MICROECONOMICS II Gisela Rua 2,5 hours

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1 MICROECONOMICS II st Test Fernando Branco Gisela Rua,5 hours I (6,5 points) James has an income of 0, which he spends in the consumption of goods and whose prices are and 5, respectively Detective Smith was asked to study James s choices but there was not much information to help him Earlier in that morning he had found a paper with James handwriting stating that his expenditure function was e(p, p,u) = ( p p u ) Now he found a small piece of paper with some function that could be James marshallian and compensated demand functions: x m = m, x m = m!, x h = p u 4 p p p & and h x =! p a ( point) What is the compensated demand function? Why is it different from the marshallian demand function of the same good? Answer: The hicksian demand function tells us what consumption bundle achieves a target level of utility and minimizes total expenditure (the dual problem) In considering the effect of a change in price on demand with utility held constant we automatically make whatever changes in expenditure are required to compensate for the effects of the price change in real income or utility The consumer s marshallian and compensated demand functions of the same good are identical if there are no income effects on the demand It is different from the marshallian demand function since the latter considers all the consumption bundles that achieve a target level of expenditure and maximize the level of utility A change in price will automatically cause change in utility while total expenditure remain constant p u & b (,5 points) Help Mr Smith to find out if the information in the paper is true Answer: Using the expenditure function, one can get the compensated demand functions using the Shephard s lemma: x h =!e = p!p p u & ' and h x =!e = p!p To find the marshallian demand functions one must first derive the indirect utility function, using the knowledge that e p, p,v(p, p,m) ( ) = m : m = (p p v) p u & ' m 4! v = Then by Roy s identity, one can 6( p p )

2 get the marshallian demand functions: x m =! both demand functions for good is not correct v v p = m p and x m v =! = m Hence, the information about p v p m m c (,5 points) Find out James income expansion path (income offer curve) and the price offer curve of good Represent them graphically and interpret (Use the initial prices) Answer: The income expansion path can be derived from the marshallian demand functions, given that the objective is to get a function that relates the quantities of both goods x = m p! m = p x x = m p = p x p x = p p x Since the optimal quantity of each good does not depend on the price of the other good, the price offer curve m of good is an horizontal line at the level p Graphically, x IEP x m p POC x x

3 After additional research on James preferences Mr Smith found out that he was consuming 5 units of good and units of good, and that this bundle corresponded to an utility index of 00 d (,5 points) Compute the total change in the demanded quantity of good caused by a change in p from 5 to 0 Decompose it in the (hicksian) substitution effect and in the income effect (If you did not answer the previous question, use the functions given above) Answer: Substituting in the marshallian demand function or in the budget line equation, one can find the final chosen bundle: ( x, x ) = ( 5,) The total change in the demanded quantity of good is! = To decompose this change in the hicksian substitution effect and in the income effect, one must first find an intermediate point which reflects the demanded quantities if James would have been given a monetary compensation in order to attain the initial utility index This point can be found using the compensated demand function for good : x h! = 0 00 & = =,44 Hence, the hicksian substitution effect is,44! =!0,586 and the income effect is!, 44 =!0, 44 e (0,5 points) Classify good in terms of price and income effects Answer: Good is an ordinary and normal good, since the price increase caused a negative change in the demanded quantity and a negative income effect II (3 points) Yesterday, Mr Smith met Brian, a friend of James Still enthusiastic with his investigation on consumer preferences, he decided to observe Brian s behavior in two situations Firstly, when prices were p = and p = 5, Brian bought 7 units of good and 3 units of good Afterwards, when prices were p = and p = 0, Brian bought 5 units of both goods Mr Smith was very surprised with these choices wondering if Brian had rational preferences a ( point) Help Mr Smith verify if Brian s preferences are indeed rational Answer: In order to study the rationality of Brian s preferences, ie, to know if they verify the WARP, let s build the following table: Expenditures Bundle A (7,3) Bundle B (5,5) Prices in the first situation 9 35 Prices in the second situation In the first situation, only bundle A was available and Brian chose it In the second situation, he chose bundle B when he could have chosen A since both were available This means that he prefers bundle B to A

4 This decision does not violate the WARP, and thus, Mr Smith has no reason to be surprised with Brian s choices, which might be rational Still not satisfied with all his discoveries, Mr Smith decided to talk to another friend of James, Joseph Joseph is a rational consumer with continuous and monotonic preferences Joseph told Mr Smith that, for him, both good and good were inferior goods Moreover, he told our detective that both goods were substitutes Mr Smith got very much confused with Joseph s statements b (,5 points) Help Mr Smith, once more, to find out if Joseph s classifications are possible Answer: In order to help Mr Smith, one must use Engel and Cournot s aggregation conditions Using Engel s aggregation condition, one can conclude that it is not possible to have two inferior goods (! x,m < 0 and! x,m < 0 ), since that would make the equation s! x,m + s! x,m = impossible Using Cournot s aggregation condition, one can conclude that it is possible that goods are substitutes, ie,! x, p > 0 or! x, p > 0 It does not violate the equations s! x, p + s! x, p = s or s! x, p + s! x, p = s, per se

5 III (3 points) Answer one, and only one, of the following two questions (A and B) A Joanna Berger is a novel s writer who wrote the current success There is always tomorrow The editor will pay her 000 this month and 050 next month She wants to use these payments to finance two-month trip to Latin America Consider that Joanna s intertemporal preferences for this trip can be represented by the following expression: u( c,c ) = c 0,6 c 0,4, where c and c are the expenditures in the first and second travelling months, respectively She has made the following arrangement with her bank: - The bank will pay her an interest rate of r = 5 on any savings she makes on the first month; - If instead, in the first month, she needs to borrow from the bank, this will charge her an interest rate of r = 0 However, there is a limit of 00 on the amount she can borrow from the bank a (,5 points) Represent Joanna s budget line for her two-month trip to Latin America graphically in the space ( c, c ) Answer: Joanna s budget line is given by the following expression: Graphically,,05c + c = 00 if c! 000,c + c = 50 if 000 < c! 00 c c (The points where c = 00 and c < 940 are not considered as part of the budget line, since they correspond to bundles where not all the income is spent) b (,5 points) Find the optimal level of expenditures in both months Calculate the subjective rate of intertemporal preference in the optimum found above, and compare it with the interest rate Justify Answer: Solving Joanna s decision problem, one will get the optimal level of expenditures in both months:

6 maxu( c,c ) = c 0,6 0,4 c st,05c + c = 00 if c! 000,c + c = 50 if 000 < c! 00 Two solutions : st equation :(00, 840) impossible! nd equation : (7,(7); 860) impossible! Optimal level of expenditures is: ( c,c ) = ( 00,940) There is no equality between the MRS and +r, since one got a corner solution If there were no credit restriction, Joanna would have spent more in the first month, because MRS > + r B Christian lives in Lisbon and he is considering the hypothesis of working for a big firm, which has several affiliates all over the city Christian s preferences are described by the utility index u c,l ( ) = c 0, l 0,8, where c is the quantity of a composite consumption good and l is the daily number of hours of leisure Christian has 80 hours per week to spend with work or leisure and receives an allowance of 50 per week, paid by his rich uncle Christian received a proposal to work at the affiliate in Amadora The hourly wage rate is 0 and the price of the composite consumption good is Christian will use public transportation to go to work After some calculations, Christian concluded that he would spend 7 per week using public transportation to go to work (Consider that the number of hours spent traveling has already been discounted and not included in the 80 hours available) a (,5 points) Represent Christian s budget line and find his optimal choice Answer: Christian s budget line is given by: *80!0l! 7 = 943!0l if l <80 c = 50 if l = 80 The optimal choice is the solution of the following problem: maxu C,l ( ) = C 0, l 0,8 st C = st equation : 943!0l if l < if l = 80 4 C FOC : l = 0 & C = 88,6 with anutilitylevel of u(88, 6; 75, 44) = 90, 63 l = 75,44 C = 943!0l nd equation : u(50, 80) = 90, 77 Since Christian has a higher level of utility if he does not work, he will choose not to work b (,5 points) Compute the amount of weekly expenditures on public transportation Christian must face in order to be indifferent between working and not working Answer: Christian will be indifferent between working and not working if he gets the same utility level in both situations Hence,

7 c 0, l 0,8 = 90, 77 4 c l = 75,57 l = 0 & c = 88,87 z = 5,93 c = 950! z!0l where z is the transportation cost IV (3 points) In the Neverlands the production of tulips is a very important economic activity Each producer of tulips has the following profit function:! ( w, w, p) = p, where w 9 w w and w represent the cost of the two inputs x and x, and p represents the price of each bunch of tulips (y) a (0,5 points) Find the value of!, justifying your answer Answer: Since the profit function is homogeneous of degree one, then! = 3 b ( points) Using the information given until now, find the supply function of each firm Justify Answer: One can find the supply function using Hotelling s lemma: y( p,w,w ) =!!p = p 3w w c (,5 points) Check if the production function y = (x x ) 3 is compatible with the profit function above Answer: To verify if the production function is compatible with the profit function, one will get the unconditional demand functions for inputs x and x, using the Hotelling s lemma x (p, w, w ) =! w = p 3 9w w x (p, w, w ) =! p 3 = w 9w w Substituting in the production function given, one conclude that it is not compatible: ( ) 3 = p 3 y = x x! p 3 3 p p 9w w 9w w & = ' 8 3 3w w w w Another way to find the unconditional demand functions is to solve the producer s problem max p(x x ) 3! w x! w x p 3 x! 3 x 3 = w = w x w FOC : p & 3 x! 3 x 3 = w x 3 = p! ' w * 3! 3( ) w +, w and substituting in the objective function:!(w,w, p) = 7 p3 p 3 7w w 9w w Thus, the production function is not compatible x p 3 x = 9w w & p 3 x = 9w w

8 V (4,5 points) a Can the Marshallian demand and the Hicksian demand functions of a given good be equal? b What are the properties of the cost function C(w,, w n, y)? c Explain the effect of a minimum wage in a monopsony market