utility function representing this consumer s preferences. What is the between and? Show all calculations. Start with and being perceived as complements : Next we know that a unit of is a combination of so that: Now in the ( ) plane we know that: This says that 5 units of good 4 are substitutable for 2 units of good 4. Thus:. Now in the ( ) plane we know that: This says that a unit of good 5 is substitutable for 2 units of good 4. Thus: that we had earlier we could do:. To reconcile this with the fact This still says that a unit of good 5 is substitutable for 2 units of good 4. Thus:. Combining these we have: Notice that in the ( ) plane: This says that 4 units of good 3 are substitutable for 5 units of good 5. The utility function is: Page 14 of 26

15 (d) [4 POINTS] A consumer perceives as a bad good and as a neutral good. Write down a utility function representing this consumer s preferences and graph the indifference curve for an arbitrary level of utility. Show all calculations. The utility function defined over { } Notice that: The slope of the indifference curve is: Page 15 of 26

16 (e) [4 POINTS] For the consumer in part (d), could the optimal choice be the bundle? What about? Explain briefly. Hint: Feel free to use a graphical argument. Since good 1 is a bad good, the optimal choice will have and since good 2 is a neutral good we can have any quantity where. Thus, it is possible for to be optimal so long as. Page 16 of 26

17 Question 5 [15 POINTS] (a) [5 POINTS] Consider a UMP where the utility function is defined on the consumption set { }. Prove that if the consumer has monotone preferences then her marginal utility of income must be strictly positive. Consider a general UMP: From the envelope theorem we know that: We need to show that if the consumer has monotone preferences then first two FOCs where noting that and that : This is indeed the case from the [ ] Of course this also implies that expenditure = income. Page 17 of 26

18 (b) [10 POINTS] Consider a general UMP where the utility function is defined on the consumption set { }. Prove that if the optimal choice is in the interior of the consumption set then at the optimal bundle the indifference curve must be tangent to the budget line. Show all calculations. Once again consider a general UMP: If the optimal choice is in the interior then: The KT conditions, especially the animals imply that As such the FOCs become: Equating yields: This says that at the interior solution, the indifference curve must be tangent to the budget line. Page 18 of 26

19 Question 6 [20 POINTS] (a) [5 POINTS] Solve the following problem in two separate ways: You are expected to use the appropriate constrained optimization methods. Show key calculations and state assumptions. The UMP is: Method #1 It s more convenient to take a positive monotonic transformation and work with: Now: Now notice that: { To see whether there could be a boundary solution we check: The UMP is: The FOCs are: Page 19 of 26

20 Now: Sub this in the budget constraint: Assume so that: Notice that expenditure on good 1 is a constant fraction of income. Next, from: Notice expenditures on goods 2 and 3 are also constant fractions of income. Page 20 of 26

21 Finally, let s solve for.we know that: Method # 2 Now: Now notice that: { To see whether there could be a boundary solution we check: The UMP is: The FOCs are: Now: Page 21 of 26

22 Sub this in the budget constraint: Assume so that: Notice that expenditure on good 1 is a constant fraction of income. Next, from: Notice expenditures on goods 2 and 3 are also constant fractions of income. Finally, let s solve for.we know that: We assumed that so that: Page 22 of 26

23 Hate (b) [5 POINTS] Suppose Without re-solving the problem, calculate the impact on optimal demands due to a 1% income tax. Show key calculations and state assumptions. To use the expressions above we have to re-scale so that. Re-define. Before the income tax, the consumer s demands are: We also know that for either good: Thus, a 1% income tax will reduce demands of both goods by so that: One would get the same answer by subbing in the new income into the demand expressions: Page 23 of 26

24 (c) [5 POINTS] Suppose Without re-solving the problem, calculate the impact on optimal utility due to an 1% income tax in two separate ways. Show key calculations and state assumptions. We can compute the change in approach. due to an income tax in two ways: by the envelope theorem and the value function The Envelope Theorem Approach First, write down the objective in terms of parameters (we use the more convenient log linear Cobb-Douglas UMP): Second, differentiate with respect to the parameter, which in this case is : Third, evaluate at the optimal solution: Noting that: [ ] Implies: The Value Function Approach First, write down the objective in terms of parameters (we use the more convenient log linear Cobb-Douglas UMP): Second, sub in the optimal solutions expressed in terms of parameters: [ ] Third, differentiate with respect to the parameter, which in this case is : Page 24 of 26

25 [ ] Nice. Page 25 of 26

26 (d) [5 POINTS] Suppose Suppose the government imposes an excise tax on good 2 (in dollars per unit) that is designed to raise the same amount of tax revenue as a 1% income tax. Calculate this excise tax rate on good 2 (dollars per unit). Which tax scheme hurts consumers the least? Show key calculations and state assumptions. Revenues from the 1% income tax are: Now, revenues from an excise tax on good 2 will be: Now we want: Thus: Post excise tax price of good 2 where Let s check if this is right: Page 26 of 26