Chapter 5 – The Theory of Demand • Thus far we have studied supply and demand and their equilibrium • In this chapter we will see how the demand curve arises out of consumer theory • Shifts in demand will be dissected and consumer choices will be investigated further
Chapter 5 – The Theory of Demand • In this chapter we will study: 5.1 Price Consumption Curve 5.2 Deriving the Demand Curve 5.3 Income Consumption Curve 5.4 Engel Curve 5.5 Substitution and Income Effects 5.6 Consumer Surplus
Chapter 5 – The Theory of Demand 5.7 Compensating & Equivalent Variation 5.8 Market Demand 5.9 Labor and Leisure 5.10 Consumer Price Index (CPI)
Demand and Optimal Choice Y (units) At a given income and faced with prices Px and Py,an individual will maximize their utility given the Tangency condition, resulting in a consumption of Good x as seen below: 10 • PX = 4 X (units) 0 XA=2 XB=10
Demand and Optimal Choice Y (units) When the price of x decreases, a consumer will maximize given the new budget line and a new amount of x will be consumed. 10 • • PX = 2 PX = 4 X (units) 0 XA=2 XB=10 20
5.1 The Price Consumption Curve Y (units) The price consumption curve for good x plots all the utility maximization points as the price of x changes. This reveals an individual’s demand curve for good x. 10 Price consumption curve • • • PX = 1 PX = 2 PX = 4 X (units) 0 XA=2 XB=10 XC=16 20
Example: Individual Demand Curve for X PX The points found on the price consumption curve produce the typically downward-sloping demand curve we are familiar with. • PX = 4 • PX = 2 • U increasing PX = 1 X XA=2 XB=10 XC=16
5.2 Deriving the Demand Curve Algebraically, we can derive an individual’s demand using the following equations: Pxx + Pyy = I (budget constraint) MUx/Px = MUy/Py (tangency point) 1) Solve (2) for y 2) Substitute y from (2) into (1) 3) Solve for x
General Example: Suppose that U(x,y) = xy. MUx = y and MUy = x. The prices of x and y are Px and Py, respectively and income = I. • x/Py = y/Px • y = xPx/Py • 2) Pxx + Py(Px/Py)x = I • Pxx + Pxx = I • 3) x= I/2Px
Specific Demand Example Let U=xy, therefore MUx=y and MUy=x Let income=12, Py=1. Graph demand as Px increases from $1 to $2 to $3. Step 1: Pxx+Pyy=I Pxx+y=12 Step 2: MUx/MUy=Px/Py y/x=Px y=Pxx
Demand Example Step 1: Pxx+y=12 Step 2: y=Pxx Step 3: Pxx+Pxx=12 x=6/Px X(1)=6 X(2)=3 X(3)=2
PX Demand Example Maximizing at each point, we arrive at the following demand curve: • PX = 3 • PX = 2 U increasing • PX = 1 X 2 3 6
Y (units) Demand, Choice and Income At a given income, a consumer maximizes using tangency as seen below: 10 • I = 10 X (units) 0 XA=2 XB=10
Y (units) Demand, Choice, and Income When income increases the budget line shifts out, resulting in a new equilibrium 10 • • I=12 I = 10 X (units) 0 XA=2 20 XB=3
5.3 The Income Consumption Curve Y (units) The income consumption curve for good x plots all the utility maximization points as income changes. This is shown by shifting the demand curve for x. 10 Income consumption curve • • • X (units) 0 20
The Income Consumption & Demand Curves Y (units) I=92 I=68 U3 Income consumption curve I=40 U2 U1 0 X (units) 10 18 24 PX $2 I=92 I=68 I=40 10 18 24 X (units)
5.4 The Engel Curve The income consumption curve for good x also can be written as the quantity consumed of good x for any income level. This is the individual’s Engel Curve for good x. When the income consumption curve is positively sloped, the slope of the Engel curve is positive.
Normal and Inferior Goods • If the income consumption curve shows that the consumer purchases more of good x as her income rises, good x is a normal good. • Equivalently, if the slope of the Engel curve is positive, the good is a normal good. •If the income consumption curve shows that the consumer purchases less of good x as her income rises, good x is an inferior good. • Equivalently, if the slope of the Engel curve is negative, the good is an inferior good.
I ($) Engel Curve Graph “X is a normal good” Engel Curve 92 68 40 X (units) 0 10 18 24
Normal and Inferior. • Some goods are normal or inferior over different income levels Example: Kraft Dinner a) at extreme low incomes, Kraft dinner consumption goes up as income increases (because starving is bad) -Kraft Dinner is a normal good at extreme low incomes b) as income rises, people substitute away from Kraft Dinner to “real foods” -Kraft dinner is an inferior good at most incomes
Y (units) Example: I=400 A good can be normal over some ranges and inferior over others I=300 U3 I=200 U2 U1 0 X (units) 13 16 18 I ($) Example: Backward Bending Engel Curve 400 Engel Curve 300 200 X (units) 13 16 18
5.5 Substitution and Income Effects When the price of a good decreases, two effects occur: 1) The good is cheaper compared to other goods; consumers will substitute the cheaper good for more expensive goods 2) Consumers experience an increase in purchasing power similar to an increase in income
Income Effect • Definition: As the price of x falls, all else constant, purchasing power rises. This is called the income effect of a change in price. • The income effect may be negative (normal good) or positive (inferior good).
Substitution Effect: As the price of x falls, all else constant, good x becomes cheaper relative to good y. This change in relative prices alone causes the consumer to adjust his/ her consumption basket. This effect is called the substitution effect. • The substitution effect always is negative Usually, a move along a demand curve will be composed of both effects. • Graphically, these effects can be distinguished as follows…
Y (units) Example: Normal Good: Income and Substitution Effects BL2 Let Px decrease BL1 A • C • • B U2 BLd U1 Substitution Income X (units) 0 XA XB XC
Y (units) Example: Inferior Good: Income and Substitution Effects BL2 “X is an inferior good” • C BL1 A • U2 BLd B • U1 Income Substitution XA XC XB X (units) 0
Finding the DECOMPOSITION Budget Line The decomposition budget line (BLd) that satisfies 2 conditions: 1) The budget line represents a change in the price ratio; it must be parallel to the new budget line (BL2) 2) The budget line must be tangent to the old indifference curve (U1)
Y (units) Budget line slopes Slope of B1 = -Px1/Py Slope of B2 = -Px2/Py Slope of Bd = -Px2/Py BL2 BL1 A • C • BLd • B U2 U1 Substitution Income X (units) 0 XA XB XC
Steps to Finding Substitution and Income Effects: • Using initial prices (and tangency), find • start point (xa, ya) • start utility (Ua) • Using final prices (and tancency), find • end point (xc, yc) • end utility (Uc)
Steps to Finding Substitution and Income Effects: 3) Using final prices and start utility for • decomposition point (xb, yb) 4) Solve: a) Substitution Effect: xB-xA b) Income Effect: xC-xB
Substitution and Income Effect Example: • Suppose U(x,y) = 2x1/2 + y. • MUx = 1/x1/2MUy = 1. • Py = $1 and I = $10. • Suppose that Px = $0.50. What is the (initial) optimal consumption basket? • Tangency Condition: • MUx/MUy = Px/Py • 1/x1/2 = Px
Solving for x: • x = 1/(Px2) • x = 1/(0.5)2 • x = 4 • Substituting, xA = 4 into the budget constraint: • Pxx + Pyy = 10 • 0.5(4) + (1)y = 10 • yA = 8 • UA = 2xA1/2 +yA • UA=2(41/2)+8 • UA=12
2) Suppose that px = $0.20. What is the (final) optimal consumption basket? • Using the demand derived in (a), • x = 1/(Px2) • xc = 1/(0.2)2 • xc = 25 • Pxx + Pyy = 10 • 0.2(25) + (1)y = 10 • yC =5 • UC=2xC1/2+yC • UC=2(251/2)+5 • UC=15
3) What are the substitution and income effects that result from the decline in Px? • Decomposition basket (New Prices, Old Utility) • Tangency: • MUx/MUy = Px/Py • 1/x1/2 = .2 • xb=25 U = 2x1/2 + y 12 = 2(25)1/2 + y yB = 2 Substitution Effect: xB-xA = 25 - 4 = 21 Income Effect: xC-xB = 25 - 25 = 0
Giffen Goods • If a good is so inferior that the net effect of a price decrease of good x, all else constant, is a decrease in consumption of good x, good x is a Giffen good. • For Giffen goods, demand does not slope down. • When might an income effect be large enough to offset the substitution effect? The good would have to represent a very large proportion of the budget. (Some economists debate the existence of Giffen Goods)
Y (units) Example: Giffen Good: Income and Substitution Effects BL2 “X is a Giffen good” C • BL1 A U2 • • B Income U1 Substitution X (units) 0 XC XA XB
5.6 Consumer Surplus • The individual’s demand curve can be seen as the individual’s willingness to pay curve. • On the other hand, the individual must only actually pay the market price for (all) the units consumed. • For example, you may be willing to pay $40 for a haircut, but upon arriving at the stylist, discover that the price is only $30 • The difference between willingness to pay and the amount you pay is the Consumer Surplus
Example: Consumer's Surplus Definition: The net economic benefit to the consumer due to a purchase (i.e. the willingness to pay of the consumer net of the actual expenditure on the good) is called consumer surplus. The area under an ordinary demand curve and above the market price provides a measure of consumer surplus. Note that a consumer will receive more surplus from the first good than from the last good.
Consumer Surplus Equilibrium Or market Price Consumer Surplus Consumer Surplus: The difference between what a consumer is willing to pay and what they pay for each item Price P* D Quantity Q*
Consumer Surplus This calculation Only works for A linear demand curve Efficiency of the Equilibrium Quantity Price Consumer Surplus = area of triangle =1/2bh =1/2(16-8)(10) =40 $16 $8 D Quantity 10
Consumer Surplus Example 1 Craig’s demand for model cars is given by the demand curve P=20-Q. If model cars cost $10 each, how much consumer surplus does Craig have? P=20-Q 10=20-Q 10=Q, Craig buys 10 model cars Consumer Surplus =1/2bh =1/2(10)(20-10) =50
5.7 Compensating and Equivalent Variation • In practice, a consumer’s demand curve is difficult to estimate • Consumer Surplus can be estimated using the optimal choice diagram (budget lines and indifference curves) • Since utility is difficult to measure, consumer surplus is measured through the money needed when a price change occurs:
Compensating and Equivalent Variation COMPENSATING VARIATION: The minimum amount of money a consumer must be compensated after a price increase to maintain the original utility. -The consumer’s ORIGINAL Utility is important. EQUIVALENT VARIATION: The change in money to give a equivalent utility to a price change. -The consumer’s FINAL Utility is important.
Compensating Variation Y (units) -A change in the price of x shifts BL1 to BL2 -Consumption moves from point A to point C -A BL at new prices that would maintain original utility is parallel to BL2 N • A C -NM represents the money required to return a consumer to their original utility, consuming at B • M U2 B • U1 BL2 BL1 X (units) O
Equivalent Variation Y (units) Q -A change in the price of x shifts BL1 to BL2 -Consumption moves from point A to point C -A BL at old prices that would make the equivalent move to the new utility is parallel to BL1 D • N • A • C -NQ represents the money equivalent to a price change, resulting in consumption at D U2 U1 BL2 BL1 X (units) O
Y (units) Compensating and Equivalent Variation Q -Here a price DECREASE occurs -MN is the max amount a consumer would PAY for this price decrease -NQ is the amount a consumer would be PAID instead of a price decrease D • N • A B • M U2 • C U1 BL2 BL1 X (units) O
CV and EV Steps • Calculate ORIGINAL and NEW consumption points that maximize utility. (Use tangency condition.) • Calculate ORIGINAL and NEW utility. • 3a) Compensating Variation: • With ORIGINAL UTILITY and NEW PRICES, minimize expenditure ECV • CV=I-ECV • 3a) Compensating Variation: • With FINAL UTILITY and ORIGINAL PRICES, minimize expenditure EEV • EV=EEV-I
Consumer Surplus Example 2 Hosea’s utility demand for mini xylophones and yogurt (x and y) is represented by U=x2+y2 MUx=2x MUy=2y. Hosea has $20. Mini xylophones originally cost $2 while yogurt cost $1. Due to an outbreak of mad xylophone disease, price of healthy mini xylophones decreased to $1 each. Calculate compensating and equivalent variation.
Consumer Surplus Example 2 Originally (at point A): MUx/Px=MUy/Py 2x/2=2y/1 2X=4Y X=2Y PxX+PyY=I 2X+Y=20 5Y=20 Y=4 X=2Y X=8 After price change (at point C): MUx/Px=MUy/Py 2x/1=2y/1 X=Y PxX+PyY=I X+Y=20 X=10 X=Y Y=10
Consumer Surplus Example 2 Originally (at point A): Y=4 X=8 U(A) =42+82 =16+64 =80 After price change (at point C): X=10 Y=10 U(B) =102+102 =100+100 =200 Decrease in price causes an increase in utility. Here we have maximized utility given a budget constraint.