Income AND Substitution PDF

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MICROECONOMICS 1

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INCOME AND SUBSTITUTION SLUTSKY EQUATION In this chapter we are considering how a consumer choice responds to changes in its price. It’s natural to think that when the price of a certain good goes up, the amount demanded will fall but, as we have seen, Giffen goods do not follow this rule. è Giffen goods are primarily a theoretical curiosity but there are some other cases where changes in prices can have rare effects too. Example: we may think that if we have a higher wage, we will work more but… What if our income goes form 10€ x hour to 1000€ x hour? In this case, would we work for more? Would we want to work less and spend money on other things?...

THE SUBSTITUTION AND INCOME EFFECT When the price of a good changes, there are 2 effects: 1. THE RATE AT WHICH YOU CAN EXCHANGE ONE GOOF FOR ANOTHER CHANGES: If good 1 becomes cheaper, you have to give up less units of good 2 in order to get more units of good 1. 2. THE TOTAL PURCHASING POWER OF YOUR INCOME IS ALTERED: If good 1 becomes cheaper, now you can buy more of good 1 with the same income. The purchasing power of your money has gone up even the total amount of income is the same. Ø SUBSTITUTION EFFECT is the first part: the change in demand due to a change in the rate of exchange between two goods. Ø INCOME EFFECT is the second part: the change in demand due to having more purchasing power1. To appreciate the effects of this measure what we will INTO 2 STEPS: 1. Let the relative prices change and 2. adjust income to maintain the purchasing power constant. PIVOT

1

do is to BREAK THE PRICE MOVEMENT Let the purchasing power adjust while holding relative prices constant. SHIFT

Can buy more with the same income

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This is what we call “THE PIVOT SHIFT” operation: 1. The first step is to pivot. The slope of the budget line changes while the purchasing power stays the same. 2. The second movement is where the slope of the new budget line remains constant; now the new relative prices are fixed and what changes is the budget line, the purchasing power.

Economic meanings of the “pivot shift effect” THE PIVOTED LINE: -

The pivoted line is a budget line that has the same slope and the same relative prices as the final budget line. THE MONEY ASSOCIATED WITH BOTH BUDGET LINES IS DIFFERENT . We can still find the original optimal bundle in the pivoted line, so this consumption bundle is just affordable. The purchasing power remains the same as the old optimal is still affordable.

HOW CAN WE CALCULATE HOW MUCH WE HAVE TO ADJUST MONEY INCOME IN ORDER TO KEEP THE OLD BUNDLE JUST AFFORDABLE? We know that our initial budget line is m=p1· x1 + p2· x2 The new budget line (with a change in x1’s price) is m’=p1’· x1+p2· x2 è If we subtract (restar) the second equation from the first it gives us: m’-m=x1· (p1’- p1) è Which is the same as: ∆𝒎 = 𝒙𝟏 ·' ∆𝒑𝟏 o o

If prices go up, we will have to raise income to keep the original bundle affordable. If prices go down, now purchasing power of the consumer goes up so, to keep the original bundle affordable, we will have to decrease consumer’s income.

ALTHOUGH THE ORIGINAL BUNDLE IS STILL AFFORDABLE, AS I SAID, THERE ARE SOME OTHER NEW OPTIONS THAT LEAVE US BETTER OFF.

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THE SUBSTITUTION EFFECT: Optimal Bundle after having pivoted the Budget line

This bundle is the optimal when prices have changed and we change the income so the original bundle is just affordable.

Original optimal bundle SUBSTITUTION EFFECT

The substitution effect can be represented mathematically as: ∆'𝒙𝟏฀𝒔฀ = 𝒙𝟏''𝒑𝟏฀′฀,'𝒎฀′฀฀ − 𝒙𝟏(𝒑𝟏, 𝒎) So basically, it shows how x1 changes when price of good 1 changes and money income changes too. HOW CAN WE FIND THE SUBSTITUTION EFFECT? 1. We have to use the function demand in order to determine the optimal bundle at (p1’,m’) and (p1,m). 2. We have to held p2 constant. The substitution effect is sometimes called: the change in compensated demand. The idea is that consumer is being compensated for a price rise by having enough income given back to him to purchase his old bundle. Of course if the price goes down he is “compensated” by having money taken away from him EXAMPLE: Imagine that a consumer has a demand function for milk of the form: x1=10 + '𝑚฀10𝑝1฀ - His income is 120€ - Price of milk is 3€ x quart o We can say his total demand is 14 quarts. - Now price of milk falls to 2€ per quart. o Now his demand turns 16 quarts. è THE TOTAL CHANGE IN DEMAND IS 2. To calculate the substitution effect: 1. We have to calculate how much income has to change in order to keep the original bundle affordable: ∆𝒎 = 𝒙𝟏 ·' ∆𝒑𝟏 = 14· (2-3)= -14 a. We can see how income has to get reduced. b. Before calculating it, we could make sure that we would obtain a negative number as price of milk has gone down and we have to decrease consumer’s income to keep the original bundle affordable. 2. The level of income has to be: m’=120-14=106 3. We should calculate consumer’s demand for the new m (m’) and the new price (2); the demand is now: 10 + '106฀ 10 · 2฀=15.3 3

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4. The substitution effect now is: 15,3-14=1.3

THE INCOME EFFECT

INCOME EFFECT

We have reached a part where, after pivoting the budget line in order to get “Y”, now it’s turn for the new budget line to shift. This shift can be traduced as: a change in income while relative prices remain constant. - In this case we simply change income m to m’ keeping prices constant at (p1’, p2) INCOME EFFECT: THE SHIFT FROM “Y” TO “Z”

The income effect can be described mathematically as: è Is the change in the demand for good 1 when we change income FROM M’ TO M è We can define the income effect as the change of income we adjusted to make the original bundle just affordable into a new income, adjusted with new price of good 1 which leaves us better off. As we saw, When the price of a good decreases, we need to decrease income in order to keep purchasing power constant. Ø If the good is a normal good, then, this decrease in income will tend to reduce demand. Ø If the good is an inferior good, the decrease in income will lead to an increase in demand. EXAMPLE (using the data given before): è We have that, when price changes, (p1’, m), the demand is 16. è We have that when price and income changes (p1’,m’), the demand is now: 15.3 In order to calculate the income effect, we have to: (p1’,m’)- (p1’, m)=16-15.3=0.7 En resum…hem de calcular: 1. Demanda a (p1,m) 2. Demanda a (p1’,m’) on m’ s’obtindrà a partir del càlcul del que necesita variar m… 3. Demanda a (p1’, m)

SIGN OF THE SUBSTITUTION EFFECT In the case of income effect we have seen that it can be positive or negative depending on whether the good is a normal or an inferior good. In the case of substitution effect… Ø If the price goes down, the change in demand must be nonnegative; p1>p1’ then we must have that: x1(p1’,m’) '≥x1(m,p1) o The reason why this happens is because, as p1 decreases, the demand is higher and x1(p1,m) represents the demand. o

And so:

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Ø If prices go up, we would obtain the contrary effect. WE CAN SHOW IT MATHEMATICALLY! MATHEMATICAL DEMONSTRATION: We have to look at all the points on the pivoted budget line where the consumption of x1 is smaller than “X”. Those points were affordable in the original budget line but weren’t the ones that made consumer better off so he chose “X” instead. è If the consumer is always choosing the best bundle he can afford, then X must be preferred to all the bundles of the yellow area of the pivoted BL. This means that… The optimal choice of the pivoted budget line can’t neither be one of the bundles that lie underneath the original budget line; in this case, as prices have changed, consumer would prefer X or a point to the right of X, as they are better. è As the new bundle of the pivoted line has to be equal to X or bigger, we have to consume an amount of x1 equal or higher than before. o In the graph, the optimal point is Y, which implies a bigger consumption of X1!! As a conclusion we could say that… - X1 always moves opposite to the price movement. We say that the substitution effect is negative, since the change in demand due to the substitution effect is opposite to the change in price: if the price increases, the demand for the good due to the substitution effect decreases.

THE TOTAL CHANGE IN DEMAND The total change in the demand ∆𝑥1 is the change in demand due to the change in price, holding income constant: And we have seen how this change in demand can be discomposed in two changes: SLUTSKY IDENTITY And we could rewrite it as:

PAY SPECIAL ATTENTION TO: ' − ''𝑆𝑈𝐵𝑆𝑇𝐼𝑇𝑈𝑇𝐼𝑂𝑁'𝐸𝐹𝐹𝐸𝐶𝑇: 𝑤𝑒'𝑢𝑠𝑒'𝑝𝑟𝑖𝑐𝑒'𝑎𝑛𝑑 '𝑖𝑛𝑐𝑜𝑚𝑒'𝑐ℎ𝑎𝑛𝑔𝑒𝑑'𝑎𝑛𝑑 '𝑝𝑟𝑖𝑐𝑒'𝑎𝑛𝑑'𝑖𝑛𝑐𝑜𝑚𝑒'𝑢𝑛𝑐ℎ𝑎𝑛𝑔𝑒𝑑'' −'𝐼𝑁𝐶𝑂𝑀𝐸'𝐸𝐹𝐹𝐸𝐶𝑇:'𝑤𝑒'𝑢𝑠𝑒'𝑝𝑟𝑖𝑐𝑒'𝑎𝑛𝑑'𝑖𝑛𝑐𝑜𝑚𝑒'𝑢𝑛𝑐ℎ𝑎𝑛ℎ𝑎𝑛𝑔𝑒𝑑'𝑎𝑛𝑑'𝑝𝑟𝑖𝑐𝑒'𝑐ℎ𝑎𝑛𝑔𝑒𝑑฀฀ We can use what we know about income and substitution effects to determine the sign of the total demand change: Ø Substitution effect must be always negative. (opposite the change to price) Ø Income effect can be positive or negative. o THE TOTAL EFFECT CAN BE POSITIVE OR NEGATIVE, DEPENDING ON THE INCOME EFFECT SIGN. 5

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Ø If we have a normal good, income and substitute effect go the same way; What we want to say is that: • If price goes up, the demand will go down due to the substitute effect. • If price goes up, we will have less purchasing power so it will be like a decrease in income, which will imply a reduction in demand too. But there are some other rare cases… -

An increase in price implies a decrease in the purchasing power, which, for a NORMAL GOOD, would imply a decrease in demand. If we have an INFERIOR GOOD, it may happen that the income effect outweighs the substitution effect: the total change in demand associated with the price is positive.

This would be the case where:

If the income effect is large enough, the total change in demand could be positive!

If this happened, we would be talking about a Giffen good!!! An increase in price would turn out into an increase in demand. But… the Slutsky identity shows that this kind of perverse effect can only occur in inferior goods; if a good is normal, then income and substitution effects reinforce each other. Then, a giffen good must be an inferior good but… this is not true as there are goods that may be inferior but not necessarily giffen the income effect not only has to be of the “wrong” sign, it also has to be large enough to outweigh the “right” sign of the substitution effect. This is why Giffen goods are so rarely observed in real life: they would not only have to be inferior goods, but they would have to be very inferior.

• In Figure 8.3A, the income effect is large enough to outweigh the substitution effect and produce a Giffen good.

• In Figure 8.3B, the income effect is smaller, and thus good 1 responds in the ordinary way to the change in its price. We can see how, even the price of p1 has decreased and income should become smaller so demand would go down, in the final line, we still choose an amount of x1 which is bigger than the original optimum.

RATES OF CHANGE As we have seen, income and substitute effects can be described: 1. GRAPHICALLY 2. ALGEBRICALLY; by the Slutsky identity (the total change in demand is the substitution plus the income effect).

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When we talk about the Slutsky identity, we refer to absolute changes but it is more common to express it in terms of rates of change. When we express the Slutsky identity in rates of change, it turns out convenient to define:

Given this definition, the Slutsky identity becomes:

And if we divide each side of the identity by p1 we have:

As we have a change in income in the numerator, following the rule of (*), we should have an income change in the denominator too.

This is the rate of change of the demand when prices change and income is adjusted to keep the old bundle affordable. (*) In this case we have an increase in price in the numerator & in the denominator.

HOW CAN WE ADD AN INCOME CHANGE IN THE DENOMINATOR?

This is the formula we use to calculate how much we have to adjust money income in order to keep the old bundle just affordable.

And we can interpret each term of the formula as: The rate of change in demand as prices change holding income fixed

The rate of change in demand as prices change adjusting income to keep the original bundle affordable; the substitution effect

The rate of change in demand holding new prices fixed and adjusting income to the new optimum; the income effect.

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Income effect by itself is composed of two pieces: - How demand changes as income changes, times the original level of demand

(this last term, we already know that comes from the expression: ∆𝒎 = 𝒙𝟏 ·' ∆𝒑𝟏 ) -

So the final formula of change in demand due to the income effect reduces to:

THE LAW OF DEMAND The law of demand comes directly from the Consumer Theory. As we have seen, despite the concerns, this theory lets us know: 1. An optimizing consumer must satisfy the Strong Axiom of revealed Preferences. 2. Any price change can be divided in two effects: a. Substitution effect: which is always negative b. Income effect: which sign depends on whether the good is a normal or an inferior good. Ø Consumer theory shows us how demand and prices interact. THE LAW OF DEMAND: If the demand for a good increases when income increases, then the demand for that good must decrease when its price increases.

EXAMPLES OF INCOME AND SUBSTITUTION EFFECTS Perfect complements When we pivot the budget line around the chosen point, the optimal choice at the new budget line is the same as at the old one—this means that the substitution effect is zero. è THE CHANGE IN DEMAND IS DUE ENTIRELY TO THE INCOME EFFECT.

Perfect substitutes In this case we see that, when we pivot the budget line, the demand jumps from the vertical axis to the horizontal axis so now, the optimal choice is some point of the horizontal axis. In this case, there is no shifting left to do. Income effect is zero. Because we change directly from consuming one good (2) to good (1). è THE CHANGE IN DEMAND IS DUE ENTIRELY TO THE SUBSTITUTION EFFECT.

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Quasilinear preferences In this case, we know that in quasilinear preferences, a shift in income causes no change in the demand for good 1. This means that the income effect is 0. è THE ENTIRE CHANGE IS DUE TO THE SUBSTITUTION EFFECT. We see that even with the shift, we are still consuming the same amount of good 1.

Example: Rebating a tax

(x,y) is in a higher indifference curve

The tax makes good 1 more expensive, and the rebate increases money income. The original bundle is no longer affordable, and the consumer is definitely made worse off. The consumer’s choice under the tax-rebate plan involves consuming less gasoline and more of “all other goods.” What can we say about the amount of consumption of gasoline? The average consumer could afford his old consumption of gasoline, but because of the tax, gasoline is now more expensive. In general, the consumer would choose to consume less of it.

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ANOTHER SUBSTITUTION EFFECT: We know that substitution effect can be described as: “the change in demand when prices change but consumer’s purchasing power remains the same so that the original bundle is still affordable” – Slutsky substitution effect But… there is another way to define the substitution effect which is also useful; we are talking about “the Hicks substitution effect”.

THE HICKS SUBSTITUION EFFECT: In this case, after good 1 changes its prices, we pivot the new budget line around the indifference curve where we find the original optimal. The new budget line presents the same relative prices as the final budget line but different income. • In the pivoted budget line, the consumer does not have enough purchasing power to get his original bundle, but he can get a bundle that gives him the same level of satisfaction. The consumer now is indifferent between buying his original bundle and the pivoted optimal.

PROPERTIES OF THE HICKS SUBSTITUTION EFFECT: Ø It keeps utility constant rather than keeping purchasing power constant. Ø the Slutsky substitution effect gives the consumer enough money to get back his original bundle (we find m’, which gives us enough monetary income to keep the original bundle just affordable); the Hicks effect gives the consumer enough money to get back to his old indifference curve. Ø The Hicks substitution effect has to be also negative.

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COMPENSATED DEMAND CURVES We have seen how the quantity demanded changes as a price changes in three different contexts: 1. Holding income fixed (the standard case) -> STANDARD DEMAND CURVE 2. Holding purchasing power fixed (the Slutsky substitution effect) -> SLUTSKY DEMAND CURVE. 3. Holding utility fixed (the Hicks substitution effect)-> HICKS DEMAND CURVE = THE COMPENSATED DEMAND CURVE: the consumer is “compensated” for the price changes, and his utility is the same at every point on the Hicksian demand curve. As we have seen: - Slutsky and hicks demand curve are downward sloping curves. - Standard demand curve is also downward sloping if we refer to normal goods. o The analysis of giffen good shows us that sometimes, the standard demand curve can be upward sloping too.

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