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Intermediate Microeconomics W3211
Lecture 6: Introduction The Effect of Changing Prices
Columbia University, Spring 2016
Mark Dean: [email protected] 2
3 4 The Story So Far…. Today’s Aims
• We have now learned how to solve the consumer’s 1. Discuss how demand for a good is affected by a change in its problem own price Giffen Goods • Used this to identify the demand function Income and substitution effects • How the consumer’s choice depends on prices and income Compensated demand and the Slutsky equation Varian Ch. 6, and 8 Feldman and Serrano Ch. 4 • Discussed how demand changes with income • Introduced the concept of the income elasticity of demand
2. Discuss how demand for a good changes with the price of other goods complements and substitutes Varian Ch. 6, Feldman and Serrano Ch. 4
6 Own-Price Changes
We have seen how demand changes with income
How about with prices? Demand and Own Price Changes Suppose only p1 increases, from p1’ to p1’’ and then to p1’’’. 1: Introduction
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7 8 Own-Price Changes Own-Price Changes
Fixed p2 and y. Fixed p2 and y. x2 x2
p1x1 + p2x2 = y p1x1 + p2x2 = y
p1 = p1’ p1 = p1’
p1= p1’’
x1 x1
9 10 Own-Price Changes Own-Price Changes
Fixed p2 and y. Fixed p2 and y. x2
p1x1 + p2x2 = y
p1 = p1’
p1= p = p ’’ p1’’’ 1 1
x1
11 p1 12 Own-Price Changes Own-Price Changes
Fixed p2 and y. Fixed p2 and y.
p1 = p1’ p1 = p1’
p1’
’ x1*(p1 ) x1*
x1*(p1’) x1*(p1’)
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p1 13 p1 14 Own-Price Changes Own-Price Changes
Fixed p2 and y. Fixed p2 and y.
p1 = p1’’ p1 = p1’’
p1’ p1’
’ ’ x1*(p1 ) x1* x1*(p1 ) x1*
x1*(p1’) x1*(p1’)
x1*(p1’’)
p1 15 p1 16 Own-Price Changes Own-Price Changes
Fixed p2 and y. Fixed p2 and y.
p1 = p1’’’
p1’’ p1’’
p1’ p1’
’ ’ x1*(p1 ) x1* x1*(p1 ) x1*
x1*(p1’’) x1*(p1’’)
x1*(p1’) x1*(p1’)
x1*(p1’’) x1*(p1’’)
p1 17 p1 18 Own-Price Changes Own-Price Changes
Fixed p2 and y. Fixed p2 and y. p1’’’
p1 = p1’’’
p1’’ p1’’
p1’ p1’
’ ’ x1*(p1 ) x1* x1*(p1’’’) x1*(p1 ) x1*
x1*(p1’’) x1*(p1’’)
x1*(p1’’’) x1*(p1’) x1*(p1’’’) x1*(p1’)
x1*(p1’’) x1*(p1’’)
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p 19 p 20 1 Ordinary 1 Ordinary Own-Price Changes demand curve Own-Price Changes demand curve Fixed p2 and y. p1’’’ for commodity 1 Fixed p2 and y. p1’’’ for commodity 1
p1’’ p1’’
p1’ p1’
’ ’ x1*(p1’’’) x1*(p1 ) x1* x1*(p1’’’) x1*(p1 ) x1*
x1*(p1’’) x1*(p1’’)
x1*(p1’’’) x1*(p1’) x1*(p1’’’) x1*(p1’)
x1*(p1’’) x1*(p1’’)
p 21 22 1 Ordinary Own-Price Changes Own-Price Changes demand curve Fixed p2 and y. p1’’’ for commodity 1
The curve containing all the utility-maximizing bundles traced p ’’ out as p1 changes, with p2 and y constant, is the p1- price offer 1 curve. p1 price
offer p1’ The plot of the x1-coordinate of the p1- price offer curve against p is the ordinary demand curve for commodity 1. curve 1 ’ x1*(p1’’’) x1*(p1 ) x1*
x1*(p1’’)
x1*(p1’’’) x1*(p1’)
x1*(p1’’)
23 24 Own-Price Changes Own-Price Changes
What does a p1 price-offer curve look like for Cobb-Douglas What does a p1 price-offer curve look like for Cobb-Douglas preferences? preferences?
Take ab Ux(,12 x ) x 12 x .
Then the ordinary demand functions for commodities 1 and 2 are
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25 26 Own-Price Changes Own-Price Changes * a y xppy(,,) p 112 1 Ordinary ab p1 and demand curve * b y for commodity 1 xppy212(,,) . is ab p2 * ay x1 ()abp 1
x1*
27 28 Own-Price Changes Perfect Complements
What does a p1 price-offer curve look like for a perfect- What does a p1 price-offer curve look like for a perfect- complements utility function? complements utility function?
Ux(,12 x )min, x 12 x . Then the ordinary demand functions for commodities 1 and 2 are
29 30 Own-Price Changes Own-Price Changes ** y ** y xppy112(,,) xppy 212 (,,) . xppy112(,,) xppy 212 (,,) . pp12 pp12
With p2 and y fixed, higher p1 causes smaller x1* and x2*.
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31 32 Own-Price Changes Own-Price Changes ** y ** y xppy112(,,) xppy 212 (,,) . xppy112(,,) xppy 212 (,,) . pp12 pp12
With p2 and y fixed, higher p1 causes With p2 and y fixed, higher p1 causes smaller x1* and x2*. smaller x1* and x2*. ** y ** y As pxx1120,. As pxx1120,. p2 p2 ** As pxx112,. 0
33 p1 34 Own-Price Changes Own-Price Changes
Fixed p2 and y. Fixed p2 and y. x2 x2 p1 = p1’ y/p2 * x2 p1’ y ’ pp12 * y x1* x1 pp12’
y x x* x 1 1 ’ 1 pp12
p1 35 p1 36 Own-Price Changes Own-Price Changes
Fixed p2 and y. Fixed p2 and y. p1’’’ x2 x2 p1 = p1’’ p1 = p1’’’ y/p2 p1’’ y/p2 p1’’
p ’ p ’ * 1 1 x2 x* y y x * 2 * y x * x* 1 x 1 pp’’ 1 y 1 ’’’ 12 pp12’’ pp12 ’’’ pp12 y y x* x x* x 1 ’’ 1 1 ’’’ 1 pp12 pp12
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p 37 38 1 Ordinary Ordinary Goods Own-Price Changes demand curve Fixed p2 and y. p1’’’ for commodity 1 x2 is y So far, all the demand curves have been downward sloping x* . y/p p ’’ 1 2 1 pp12 Question: does the quantity demanded of a good have to * increase as a price decreases? x2 p1’ Surely it does? y Sigh. pp12 y x1* p A good is called ordinary if the quantity demanded of it always 2 increases as its own price decreases.
* y x1 x1 pp12
39 40 Ordinary Goods Ordinary Goods
Fixed p2 and y. Fixed p2 and y. x2 x2
p1 price offer curve
x1 x1
41 42 Ordinary Goods Giffen Goods
Fixed p2 and y. Downward-sloping demand curve x2 p1
If, for some values of its own price, the quantity demanded of a p1 price good rises as its own-price increases then the good is called offer Giffen. Good 1 is curve ordinary
x1*
x1
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43 44 Ordinary Goods Ordinary Goods
Fixed p2 and y. Fixed p2 and y. x2 x2 p1 price offer curve
x1 x1
45 46 Ordinary Goods Giffen goods Fixed p and y. Demand curve has 2 a positively Ok, so some goods are weird! x2 p price offer p1 sloped part 1 Prices increases and so does demand!!! curve What is going on? Increase in the price of the good should cause the consumer to switch Good 1 is to the other good and consume less (substitution effect) Giffen However, price increases make the consumer poorer (income effect) If the good is inferior, they will consume more If they are really inferior, then the income effect may outweigh the substitution effect
x1* Sounds crazy, and realistically this is probably an extreme effect
Could you think of some examples? x1 Potato in XIX century Ireland Rice/Sorghum in some developing countries
48 More about price changes
In order to understand the effect of price changes on demand, we can “decompose them”
What happens when a commodity’s price decreases?
1. Substitution effect: the commodity is relatively cheaper, so Demand and Own Price consumers substitute it for now relatively more expensive other commodities. Changes 2: Income and Substitution Effects 2. Income effect: the consumer’s budget of $y can purchase more than before, as if the consumer’s income rose, with consequent income effects on quantities demanded.
We will now analyze these two effects separately 47
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49 50 Effects of a Price Change Effects of a Price Change
Consumer’s budget is $y Consumer’s budget is $y x x 2 2 Lower price for commodity 1 y Original choice y pivots the constraint outwards p2 p2
x1 x1
51 52 Effects of a Price Change Effects of a Price Change
Consumer’s budget is $y x2 Lower price for commodity 1 Changes to quantities demanded due to this ‘extra’ income y pivots the constraint outwards are the income effect of the price change ’ p2 Now only $y x1 53 54 Pure Substitution Effect Pure Substitution Effect Only x2 We will now calculate these two effects separately How? x2’ We ‘break up’ the movement of the budget line in 2 First, we move the budget line with the new slope but so that the original bundle was just affordable This is gives us a new budget set We compute the bundle that would be chosen by the decision maker for this set ’ x1 x1 9 3/1/2016 55 56 Pure Substitution Effect Only Pure Substitution Effect Only x2 x2 We break up the movement of the budget set in two steps x2’ x2’ ’ ’ x1 x1 x1 x1 Budget Set with new prices but where Previous bundle is just affordable 57 58 Pure Substitution Effect Only Pure Substitution Effect Only x2 x2 Optimal choice in this new budget set x2’ x2’ x2’’ x2’’ ’ ’ x1 x1’’ x1 x1 x1’’ x1 Budget Set with new prices but where Previous bundle is just affordable 59 60 Pure Substitution Effect Only Pure Substitution Effect Only x2 x2 Lower p1 makes good 1 relatively Lower p1 makes good 1 relatively cheaper and causes a substitution cheaper and causes a substitution from good 2 to good 1 from good 2 to good 1. (x1’,x2’) (x1’’,x2’’) is the pure substitution effect. x2’ x2’ x2’’ x2’’ ’ ’ x1 x1’’ x1 x1 x1’’ x1 10 3/1/2016 61 62 And Now The Income Effect And Now The Income Effect x2 x2 The income effect is (x1’’,x2’’) (x1’’’,x2’’’). x2’ (x1’’’,x2’’’) x2’ (x1’’’,x2’’’) x2’’ x2’’ ’ ’ x1 x1’’ x1 x1 x1’’ x1 It is just like a change in income! 63 So: 64 The Overall Change in Demand We can separate the change due to a price change into: x2 The change to demand due to lower p1 is the sum of the income and substitution effects, Substitution effect: computed as the change to the optimal choice (x1’,x2’) (x1’’’,x2’’’). in a budget set with new prices but in which old bundle is just affordable x2’ (x1’’’,x2’’’) Income effect: the change from that optimal choice to the new one – just like a change in income x2’’ Total effect is just the sum of the two effects ’ x1 x1’’ x1 What are the signs of these effects?65 66 Pure Substitution Effect Only Are these effects always positive, negative ? x2 Can the substitution effect First, how about the substitution effect? mean that the optimal bundle moves to x’’? It must be always positive: lower price lead to more demand in the substitution effect x2’’ x ’ Why? See graph 2 x ’’ ’ 1 x1 x1 11 3/1/2016 67 What are the signs of these effects?68 Pure Substitution Effect Only Are these effects always positive?ky negative, ? x2 Can the substitution effect mean that the optimal First, how about the substitution effect? bundle moves to x’’? X’’’ No: otherwise x’’ weakly It must be always positive: lower price lead to more demand in the x2’’ substitution effect x ’ better than x’ but (by 2 monotonicity)a bundle Why? See graph like x’’’ is strictly better than x’’ This means x’ cannot have been optimal in the How about the income effect? fist budget set It’s just like a change in income Does demand go up or down with income? x ’’ ’ 1 x1 x1 If good is inferior -> negative If good is normal -> positive What are the signs of these effects?69 Example, normal good 70 So: x2 Good 1 is normal because Substitution is always positive and income could be positive or higher income increases negative (in the case of inferior goods) demand Total effect is always the sum x2’ (x1’’’,x2’’’) Means that: for normal goods, total effect is positive For inferior goods: if income effect is stronger than substitution x2’’ effect, then total effect can be negative This is what happens with Giffen goods Question: Can Giffen goods be normal? ’ x1 x1’’ x1 Effects for Normal Goods 71 Example: Inferior Goods 72 x2 Good 1 is normal because x2 higher income increases demand, so the income and substitution x ’ (x ’’’,x ’’’) effects reinforce x ’ 2 1 2 each other. 2 x2’’ ’ ’ x1 x1’’ x1 x1 x1 12 3/1/2016 Example: Inferior Goods 73 Example: Inferior Goods 74 x2 x2 x2’ x2’ ’ ’ x1 x1 x1 x1 Example: Inferior Goods 75 Example: Inferior Goods 76 The pure substitution effect is as for a normal x2 x2 good. But … x2’ x2’ x2’’ x2’’ ’ ’ x1 x1’’ x1 x1 x1’’ x1 Example: Inferior Goods 77 Example: Inferior Goods 78 The pure substitution effect is as for a normal The pure substitution effect is as for a normal x2 x2 good. But, the income effect is good. But, the income effect is in the opposite direction. in the opposite direction. Good 1 is income-inferior (x ’’’,x ’’’) (x ’’’,x ’’’) 1 2 1 2 because an ’ ’ x2 x2 increase to income causes demand to x2’’ x2’’ fall. ’ ’ x1 x1’’ x1 x1 x1’’ x1 13 3/1/2016 Example: Inferior Goods 79 Effects for Giffen Goods 80 x2 x2 The overall changes to demand are A decrease in p1 causes the sums of the substitution and quantity demanded of income effects good 1 to fall. (x1’’’,x2’’’) x2’ x2’’ x2’ ’ ’ x1 x1’’ x1 x1 x1 Effects for Giffen Goods 81 Effects for Giffen Goods 82 x2 x2 A decrease in p1 causes A decrease in p1 causes quantity demanded of quantity demanded of good 1 to fall. good 1 to fall. x2’’’ x2’’’ x2’ x2’ x2’’ ’’’ ’ ’’’ ’ x1 x1 x1 x1 x1 x1’’ x1 Substitution effect Income effect 84 The Slutsky Equation There is another, more elegant way of separating out the income and substitution effects You will like it…. But it means we will have to go through some new concepts a Demand and Own Price little quickly Changes (Sorry) 3: The Slutsky Equation In my defense, these concepts which may seem a little weird now, will seem very sensible when we start talking about firms 83 14 3/1/2016 85 86 Ordinary and Compensated Ordinary and Compensated Demand Demand Here is the standard consumer problem Here is a related problem, sometimes called the ‘dual’ problem 1. CHOOSE a consumption bundle 1. 2. IN ORDER TO MAXIMIZE preferences CHOOSE a consumption bundle 3. SUBJECT TO the budget constraint 2. IN ORDER TO MINIMIZE expenditure 3. This gives rise to demand functions: amount of the good SUBJECT TO utility being equal to some u* consumed given prices and income , Can also give rise to the indirect utility function: the maximum utility that can be achieved given prices p and income y , , , , 87 88 The Dual Problem The Dual Problem x2* u=u* u=u* x1* 89 90 Ordinary and Compensated Relationship between the Two Demand Problems Here is a related problem, sometimes called the ‘dual’ problem The previous picture should give you the idea that there is a strong relationship between the two problems 1. CHOOSE a consumption bundle Fact: 2. IN ORDER TO MINIMIZE expenditure , e , = , 3. SUBJECT TO utility being equal to some u* In words: This gives rise to compensated demand functions: amount of the good consumed given prices and utility Figure out the amount of I would use to minimize expenditure while achieving utility u , Figure out the amount of expenditure e I need to achieve u Can also give rise to the expenditure function: the minimum Figure out the amount of I would use to maximize utility if you expenditure that can be achieved given prices p and utility u gave me e These two demands are the same e , , , 15 3/1/2016 91 92 The Dual Problem Relationship between the Two Problems , e , = , We can use this to split out income and substitution effects , , , , = Where , Rearranging x2* , , , , = u=u* This already looks like an income and substitution effect However, we can do better x1* Makes sense: Same bundle minimizes y with respect to u and maximizes u with respect to y 93 94 The Envelope Theorem The Envelope Theorem , What about ? , , Why is + 0? Well, we know that e , , , , so First notice that, because , , , , , , = , + 0 Claim: these last two terms are zero, Next, take the tangency condition , This means = , Follows because is the solution to an optimization problem = This is the ‘envelope theorem’ You won’t get this on your first time through Make sure you review Implies Take a deep breath.. 95 96 The Envelope Theorem The Envelope Theorem So This tells us that 0 , = , And so Implies 0 , , , , = Implies 0 Becomes Implies 0 , , , = , Assuming strict monotonicity =0 This is the Slutsky Equation 16 3/1/2016 97 Hicksian Demand and 98 The Slutsky Equation Increase in Price of Good 1 This is the Slutsky Equation , , , = , Second term is the income effect First term is the ‘substitution effect’ Impact of prices on Hicksian Demand i.e. while keeping utility constant (Slightly different from the previous substitution effect – keeping u=u* income constant) Always negative Hicksian Demand and 99 Hicksian Demand and 10 0 Increase in Price of Good 1 Increase in Price of Good 1 u=u* u=u* 10 Cross-Price Effects 2 What happens to demand of one good as I change the price of the other good? Think about it: If the price of cars increases does demand for petrol increase or decrease? If the price of cars increases, does demand for bicycles increase or Demand and Own Price decrease? Changes The first is the case of complements: things that are generally consumed together The second a case of substitutes: things that are generally consumed instead of each other 10 1 17 3/1/2016 10 10 Cross-Price Effects 3 Cross-Price Effects 4 Formally A perfect-complements example: Price increase for commodity 2 increases demand for commodity * y 1 then commodity 1 is a gross substitute for commodity 2. x1 Same as saying the cross-elasticity of demand is positive pp , , so 12 0 x* y 1 0. Price increase for commodity 2 decreases demand for 2 commodity 1 then commodity 1 is a gross compliment for p2 pp commodity 2. 12 Same as saying the cross-elasticity of demand is negative Therefore commodity 2 is a gross , , 0 complement for commodity 1. That’s why it’s called perfect complement 10 10 Cross-Price Effects 5 Cross-Price Effects 6 A Cobb- Douglas example: A Cobb- Douglas example: by by x* x* 2 ()abp 2 ()abp so 2 so 2 x* 2 0. p1 Therefore commodity 1 is neither a gross complement nor a gross substitute for commodity 2. 10 Summary 8 Today we have done the following 1. Discuss how demand for a good is affected by a change in its own price Giffen Goods Income and substitution effects Compensated demand and the Slutsky equation 2. Discuss how demand for a good changes with the price of Summary other goods complements and substitutes 10 7 18