Topic 2: Household ehaviour Introductory Microeconomics (ES11) Topic 2: Consumer Theory Exercise 4: Suggested Solutions 1. Which of the following statements is not valid? utility maximising consumer chooses to be at a point of tangency between his budget line and an indifference curve because: (a) this is the highest indifference curve that can be obtained; (b) at any point to the left of the budget line some income would be unused; (c) all combination of goods that lie to the right of his budget line are unreachable, given money income; (d) this point represents the most favourable relative prices; (e) at any other point on the budget line he will gain less utility. (d) 2. Is it possible when there are only two goods, x and y, (a) for y to be both a substitute for x and a normal good; (b) for y to be both a complement to x and an inferior good? 2. (a) is certainly possible; (b) however, is not. See Figures 1 and 2 below in which we have modelled a rise in the price of good x, causing the budget line to pivot from to and moving the consumer from equilibrium point E to. Moving the new budget constraint out parallel until it is just tangent to the consumer s original indifference curve allows us to decompose the movement from E to into a substitution effect (E to E 2 ) and an income effect (E 2 to ). It is apparent that in both cases the substitution effect causes the consumer to move away from good x to good y such that point E 2 must lie to the north-west of E. For y to be both a substitute for x and a normal good we simply require point to lay: (i) above point E in terms of the y-axis, so that more y is purchased as the price of x rises (i.e. so that y is a substitute for x); (ii) below point E 2 in terms of the y-axis, so that more y is demanded as income increases (i.e. so that y is a normal good). This is certainly feasible see Figure 1. For y to be a complement to x, however, we require its consumption to decline with the increase in the price of x implying that point should lay below point E in terms of the y- axis. Since E 2 must lie to the northwest of E, good y must, in this case be a normal good. 1
Topic 2: Household ehaviour y E 2 E I 2 x Figure 1 y E 2 E I 2 x Figure 2 Intuitively, for y to be a substitute for x its consumption must increase with the price of x. If the consumer only has two goods from which to choose, the rise in p x must have induced a 2
Topic 2: Household ehaviour substitution towards y. The rise in p x will also have induced a decline in real income and if y is normal this effect will act to reduce the consumption of y. Providing, however, that the substitution effect is dominant then the net effect will be for the consumption of y to rise as per part (a). For y to complement x, however, its consumption must fall with the rise in p x. Since the substitution effect would have induced an increase in the consumption of y the only way for the consumption of y to decline is for the fall in real income (recall p x has risen) to have induced a fall in the consumption of y, which requires y to be a normal good. 3. Grace spends her entire budget and consumes 19 units of x and 18 units of y. The price of x is twice the price of y. Her income doubles and the price of y doubles, but the price of x stays the same. If she continues to buy 18 units of y, what is the largest number of units of x that she can afford? Consider first Grace s original budget constraint: 19 p x +18p y = m 38p y +18p y = m m = 56 p y since p x = 2 p y. Now, let θ denote the largest number of units of x that Grace can afford if she continues to buy 18 units of good y when the price of good y and her money income double: θ p x + 36 p y = 2m 2θ p y + 36 p y = 112 p y 2θ + 36 = 112 2θ = 76 θ = 38 4. n individual is faced with a choice of buying housing in one of two markets; the private market where he may buy any amount of housing he pleases at the going price, and the local authority market where he will be offered, on a take it or leave it basis, a particular amount of housing. Will he necessarily choose the local authority house? If he does, may we conclude that he will consume more housing than he would have purchased had he been forced to buy on the private market? ssume that the individual is indifferent between public and private housing and prefers more housing to less. lso, she may only buy either public or private housing, and cannot mix the two. The question implies that the supply curve of private housing is perfectly elastic at some price, say p 2, and that the supply of public housing is perfectly inelastic at some quantity, say q 2 see Figure 3. It is apparent that there are only two situations in which the individual will choose to consume public housing. First, if demand is as shown by D 1 then the individual is faced with the choice of either q 1 private housing at the going price p 2, or of q 2 public housing at the price p 1 < p 2. Given that the individual is indifferent between public 3
Topic 2: Household ehaviour and private housing, and given that he prefers more housing to less, she will prefer the latter option and will certainly consume more housing than she would had she been constrained to the private sector. If her demand schedule is of the type D 2, then she will be indifferent between public and private housing, either option offering her q 2 housing at price p 2. If the individual s demand curve lays above D 2, then she would prefer private housing to public housing. If demand is given by D 3, for example, then q 3 private housing at price p 2 is preferable to q 2 public housing at price p 3 housing the former option offering p S Public p 3 p 2 S Private p 1 D 2 D 3 q 1 q 2 q 3 q D 1 Figure 3 5. n individual is known to increase the hours per week he works when his non-wage income is decreased. What will happen to the hours he works if: (a) a proportional income tax is levied on his wage income; (b) a proportional income tax is levied on his total income; (c) a proportional income tax on his wage income is used solely to finance an increase in his non-wage income? Would any of your answers differ if the individual were known to decrease his working hours when his non-wage income decreased? Figures 4a, 4b, and 4c, illustrates the effects of various forms of taxation on an individual s supply of labour. T represents the total hours available to an individual, and y represents the individual's unearned (i.e. non-wage) income. We assume the individual is paid a constant wage per hour, and this gives the slope of the budget constraint. We know that the individual responds to a fall in his non-wage income (i.e. a decline in the intercept T) by increasing (reducing) his hours of work (leisure). This is essentially an income effect, as the portion of the budget constraint shifts inward, and so we can deduce that the individual regards work (leisure) as an inferior (normal) good. (a) proportional income tax levied on the individual's wage income pivots the budget constraint inward from to, as shown in Figure 4a. The substitution effect, from E to E 2, must be positive - a fall in the take home wage (i.e. the opportunity cost or price of 4
Topic 2: Household ehaviour leisure), holding utility constant, will increase the individual s demand for leisure and reduce the individual s supply of labour. Since we assume leisure to be a normal good, the income effect will work in the opposite direction, the individual tending to decrease his demand for leisure (and thus increase his supply of labour) as his real income, holding relative prices constant, is reduced by the imposition of the tax. The final equilibrium must lay along to the left of. However we cannot say whether it will lay to the left or right of E. lthough leisure is a normal good, it may still be Giffen. Y C E E 2 C T L Figure 4a (b) The effects of a proportional income tax on total income are illustrated in Figure 4b. The tax alters the individual's constraint, from T to T, as both wage and non-wage income are reduced. For the same reason as in case (a), we are unable to determine where the final equilibrium will be, although it must be to the left of. 5
Topic 2: Household ehaviour Y C E E 2 C Y T L Figure 4b (c) proportional tax levied on wage income, which is used solely to finance an increase in non-wage income, would have the effect of altering the individual s constraint from T to T in Figure 4c. The original bundle of income and leisure will still be available to the individual, and so the two constraints intersect at E. However, assuming conventional tastes between income and leisure, the individual will move to a point such as, reducing his supply of hours and increasing his consumption of leisure. Y I (1-t)w Y E Y Y w L T L Figure 4c 6
Topic 2: Household ehaviour The proof that the two budget constraints intersect at E, is as follows: ssume that the original wage rate is w such that the individual s pre-tax income is given by: ( ) y = y+ w T L where w is the individual s original level of unearned income. If, after the introduction of the tax and the increase in unearned income the new budget constraint intersects at E, then: ( 1 )( ) y = y+ w t T L Thus: ( ) ( 1 )( ) y+ w T L = y+ w t T L y+ wt w L = y+ wt w L wtt + wtl y = y wtt + wtl ( ) ( ) Δy = y y = wt T L The RHS of the above equation denotes the increase in unearned income and the LHS denotes the tax revenue. Note that the move from E to is essentially a Slutsky ( ) with real income being held compensating variation around the initial bundle L, y constant since the increase in unearned income is being financed by the income tax paid. Since no income effects are involved, the movement from E to, and the subsequent increase (decrease) in the demand for leisure (supply of labour) will definitely occur. Note also that the maximum possible income available to the individual will decline as a result of the policy. Denote maximum income available before and after the scheme as y max and y max respectively. Then: y max = y + w T nd: y max = y + w ( 1 t)t Thus: 7
Topic 2: Household ehaviour Δy max = y max y max Δy max = y + w T y w T + w tt ( ) Δy max = w tt y y Δy max = w tt Δy ( ) Δy max = w tt w t T L Δy max = w tt w tt + w tl Δy max = w tl > 6. It is known that, were his current period s income increased by 1, an individual would increase his current consumption by 8. If the rate of interest is 1%, what will be the effect on the same individual s current consumption of a guaranteed increase in his next period s income of 11? The current period budget constraint may be written as Y = Y 1 + (1/1+r) Y 2. n increase in Y 1 of 1 increases Y by an equivalent amount and so ΔC1 =. 8ΔY. If the rate of interest is 1%, a guaranteed increase in Y 2 of 11 would increase the budget constraint by (1/1+.1) 11 = 1. Current consumption would therefore rise by ΔC =.8ΔY =.8 1 8. 1 = 7. Show (a) that if an individual borrows by selling bonds at a particular value of the rate of interest, a fall in the rate of interest will make him borrow more; (b) that if current and future consumption are both normal goods, then if he borrows by taking out a fixed capital value loan, a fall in the rate of interest will also make him borrow more? (a) Figure 5a shows the effects of a fall in the rate of interest on an individual who borrows by selling bonds at a particular value of the rate of interest. The individual has y 1 current income and y 2 future income, and is assumed to live for two periods only. His preferred consumption point is E, consuming c 1 in the current period and c 2 in the following period. If the individual borrows by selling a bond (i.e. a right to receive a given sum of money in the following period) then he is able to increase his current consumption beyond his current income by an amount (c 1 - y 1 ) in return for guaranteeing a payment of (y 2 c 2 ) to the buyer of the bond in the following period. This allows him to move to his preferred consumption point E. The individual has promised to pay the buyer of the bond (y 2 - c 2 ) in the following period. Therefore, whatever happens to the rate of interest the individual will be able to achieve E. If, after he has sold the bond but before period two, the rate of interest falls then the budget constraint would pivot around E to. However the individual will no longer be in equilibrium at E, since he can increase utility by moving to a point such as, increasing current consumption at the expense of future consumption. He will sell an additional bond (on more favourable terms) in order to finance the desired increased in current consumption. 8
Topic 2: Household ehaviour There is an unambiguous increase in borrowing through a pure substitution effect from E to. c 2, y 2 I y 2 c 2 E y 1 c 1 c 1, y 1 Figure 5a (b) If the individual borrows by taking out a fixed capital loan then the capital value of borrowings does not change with the rate of interest. fall in the rate of interest will pivot the budget constraint around (y 2, y 1 ) from to in Figure 5b. If the both current and future consumption are normal goods, then the income and substitution effects generated by a fall in the rate of interest will tend to reinforce each other. The fall in the rate of interest will increase both real income and the cost of future consumption relative to present consumption, both effects tending to make the individual borrow more. This is shown in Figure 5b. The individual is initially faced with the constraint and maximises utility by borrowing (c 1 - y 1 ). The fall in the rate of interest pivots the constraint around, from to. The substitution effect, from E to E 2, must be positive. The fall in the rate of interest reduces the cost of present relative to future consumption. We cannot say for certain where the final equilibrium will be, although since present and future consumption are both normal goods it must lie within the range indicated along such that borrowing must rise. 9
Topic 2: Household ehaviour c 2, y 2 I y 2 E c 2 E 2 y 1 c 1 c 1, y 1 Figure 5b 1