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ECOS2001

Intermediate Microeconomics

Solutions for Tutorial 1

2.1

(a) In general the equation of the budget equation is p1 x1 + p2 x2 = m. Substituting p1 = 10, p2 = 5, and m = 40 gives 10x1 + 5x2 = 40.

(b) When x2 = 0, x1 = m/p1 = 40/10 = 4.

(d) Now, with p1 = 5, we have 5x1 + 5x2 = 40.

(e) Finally, with p1 = 5 and m = 30, we have 5x1 + 5x2 = 30.

2.4 The price of X has increased by one unit while the price of Y remained unchanged. Since he originally consumed 100 units of X, his income would need to increase by 100 for him to continue to exactly afford the same bundle as before.

(d) True: Since 15 > 100/10, (10, 15) lies above the indifference curve through (20, 5).

(e) True: We actually know that (20, 5) ～ (10, 10), so it must be the case that (20, 5) ≥ (10, 10) as well.

(f) False: The bundle (24, 4) lies below the indifference curve xB = 100/xA, while the bundle (11, 9.1) lies on it.

(g) True. You can verify that (11, 14) ～ (10, 15), and (20, 5) ～ (2, 49). But we also know from (d) that (10, 15) ～ (20, 5), and so by transitivity (11, 14) ～ (2, 49).

(h) Yes: A straight line joining any two points in the grey-shaded region will always also lie in that region.

(i) No: We can ﬁnd two points in the blue-shaded region such that a straight line join-

ing them passes outside that region.

(j) Marginal rate of substitution.

(k) Substitute xA = 10 into MRS = 一100/xA(2) to ﬁnd MRS = 一1.

(l) Substitute xA = 5 to ﬁnd MRS = 一4.

(m) Substitute xA = 20 to ﬁnd MRS = 一1/4.

(n) Yes: As the value of xA increases and we move from left to right around an indif- ference curve, the indifference curve becomes ﬂatter and the (absolute value of) the MRS diminishes.

(a) òU/òx1 = x2 + 6 and òU/òx2 = x1 + 2, so MRS = 一 = 一 . Substituting x1 = 4 and x2 = 6 gives MRS = 一2.

(b) At (4, 6), U (4, 6) = (4 + 2) (6 + 6) = 72. The set of bundles that give the same utility are (x1, x2 ) such that (x1 + 2) (x2 + 6) = 72, or x2 = 72/ (x1 + 2) 一 6. This is the equation for the indifference curve through (4, 6). Other bundles that lie on this indifference curve include (10, 0), (7, 2), and (2, 12).

(c) If he makes this trade he gains 9 units of x2 in exchange for 3 units of x1, taking him to the bundle (1, 15). Now U (1, 15) = 63 ← 72, so he should not accept this trade.

(d) If he makes the ﬁrst trade he ends up with U (3, 9) = 75 > 72, so he should accept.

If he makes the second trade he ends up with U (2, 12) = 72, so he is indifferent. (e) The bundles that make him better off are the ones between (4, 6) and (2, 12).

4.12 (a) òu/òx1 = 2x1 + 2x2 and òu/òx2 = 2x1 + 2x2, so MRS = 一 = 一1. (b) òv/òx1 = 1 and òv/òx2 = 1, so MRS = 一 = 一1.

(c) Yes: Both correspond to linear indifference curves with a slope of 一1, i.e. one-for- one perfect substitutes. Since u (x1, x2 ) = [v (x1, x2 )]2, u is a positive monotonic transformation of v.