Problem set 3 solutions, Quizzes of Economics

Download Problem set 3 solutions and more Quizzes Economics in PDF only on Docsity! Problem Set 3: Solutions ECON 301: Intermediate Microeconomics Prof. Marek Weretka Problem 1 (Cobb-Douglas Utility Functions) 1.1: Optimal fraction of income spent on (berries) x2: b a+b . Optimal fraction of income spent on (nuts) x1: a a+b . (The problem only asks for berries.) Notice how neither fraction depends on income m or the prices of the two goods, p1 and p2. This is always true for Cobb-Douglas utility but not true for all types of utility functions. 1.2: From the answer above we know that optimal total dollars spent on berries, p2x2, will be equal to b a+b of income m, so this gives us the equation p2x2 = b a+b m. Using the same reasoning (again, the problem does not ask for this), the total amount spent on nuts, x1, will be p1x1 = a a+b m. 1.3: The quantity of nuts consumed will be x1 = a a+b · m p1 (and the optimal quantity of berries consumed will be x2 = b a+b · m p2 . We can see this from (1.2): Just divide both sides of those equations by p1 to find the optimal quantities (i.e., solve for x1 and x2). 1.4: For these problems, we first find the optimal bundle (using the equations in 1.3) and plug these quantities into MRS (x1, x2), which gives the slope of the indifference curve at (x1, x2). The MRS for Cobb-Douglas utility functions is MRS (x1, x2) = −ax2 bx1 (you can verify this). • (a) Given a = 4, b = 8, p1 = 5, p2 = 10, and m = 60: – Optimal quantity of x1 is x1 = a a+b · m p1 = 4 4+8 · 60 5 = 4 – Optimal quantity of x2 is x2 = b a+b · m p2 = 8 4+8 · 60 10 = 4 – Slope of the indifference curve given by MRS (x1, x2) = −ax2 bx1 = −4·4 8·4 = −1 2 • (b) Given a = 1 3 , b = 1 3 , p1 = 4, p2 = 1, and m = 12: – Optimal quantity of x1 is x1 = a a+b · m p1 = 1/3 1/3+1/3 · 12 4 = 3 2 – Optimal quantity of x2 is x2 = b a+b · m p2 = 1/3 1/3+1/3 · 12 1 = 6 – Slope of the indifference curve given by MRS (x1, x2) = −ax2 bx1 = − 1 3 ·6 1 3 · 3 2 = −4 • (c) Given a = 1 2 , b = 3 2 , p1 = 5, p2 = 1, and m = 20: – Optimal quantity of x1 is x1 = a a+b · m p1 = 1 2 1 2 + 3 2 · 20 5 = 1 1 – Optimal quantity of x2 is x2 = b a+b · m p2 = 3 2 1 2 + 3 2 · 20 1 = 15 – Slope of the indifference curve given by MRS (x1, x2) = −ax2 bx1 = − 1 2 ·15 3 2 ·1 = −5 Problem 2 (Well-Behaved Preferences) (a) The two secrets of happiness for well-behaved preferences such as these, which we’ll use to derive demand, are: • (1) p1x1 + p2x2 = m (Benjamin spends all of his income, choosing a bundle on the budget line.) • (2) MRS (x1, x2) = −p1 p2 (He chooses the bundle where the budget line is tangent to the indifference curve furthest from the origin.) For this utility function, we get that MRS (x1, x2) = −MU1 MU2 = −x2 x1 , for condition (2) we have −x2 x1 = −p1 p2 . Solving these two equations for x1 and x2 as we’ve done before, we get x1 = m 2p1 and x2 = m 2p2 . (b) To find the price-offer curve (which lies in our x1-x2 commodity space) with p2 = 1 and m = 10, we plug these numbers into demand x2 = m 2p2 =⇒ x2 = 10 2·1 = 5. So, regardless of the price of x1, with p2 = 1 and m = 10, 5 units of x2 (MP3s) are always purchased. Plotting this out in the commodity space: x1 x2 price o↵er curve: x2 = 5 5 We found the demand function x1 = m 2p1 above, and with m = 10 we have x1 = 10 2p1 (or, equivalently x1(p1) = m 2p1 ). To plot the demand in the p1-x1 space, we use the inverse of this demand function: p1(x1) = 10 2·x1 = 5 x1 . 2 (b) For p2 = 1 and m = 10, we have the price-offer curve x2 = 2x1. (Shown below on the left.) Demand for x1 is x1 = 10 p1+2 . We solve this for p1 to get the inverse demand curve to graph: p1 = 10 x1 − 2. (Shown below on the right.) x2 = 2x1 POC: x1 x2 x1 p1 p1(x1) = 10 x1 2 5 (c) Milk, x1 is an ordinary good for Trevor. Looking at the demand function, we see that as p1 increases (decreases), quantity demanded x1 decreases (increases). (d) Given p1 = p2 = 1, the optimal choices given m (which define the Engel curve) are given by x1 = m p1 + 2p2 = m 3 and x2 = 2m p1 + 2p2 = 2m 3 We graph the Engel curves for each good in x1-m and x2-m space: x1 x2 m m m = 3x1 m = 3 2 x2 5 The income-offer curve coincides with the optimal proportion line (solve each of the demand functions to m, set them equal to each other and solve for x2). x1 x2 IOC: x2 = 2x1 The two commodities are normal rather than inferior since, as we can see in the demand functions analytically and the Engel curve graphically, as income increases, the quantity demanded for both goods increases. (e) Since ∆x1 ∆p2 < 0 (as p2 goes up, demand for x1 goes down) and ∆x2 ∆p2 < 0 (as p1 goes up, demand for x2 goes down), x1 and x2 are gross complements. Problem 4 (Perfect Substitutes) (a) Our demand functions for x1 and x2 will be depend on what the price ratio is relative to the MRS (the slope of the indifference curves, which is constant for perfect substitutes like these). Here, MRS (x1, x2) = −MU1 MU2 = −2 1 . If the budget line is less steep than the slope of the indifference curves, i.e., if | − p1 p2 | < |MRS (x1, x2)|, then Kate consumes only x1 (Red Delicious). Otherwise, if if the budget line is more steep than the indifference curves | − p1 p2 | > |MRS (x1, x2)|, Kate consumes only Jonagolds x2, as shown graphically in Problem Set 2. So we have that ∣∣∣∣−p1 p2 ∣∣∣∣ < |MRS (x1, x2)| =⇒ x1 = m p1 , x2 = 0 and ∣∣∣∣−p1 p2 ∣∣∣∣ > |MRS (x1, x2)| =⇒ x1 = 0, x2 = m p2 . 6 (b) With m = 10 and p2 = 1, we have |− p1 p2 | = |− p1 1 | = p1. Since |MRS (x1, x2)| = |−2| = 2, • For p1 > 2, demand for x1=0 and x2 = m p2 = 10 1 = 10. • For p1 < 2, demand for x1 = m p1 and x2 = 0. • For p1 = 2, MRS = −p1 p2 everywhere, so any bundle along the budget line is optimal. Tracing out all these optimal bundles in the commodity space as p1 varies, we get the price- offer curve shown below one the left, and the demand for Red Delicious apples x1 is on the right: x1 x2 5 POC x1 p1 5 2 (c) Since demand for x1 does not increase as its price p1 increases, x1 is an ordinary good. (d) Fixing p1 = 1 and p2 = 1, for this price ratio we know that Kate will always choose x1 = m p1 = m Red Delicious apples and x2 = 0 Jonagold apples. We get the following income-offer curve (left) and Engel curves for x1 and x2 (two on the right): x1 x2 IOC x1 x2 m m x1 = m x2 = 0 7 – Net demand for x1 is zero since x1(3, 2, 100) = 50 3 > 20 = ω1. (He will be a net supplier of x1.) – Net demand for x2 is given by x2(3, 2, 100)− ω1 = 25− 20 = 5 (d) The magnitude of the slope of the indifference curve at the endowment point is |MRS(20, 20)| = | − 20 20 | = 1. • (1) For p1 = 1, |MRS(20, 20)| = 1 > 1 2 = | − p1 p2 |. He is better off buying more apples (x1) and selling some of his oranges (x2). • (2) At p1 = 2, |MRS(20, 20)| = 1 = 1 = | − p1 p2 |. He can not get any greater utility from buying or selling apples and oranges; he is best off with his endowment. • (3) At p1 = 3, |MRS(20, 20)| = 1 < 3 2 = | − p1 p2 |. He is better of buying more oranges (x2) and selling some of his apples (x1). (e) Dave’s optimal bundles for the three price combinations found above analytically are shown graphically below with bundles (1), (2), and (3) labeled. The price-offer curve must pass through these points, and is shown in the second graph. You will see that at all but one point (the endowment, point (2)=(ω1, ω2)), the price offer curve lies strictly above the indifference curve passing through the endowment (where u = 400). This is significant: It means that one is always at least as well off trading as he is just consuming his endowment! 10 x1, apples x2, oranges (!1,!2) p1 = 3 p1 = 1 p1 = 2 u = 400 u = 416 2 3 u = 450 (1) (3) (2) x1, apples x2, oranges (!1,!2) p1 = 3 p1 = 1 p1 = 2 u = 400 u = 416 2 3 u = 450 (1) (3) (2) price-o↵er curve 11