Introductory Microeconomics (ES10001)

Transcription

1 Introductory Microeconomics (ES10001) Exercise 3: Suggested Solutions 1. True/False: a. Indifference curves always slope downwards to the right if the consumer prefers more to less. b. Indifference curves never intersect if the consumer has consistent preferences. c. The slope of the budget line depends only upon the relative prices of the two goods. d. The budget constraint shows the maximum affordable quantity of one good given the quantity of the other good that is being purchased. e. An individual maximises utility where the budget line cuts an indifference curve. f. A change in money income alters the slope and position of the budget line. g. All Giffen goods are inferior goods. h. All inferior goods are Giffen goods. i. The income expansion path slopes upward to the right if both goods are normal goods. j. The substitution effect of an increase in the price of a good unambiguously reduces the quantity demanded of that good. k. If, following an increase in the price of X, the substitution effect is exactly balanced by the income effect, then X is neither a normal nor an inferior good. a. True; b. True; c. True; d. True; e. False - the individual can always improve on such a point; f. False - the slope depends only on the prices; g. True - if no endowment effect; h. False; i. True; j. True; k. False - if the income effect is working against the substitution effect, then X must be an inferior good; l. False - in general, consumers potentially gain by freedom to choose. 2. Match each lettered concept with the appropriate numbered phrase: (a) Consumption bundle; (b) utility; (c) income expansion path; (d) point of consumer choice; (e) budget constraint; (f) indifference curves; (g) substitution effect; (h) individual demand curve; (i) marginal rate of substitution; (j) budget line; (k) transfers in kind; (l) utility maximisation; (m) income effect; (n) market demand curve; (o) Complementarity; (p) Giffen. Solution: (1) A curve showing how the chosen bundle of goods varies with the level of income; (2) any particular combination of goods considered for purchase by the particular consumer; (3) the sum of the demand curves of all individuals in the market; (4) the quantity of one good that the consumer must sacrifice in order to increase the quantity of the other good by one unit without increasing his total utility; (5) a situation where goods are necessarily consumed together; (6) a line representing the maximum combination of two goods that a consumer can afford to purchase; (7) the point at which consumers maximise utility, where the marginal rate of substitution between two goods is equal to the ratio of their prices; (8) An inferior good where the income effect outweighs the substitution effect, causing the demand curve to slope upwards to the right; (9) That part of a consumer s response to a price change arising from the change in the consumer s purchasing power; (10) That part of a consumer s response to a price change arising from the change in relative prices; (11) A curve showing all the consumption bundles that yield the same level of utility 1

2 to the consumer; (12) A transfer payment in some other form than cash; (13) The assumption that the consumer chooses the affordable bundle that yields the most satisfaction; (14) The set of consumption bundles that the consumer can afford, given income and prices; (15) The satisfaction a consumer derives from a particular bundle of money; (16) A curve showing the amount of money demanded by the consumer. 1-c; 2-a; 3-n; 4-i; 5-o; 6-j; 7-d; 8-p; 9-m; 10-g; 11-f; 12-k; 13-l; 14-e; 15-b; 16-h. 3. Explain the underlying assumptions made in respect of the indifference curves shown in the following diagrams. Illustrate the diagrams with examples of commodities which your believe are consistent with these preferences. (N.B. I 2 denotes a higher level of utility than I 1 etc.) y I 2 I 1 I 0 0 x Figure 1 Solution: Figure 1: Good Y is a conventional good, for which satisfaction increases with the consumption of the good, though at a diminishing rate. However X is an economic bad, the utility from which decreases as its quantity is increased. The individual requires increases in Y to compensate for increases in X, and these increases in Y are themselves larger, the larger is X. An example is income and work. 2

3 y I 2 I 1 I 0.I 3 0 x Figure 2 Figure 2 - The single point on the indifference map represents the individual's satiation, or bliss point. Further increases in either X or Y beyond this point reduce utility. If the map is divided into four quarters centred on the bliss point it will be seen that only in the south-west quarter do the indifference curves behave normally. Examples are any pair of goods that the class consider both to have a bliss point. A fatuous one is cream cakes (Y) and alcohol (X) over a short time period. y I 2 I 1 I 0 0 x Figure 3 3

4 Figure 3 - The indifference curves are concave to the origin at high (low) values of Y (X), and convex to the origin as the quantity of Y (X) falls (rises). Convexity implies that individuals prefer combinations of X and Y to extremes; concavity, that the individual prefers all X or all Y. The indifference map indicates that, above (below) a certain amount of Y (X), the individual would prefer to hold extremes of X or Y. However below (above) this amount of Y (X), the individual prefers to hold a mix of X and Y, rather than an extreme of either. The point to stress is that, though conceptually feasible, it is unlikely for an individual to behave in such a manner. Hence it is hard to think of examples. 4. A consumer begins at point A with the budget line as depicted In Figure 4. Which of the following could have transpired if the consumer later chooses to be at B? (a) a change in tastes; (b) a small increase in the price of x and a larger percentage decrease in the price of y; (c) an increase in the price of x and a smaller percentage increase in the price of y; (d) a fall in real income; (e) Equal percentage increases in money income and both prices; y y A A y B B 0 x x A x B Figure 4 If only tastes change, B remains unobtainable, so the answer cannot be (a). Option (c) and (d) both move the budget line closer to the origin; (e) leaves the budget line unchanged. Hence the answer is (b). 5. Inferiority is a necessary, but not sufficient, condition for Giffeness. Discuss! Solution: In the absence of initial endowments, this statement is true. 1 It might be worth going over the concepts of necessary and sufficient conditions with students. The concepts of necessary condition and sufficient condition are used frequently in economics and it is essential to understand their precise meanings. 1 An example of initial endowments rendering the statement untrue is illustrated in the case of a backward-bending supply curve of labour. More general examples are considered in ES20011 Intermediate Microeconomics 1 and ES20012 Intermediate Microeconomics 2 4

5 A Necessary Condition is in the nature of a prerequisite: Suppose that a statement p is true only if another statement q is true; then q constitutes a necessary condition of p. We can use symbols to express this relationship as follows: p q which is read: p only if q if p then q p implies q Example 1: Let p be the statement a person is a father and q be the statement a person is male. Then the logical statement p q applies. A person is a father only if he is male, and to be male is a necessary condition for fatherhood. Note, importantly, that the converse is not true: Fatherhood is not a necessary condition for maleness. A Sufficient Condition is used to describe a situation in which a statement p is true if q is true, but p can also be true when q is not true. The truth of q suffices for the establishment of the truth of p, but it is not a necessary condition for p. This case is expressed symbolically by: p q which is read: p if q if q then p q implies p Example 2: Let p be the statement one can get to North America and q be the statement one takes a plane to North America. Then p q. Flying can serve to get one to North America, but since ocean transportation is also feasible, flying is not a prerequisite. Thus we can write p q but not p q. A Necessary and Sufficient Condition is used to describe a situation when p is true if and only if q is true; that is, q is both necessary and sufficient for p. Here we write: p q which is read: p if and only if q p iff q Thus p implies, and is itself implied, by q. 5

6 Example 3: Let p be the statement there are less than 30 days in month and q be the statement it is the month of February. Then p q. To have less than 30 days in the month, it is necessary that it is February. Conversely, the specification of February is sufficient to establish that there are less than 30 days in the month. Thus q is a necessary and sufficient condition for p. In summary, note that in order to prove p q, it needs to be shown that q follows logically from p. Similarly, to p q, requires a demonstration that p follows logically from q. But to prove p q necessitates a demonstration that p and q follow from each other. 6. Ben consumes two goods, x and y, over which he has standard convex preferences and for which he pays exogenous prices of and p y per unit respectively. The government introduces a scheme whereby the first x units of good x that Ben buys is subsidised and any further amounts are taxed. If Ben is made neither better nor worse off as a result of the scheme, then it must be the case that the total amount of tax he pays is equal to the subsidy he receives: TRUE / FALSE Solution: Before the introduction of the scheme, Ben faces budget constraint AB in Figure 5. When the scheme is in operation, the subsidy reduces the effective price of the first x units of good x. Assuming that the market price of good x remains unchanged, the new budget constraint must therefore lay above and have a gentler slope than AB. For quantities of good x in excess of x, the effective price per unit has risen, and so the new budget constraint is steeper than AB. Overall, the new budget constraint is shown as ACD. If Ben s utility is unchanged as a result of the scheme, then it must be the case that the same indifference curve is tangential to both the new and old budget constraints; for example, at E 0 and E 1. As can be seen from Figure 1, this is only possible if his consumption of good x falls, in this case from x 0 to < x 0. Since E 1 lays above AB, it is apparent that Ben is able to afford a larger consumption bundle than before the scheme was introduced i.e. given, Ben is able to afford y 1 of good y with the scheme but only y 1 < y i without the scheme. The expenditure required to purchase this additional quantity of good y, p y y 1 y 1 ( ) must therefore represent the net subsidy received by Ben. Note also that y 1 > y 1 i.e. the net subsidy must be strictly positive. An alternative approach is to focus on the consumption bundle that is affordable under both schemes. The initial and post-scheme budget constraints are Initial budget constraint AB: x + p y y = m Post-scheme budget constraint ACD: ( s)x + p y y = m x x ( + t)x + p y y = m x > x Thus, denoting the bundle where the two budget constraints intersect as (!x,!y ), we have: 6

7 !x + p y!y = m = ( s)x + ( + t) (!x x ) + p y!y!x = ( s)x + + t ( )(!x x )!x = x sx +!x x + t!x tx ( ) sx = t!x x That is, at the intersection the subsidy received must equal the tax paid. But given convex preferences, Ben would prefer a bundle to the left of the kink such that x <!x so that it must be the case that sx > t x x ( ), x < x <!x. y A C y 1 y 1!y E 1 E 0 I 0 0 x!x x 0 D B x Figure 5 7. Ben consumes two goods, x and y, and always devotes 45 per cent of his income to the consumption of the former. His income elasticities of demand for the two goods x and y are respectively E xm = 0.45 and E ym = 0.55: A: TRUE B: FALSE Solution: Consider good x: 7

8 x = 0.45m x = 0.45 m Thus: x m = 0.45 Ε xm x m m x = 0.45 m x Ε xm x m m x = 0.45 m 0.45m ( ) = 1 And similarly for good y. Alternnatively, without calculus: x = 0.45m x = 0.45 m Define: x 0 = 0.45 m 0 and: = 0.45 m 1 Thus: Δx = x 0 = 0.45 ( m 1 m 0 ) Δx = 0.45 Δx Δm = 0.45 Δm 8

9 Thus: Ε mx = Δx Δm m x = 0.45 m x = 0.45 And similarly for good y. m 0.45 m = 1 8. Ben lives in a two-good world. An increase in the price of good 1 will increase his demand for good 2 providing his own-price elasticity of demand for good 1, ( )( p 1 ) > 0, is less than 1: Ε 11 = Δ A: TRUE B: FALSE Solution: The two-good budget constraint is p 1 + x 2 = m From the question, we wish to investigate how and x 2 change as p 1 changes and and m are held constant. Hence m = p x 2 = 0 m = p x 2 = 0 m = Ε x 2 = 0 Thus x 2 = ( Ε 11 1) Since > 0 and > 0, then ( x 2 ) < 0 if Ε 11 < 1. Alternatively without calculus, again write the two-good budget constraint as: p 1 + x 2 = m When the price of good 1 changes, we have: m = p 1 + x 2 9

10 Thus: ( p x ) p 1 + x 2 p 1 p 1 + x 2 x 2 = p 1 + p 1 ( ) = ( m m) ( ) = 0 ( ) = p 1 + p 1 + p 1 p 1 = p 1 ( ) ( p 1 p 1 ) = p 1 Δ Δx Δx 2 = p 1 x 1 1 = Δx p If the change in price and quantity is very small then p 1 p 1 and such that: = Δ p 1 1 Ε 12 = Ε 11 1 Thus, since > 0 and > 0, it must be that case that > 0 iff Ε 11 > 1. Intuitively, if demand for one good is elastic, then an increase in the price of that good will result in a decrease in expenditure on that good. If the money income and the price of the other good are held constant, then it must be the case that spending on, and thus demand for, the other good will increase. 10