micro economic
Problem Set 1 Due on June 10, 2019
The goal of the problem set is to give you practice in mastering the course material. Try to give short, but precise answers, and always justify your solution.
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STUDENT 1 (NAME AND STUDENT ID NUMBER)
STUDENT 2 (NAME AND STUDENT ID NUMBER)
STUDENT 3 (NAME AND STUDENT ID NUMBER)
STUDENT 4 (NAME AND STUDENT ID NUMBER)
Microeconomics A/I - Waseda University Hideo Owan
Problem Set 1 Due on June 10, 2019
1. (40 points) Martha, John’s neighbour, consumes only exes and whys, just like everyone else in the neighbourhood. The consumption bundle where Martha consumes x units of exes per week and y units of whys per week is written as (x, y).
Martha has quasilinear preferences that are represented by the utility function u(x, y) = 2 √ x + y.
(a) In the graph below, draw two of Martha’s indifference curves: one that passes through the point (4, 14) and an other that passes through (36, 0).
4 points
Microeconomics A/I - Waseda University Hideo Owan
Problem Set 1 Due on June 10, 2019
Let us find Martha’s demand function for exes and whys. Suppose that the unit price of exes is px, the unit price of whys is py and Martha’s income is m.
(b) Compute and set the slope of Martha’s indifference curve equal to the slope of the budget line, and solve for x.
4 points
(c) Use your previous result for x and plug it into the budget equation to find y.
2 point
(d) What is the minimum income level that makes sure that Martha consumes a non- negative amount of both goods?
4 point
(e) What is the optimal level of consumption of exes and whys if Martha’s income is less than the minimum above? Is it an interior solution or a corner solution? Is the marginal rate of substitution at the consumption choice higher or lower than the slope of the budget line?
5 points
Microeconomics A/I - Waseda University Hideo Owan
Problem Set 1 Due on June 10, 2019
(f) Write (the mathematical expression for) Martha’s demand function for exes and her demand function for whys.
5 points
(g) Use your answer to the previous question and complete the table below with Martha’s optimal bundle for different price-income combinations.
px py m x ∗ y∗
2 8 32
4 8 32
2 8 24
2 8 6
4 points
(h) Assume now that px = 1 and py = 2, and draw the following functions in the two graphs below.
i. Martha’s Engel curve curve for exes. ii. Martha’s Engel curve curve for whys. 4 points
Microeconomics A/I - Waseda University Hideo Owan
Problem Set 1 Due on June 10, 2019
Assume again that the unit price of exes is 1, the unit price of whys is 2, and that Martha’s income is 32. Now the local government levies a new tax on exes that increases the unit price of exes to 2.
(i) How does Martha’s consumption of exes change as a consequence of the new tax?
4 point
(j) What part of the above change in demand is due to the substitution effect, and what part is due to the income effect?
4 points
Microeconomics A/I - Waseda University Hideo Owan
Problem Set 1 Due on June 10, 2019
2. (23 points) Katie, a consumer with a fixed income (m), only consumes burgers and pickled cucumbers (x1 and x2) and firmly believes that they should be consumed in fixed proportions: one burger with two pickled cucumbers.
(a) Find the utility function that represent Katie’s preferences. Are they homothetic? Explain.
5 points
(b) Write and solve the consumers utility-maximization problem to find her demand functions, x1(p1, p2, m) and x2(p1, p2, m).
5 points
(c) Using your previous results, write the consumer’s maximum utility level as a func- tion of prices and income. This function is called the indirect utility function and is denoted by v(p1, p2, m).
4 points
Microeconomics A/I - Waseda University Hideo Owan
Problem Set 1 Due on June 10, 2019
(d) How does the consumer’s maximum achievable utility level change if ...
3 point
i. ... income increases?
ii. ... prices increase?
iii. ... both income and prices double?
(e) Now suppose that Katie is on welfare and her welfare payment is fully indexed by the Laspeyres price index. Could she be overcompensated by the full indexation? Why?
6 point
Microeconomics A/I - Waseda University Hideo Owan
Problem Set 1 Due on June 10, 2019
3. (37 points) Misty and May are monozygotic twins who have exactly the same pref- erences. They have recently got married and live in two different regions, Kanto and Alola, respectively. They both face the same problem: how many children they plan to have (n) and whether they should work (E = 1) or stay home (E = 0).
Their husbands have no intention to stay home and they leave the above choice entirely to their wives (that is, only the wives’ utility function matters). If the wife stays home, the household income is only the husband’s labor income (mh). If she works, it is the joint income (mh + mw).
Misty and May consume only exes (x) and derive utility from having children. Their preferences are quasi-linear, do not depend on E, and can be represented with the fol- lowing utility function: u(x, n) = f(n) + x, where f is a continuous, monotonically increasing concave function. Although children are indivisible, in this exercise you can treat n as a continuous variable for the sake of simplicity.
The price of exes is px while the cost of raising children is pE per child which varies depending on whether E = 1 or E = 0. The price of raising a child when the mother is working is assumed to be larger than the price when the mother stay home (p1 > p0), because a working couple will have to send their children to a daycare center, hire babysitters when necessary. etc.
(a) In Kanto, where Misty lives, px = 1, p0 = 10, p1 = 35. For Misty and her husband, mh = 55, mw = 45.
i. In the graph on the next page, draw Misty’s budget set using a black pen. Note that it is the union of the set of affordable children-exes bundles when Misty works and the set of affordable children-exes bundles when Misty does not work.
5 points
ii. Suppose that Misty chooses to work and decides to have one child. Mark this bundle and label it B. Note that she could have chosen not to work and have two children. Mark this hypothetical bundle and label it C. Has B been directly revealed preferred to C?
3 points
iii. Shade in the area representing the bundles that are directly revealed less pre- ferred to B.
3 points
Microeconomics A/I - Waseda University Hideo Owan
Problem Set 1 Due on June 10, 2019
(b) In Alola, where May lives, the regional government offers a child allowance of 5 per child. Just like for Misty, for May and her husband, mh = 55, mw = 45.
i. Using a blue pen, draw May’s budget set in the graph above. 5 points
ii. Mark the bundles that include zero, one, two, three children and label them A′, B′, C′ and D′, respectively. Given that May and Misty have the same preferences and given your answer to the questions above, could it be possible that May chooses bundle A′ over bundles B′ and C′? Explain.
4 points
Microeconomics A/I - Waseda University Hideo Owan
Problem Set 1 Due on June 10, 2019
iii. Draw two indifference curves that go through B and D′, respectively, that are consistent with the decisions made by Misty and May. Recall that they have identical preferences.
4 points
(c) Suppose that May has chosen bundle D′, and recall that Misty has chosen bundle B.
i. Does child allowance have a positive or negative total effect on the number of children that women plan to have?
3 points
ii. How much of the above choice difference between Misty and May is due to the income effect? Explain.
4 points
iii. How much of the difference is due to the substitution effect? Explain. 3 points
iv. Based on the above results, does the child allowance encourage women to continue to work after giving a birth?
3 points
Microeconomics A/I - Waseda University Hideo Owan