Consumption -Applications Hicks deco There is another way to decompose the price effect. For simplicity consider a consumer with fixed income m. The Hicks decomposition involves pivoting the budget line around the initial indifference curve rather than the initial bundle. The diagram below illustrates for a price decrease in good 1 The consumer now chooses(as before)ri on the outermost budget line. The Hicks substitution effect is the change in demand due to the relative price change given a constant level of utility rather than income r parallel budget line, the consume no longer afford r-unlike before. The Hicks substitution effect is then given by Arisrf-rf The hicks effect is Arn=s-rf. These can be different in size from the Slutsky effects. The price effect is the This means a Slutsky inferior good is not always a Hicks inferior good and The substitution effect is still always in the opposite direction to the price change. Why Consumption-Applicatiorsa Returning to Slutsky decompositions, consider the cases of perfect complements and perfect substitutes. In both cases there is a decrease in pi-The changes from rf to zh The first graph shows the perfect complement case- it is all income effect. The substitution effect is The second graph shows the perfect substitute case- it is all substitution effect. The income effect is zer Note the Slutsky and Hicks decompositions are identicnl for both cases. Why?
Consumption — Applications 1 Hicks Decomposition • There is another way to decompose the price effect. For simplicity consider a consumer with fixed income m. • The Hicks decomposition involves pivoting the budget line around the initial indifference curve rather than the initial bundle. The diagram below illustrates for a price decrease in good 1. . ................................................................................ . .................................................................................................... ............................................................................................................................................................................................................................................................................................. . . 0 x2 x1 x a 1 x b 1 x c 1 . . . . . . . . . . . . . . . . • The consumer now chooses (as before) x b 1 on the outermost budget line. The Hicks substitution effect is the change in demand due to the relative price change given a constant level of utility rather than income. • Notice the consumer is indifferent between bundles x a and x c . On this new parallel budget line, the consumer can no longer afford x a — unlike before. The Hicks substitution effect is then given by ∆x s 1 = x c 1 − x a 1 . • The Hicks income effect is ∆x n 1 = x b 1 − x c 1 . These can be different in size from the Slutsky effects. The price effect is the same of course. This means a Slutsky inferior good is not always a Hicks inferior good and vice-versa. • The substitution effect is still always in the opposite direction to the price change. Why? Consumption — Applications 2 Some Examples • Returning to Slutsky decompositions, consider the cases of perfect complements and perfect substitutes. ............................................................................................................................................................................................................................................................................................. . . ............................................................................................................................................................................................................................................................................................. . . ............................................................................................................. ............................................................................................................. ................................................................................................................................................................................................................................................................................................................ . . ........................................................................................................................................................ ...... ...... ... ...... ...... ...... ... 0 0 x2 x1 x2 x1 x a 1 x b 1 x a 1 x b 1 • • • . • . . . . . . . . • In both cases there is a decrease in p1. The consumer changes from x a 1 to x b 1 . • The first graph shows the perfect complement case — it is all income effect. The substitution effect is zero. • The second graph shows the perfect substitute case — it is all substitution effect. The income effect is zero. • Note the Slutsky and Hicks decompositions are identical for both cases. Why?
Consumption -Applications Rates of Change Mathematically, the Slutsky decomposition can be written in the following way. Ax1=△r1+△r1+△r · What is the rate of change of demand with respect to price,△r1/△ Suppose price changes from Pf to pi. The old and new amounts of money income required to a"=Pr1+p2I2 and m= PI1+paI2 respectively. Subtracting one from the other gives th quired to enable the just afford bundle z =(r1I2) · For convenience, define△r1=-△r1, the negative of the income effect.△r1/△ m is then the change in demand hen income changes.So△r"/Ap=-△m/△p1=-x1△r/△m. The endowment income effect is given by the change in demand when income changes multiplied by the change in acome when price changes. The first term is Ar" /Am · The second term is△m/△P1.Now,m4=P1+p22andm=1+p2u2.So△m=m2-m=14p Therefore△m/△p1=a1. Finally Ar1△xi,△x,A式△x ApIApI 4p,* ①P +(a1-x1) △m This is the standard way to write the slutsky equation usually with differentials. It can be applied easily. Consumption-Applicatiorsa Tax rebates The government decides to tax beer. They return all the revenue to the consumer making them no worse off The amount of beer bought before the tax was b. The amount of other consumption Make 'other consumption'the numeraire. The price of beer is p. Adding a tax to the price of beer effectively increases the price the consumer faces to p+t. The consumer will change consumption due to the price change and due to their increased income from the tax rebate. Suppose the new bundle is(0, c). The government raises th in tax. Income before the tax was m. After the tax rebate they will get m +tb. The budget line before the tax (p+t)b +e. This equation simplifies to m= pb+c So bundle(b, e)was affordable when the consumer chose bundle(6, c). They cannot prefer it and must be worse off
Consumption — Applications 3 Rates of Change • Mathematically, the Slutsky decomposition can be written in the following way. ∆x1 = ∆x s 1 + ∆x n 1 + ∆x ω 1 • What is the rate of change of demand with respect to price, ∆x1/∆p1? • Suppose price changes from p a 1 to p b 1 . The old and new amounts of money income required to purchase x are ma = p a 1x1 + p2x2 and mb = p b 1x1 + p2x2 respectively. Subtracting one from the other gives the change in money income required to enable the consumer to just afford bundle x = (x1, x2): ∆m = mb − ma = x1(p b 1 − p a 1 ) = x1∆p1 • For convenience, define ∆x m 1 = −∆x n 1 , the negative of the income effect. ∆x m 1 /∆m is then the change in demand when income changes. So ∆x n 1 /∆p1 = −∆x m 1 /∆p1 = −x1∆x m 1 /∆m. • The endowment income effect is given by the change in demand when income changes multiplied by the change in income when price changes. The first term is ∆x m 1 /∆m. • The second term is ∆m/∆p1. Now, ma = p a 1ω1 + p2ω2 and mb = p b 1ω1 + p2ω2. So ∆m = mb − ma = ω1∆p1. Therefore ∆m/∆p1 = ω1. Finally: ∆x1 ∆p1 = ∆x s 1 ∆p1 + ∆x n 1 ∆p1 + ∆x ω 1 ∆p1 = ∆x s 1 ∆p1 + (ω1 − x1) ∆x m 1 ∆m • This is the standard way to write the Slutsky equation — usually with differentials. It can be applied easily. Consumption — Applications 4 Tax Rebates • The government decides to tax beer. They return all the revenue to the consumer — making them no worse off? • The amount of beer bought before the tax was b. The amount of other consumption was c. Make ‘other consumption’ the numeraire. The price of beer is p. Adding a tax to the price of beer effectively increases the price the consumer faces to p + t. • The consumer will change consumption due to the price change and due to their increased income from the tax rebate. Suppose the new bundle is (b 0 , c 0 ). The government raises tb0 in tax. • Income before the tax was m. After the tax rebate they will get m + tb0 . The budget line before the tax was m = pb + c. After the tax it is m + tb0 = (p + t)b 0 + c 0 . This equation simplifies to m = pb0 + c 0 . • So bundle (b 0 , c 0 ) was affordable when the consumer chose bundle (b, c). They cannot prefer it and must be worse off
Consumption -Applications Labour Supply Budgets Suppose a consumer has savings m. Income is spent on c lots of consumption goods the numeraire good. TH nsumer works for L hours, earning wage w per hour. There are only L hours available. The budget line is given by c=m+wL. This can be rewritten as c+u(L-L)=m+w. This has a better interpretation. u, the wage, is the price of leisure and m aL is the endowment income get line slope=一 Leisure=L-L 0 L Notice the consumer cannot have more than l hours of leisure or work Consumption-Applicatiorsa Labour Supply - Choice Since the er values consumption and leisure, both are goods. Indifference curves are easily drawn. L-L Reading from the graph, L-L' is the optimal amount of leisure and L' is the optimal amount of labour supplied. eis the optimal amount of consumption. Notice if Leisure =L, no labour What happens when wages increase
Consumption — Applications 5 Labour Supply — Budgets • Suppose a consumer has savings m. Income is spent on c lots of consumption goods — the numeraire good. The consumer works for L hours, earning wage w per hour. There are only L hours available. • The budget line is given by c = m + wL. This can be rewritten as c + w(L − L) = m + wL. This has a better interpretation. w, the wage, is the price of leisure and m + wL is the endowment income. • Since both are goods plot leisure against consumption to give the budget set. ................................................................................................................................................................................................................................................................................. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 0 c Leisure = L − L L BS Budget line slope = −w • Endowment m m + wL ............. ............. ............. ............. ............. ............. ............. ............. ............. • Notice the consumer cannot have more than L hours of leisure or work. Consumption — Applications 6 Labour Supply — Choice • Since the consumer values consumption and leisure, both are goods. Indifference curves are easily drawn. . .................................................................................................. ................................................................................................................................................................................................................................................................................. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ................................................................................................................................ ...... ...... ... ...... ...... ...... ... ................................................................................................................................ ............... ..................... .......................................................................... ...... ...... ... ...... ...... ...... ... .......................................................................... ............... ..................... 0 c Leisure = L − L L ∗ L − L ∗ • c ∗ • . . . . . ............. ............. ............. ............. ............. ........... • Reading from the graph, L − L ∗ is the optimal amount of leisure and L ∗ is the optimal amount of labour supplied. c ∗ is the optimal amount of consumption. Notice if Leisure = L, no labour is supplied — a corner solution. • It is usually assumed that leisure is a normal good. Leisure time should increase as income increases. • What happens when wages increase?
Consumption -Applications The Labour Supply Curve As wages increase the budget line becomes steeper as in the diagram below. Indifference curves are again suppressed. en> aI Leisure L? Li L the wage rate consumer to supply However as wages rise further from az to w'g, labour supply falls from L? back to Li. A wage increase consumer to supply less labour to the market. The price effect can be decomposed to show how this happens. The endowment income effect of the wage rise outweighs the substitution effect. The consumer chooses to work less. "taking advantage"of the higher wage. chard bending labour supply Consumption-Applicatiorsa Intertemporal Budgets There are two goods, consumption today and consumption tomorrow. The consumer buys c units today and cy tomorrow at a constant price level of 1.(This will be allowed to vary later). The consumer receives an income of mi today and mz tomorrow but can save(and borrow) at an interest rater. The amount that can be consumed next period is c2 <m2+(I+r)(m1-cn) So the budget line is given by Ca Slope =-(1+r) (1+r)m1+mN Slope=-(1+r) m21 m1+m2/(1+r) Present value The first graph shows the budget set with no borrowing. The second shows the budget set when there is
Consumption — Applications 7 The Labour Supply Curve • As wages increase the budget line becomes steeper as in the diagram below. Indifference curves are again suppressed. ..................................................................................................... ... ... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... .... .............................................................................................................................................................................................................................................................................................. ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ........ ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... ................................................................................................................................................................................................................................................................................ ..................................................................................................... ............... ..................... ..................................................................................................... ...... ...... ... ...... ...... ...... ... ........................................................................................................................................................... ............... ..................... .......................................................................................................................................................... ...... ...... ... ...... ...... ...... 0 ... 0 c w Leisure L L ∗ 1 L ∗ 2 L ∗ 1 L ∗ 2 w1 w2 w3 • • • • • • • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ......... ............. ............. ............. ............. ......... • As wages increase from w1 to w2, choice of L rises initially (as might be expected) from L ∗ 1 to L ∗ 2 . An increase in the wage rate causes the consumer to supply more labour. • However as wages rise further from w2 to w3, labour supply falls from L ∗ 2 back to L ∗ 1 . A wage increase causes the consumer to supply less labour to the market. • The price effect can be decomposed to show how this happens. The endowment income effect of the wage rise outweighs the substitution effect. The consumer chooses to work less, “taking advantage” of the higher wage. • A backward bending labour supply curve is generated in the second graph. Consumption — Applications 8 Intertemporal Budgets • The model developed so far can be used to analyse choices over time — intertemporal choice. • There are two goods, consumption today and consumption tomorrow. The consumer buys c1 units today and c2 tomorrow at a constant price level of 1. (This will be allowed to vary later). • The consumer receives an income of m1 today and m2 tomorrow but can save (and borrow) at an interest rate r. • The amount that can be consumed next period is c2 ≤ m2 + (1 + r)(m1 − c1). So the budget line is given by: c1 + c2 1 + r = m1 + m2 1 + r ................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................................................................................................................. . . . . .................... 0 0 • • c2 c1 c2 c1 m1 m2 m1 m2 BS BS Endowment Slope = −(1 + r) Slope = −(1 + r) m1 + m2/(1 + r) (1 + r)m1 + m2 (Future value) (Present value) . . . . . ......... ............. ............. ............. ............. ......... ............. ............. ............. ............. • The first graph shows the budget set with no borrowing. The second shows the budget set when there is
Consumption -Applications Intertemporal Preferences The case of perfect substitutes represents someone who doesn't care whether the today or tomorrow The case of perfect complements represents someone who wants to consume exactly the same amount on each da equal amounts of consumption on each day) are preferred to extremes(a more unequal split over the two days) Consumption-Applicatiorsa Intertemporal Choice Optimal choice takes place as before. The consumer is either a lender or a borrower Ca c2 The first graph shows the case of a lender, the second a borrower. The first agent consumes less than they start with today and more tomorrow, the second agent consumes more than they start with today and less tomorrow Notice how this corresponds with net buyers and sellers when discussing endowments in earlier examples. If the consumer is a lender and r rises they are better off and they continue to be a lender. If the consumer is a borrower and r rises, they need not continue to be a borrower- but if they do they are worse off
Consumption — Applications 9 Intertemporal Preferences • The case of perfect substitutes represents someone who doesn’t care whether they consume today or tomorrow. • The case of perfect complements represents someone who wants to consume exactly the same amount on each day. • Concavity is quite appropriate — the consumer wishes to split consumption over the two days. Averages (more equal amounts of consumption on each day) are preferred to extremes (a more unequal split over the two days). . ................................................................................ . .................................................................................................... ................................................................................................................................................................................................................................................................................ . ........ ........ ..... 0 c2 c1 Consumption — Applications 10 Intertemporal Choice • Optimal choice takes place as before. The consumer is either a lender or a borrower. . ................................................................................................ . ................................................................................................ ................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................................................................................................................. 0 0 • • • • c2 c1 c2 c1 m2 m1 m2 c m1 ∗ 1 c ∗ 2 c ∗ 1 c ∗ 2 . . . . . ......... ............. ............. ............. .............. ......... ............. ............. ............. ............. . . . . . . . . . . . ............. ............. ..... . . . ............. ............. ............. ............. ............. ............. ............. • The first graph shows the case of a lender, the second a borrower. The first agent consumes less than they start with today and more tomorrow, the second agent consumes more than they start with today and less tomorrow. • Notice how this corresponds with net buyers and sellers when discussing endowments in earlier examples. • If the consumer is a lender and r rises they are better off and they continue to be a lender. If the consumer is a borrower and r rises, they need not continue to be a borrower — but if they do they are worse off
Consumption -Applications Inflation So far, consumption prices were assumed to be 1. Allowing for inflation in the price level is straightforward. The price today is 1 and tomorrow is p2. Tomorrow, the an spend p2c2 P2m2+(1+r)( The inflation rate, T, is the increase in the price level, p2=1+r. Therefore the budget line equation is e=m2+1+r Clt + Where 1+p=(1 +r)/(1+r)-p is the real interest rate. r is the nominal interest rat Rearranging to find P gives p=(r-T)/(1+x)Nr-t. Often the second approximation is used. Consumption-Applicatiorsa Present value Ignoring inflation, how much is a pound tomorrow worth in terms of pounds today? What is its present value? A pound today is worth 1 +r tomorrow(from saving it- its future tale), so the present value of a pound tomorrow is 1/(1+r). Budget sets are when"present value of consumption is at most present value of income Present value can be generalised to many periods. If there were 3 days. the present value of an income ms on day 3 rould be ma/(1+r)2.(If ma were saved on day I it would grow to ma(l +r)on day 2, and m3(1+r)2 on day 3). If there were t periods, the present value(Pv) of an income stream mI ...,mt is: Pv=m+1++a+7+…+a+1P=∑a+7 This is the correct way to value income streams. It is easy to generalise to allow for interest rates that change over time. t can be finite or infinite. This formula will appear again and again- remember it
Consumption — Applications 11 Inflation • So far, consumption prices were assumed to be 1. Allowing for inflation in the price level is straightforward. • The price today is 1 and tomorrow is p2. Tomorrow, the consumer can spend p2c2 ≤ p2m2 + (1 + r)(m1 − c1). • The inflation rate, π, is the increase in the price level, p2 = 1 + π. Therefore the budget line equation is: c2 = m2 + 1 + r 1 + π (m1 − c1) ⇐⇒ c1 + c2 1 + ρ = m1 + m2 1 + ρ • Where 1 + ρ = (1 + r)/(1 + π) — ρ is the real interest rate. r is the nominal interest rate. • Rearranging to find ρ gives ρ = (r − π)/(1 + π) ≈ r − π. Often the second approximation is used. Consumption — Applications 12 Present Value • Ignoring inflation, how much is a pound tomorrow worth in terms of pounds today? What is its present value? • A pound today is worth 1 + r tomorrow (from saving it — its future value), so the present value of a pound tomorrow is 1/(1 + r). Budget sets are when “present value of consumption is at most present value of income”. • Present value can be generalised to many periods. If there were 3 days, the present value of an income m3 on day 3 would be m3/(1 + r) 2 . (If m3 were saved on day 1 it would grow to m3(1 + r) on day 2, and m3(1 + r) 2 on day 3). • If there were t periods, the present value (PV ) of an income stream m1, . . . , mt is: PV = m1 + m2 1 + r + m3 (1 + r) 2 + · · · + mt (1 + r) t−1 ⇐⇒ PV = Xt i=1 mi (1 + r) i−1 • This is the correct way to value income streams. It is easy to generalise to allow for interest rates that change over time. t can be finite or infinite. This formula will appear again and again — remember it
