5. DemandDeriving Demand CurvesDeriving Demand CurvesExample: Deriving Demand CurvesDeriving Demand CurvesSlide Number 6Slide Number 7Deriving Demand Curves GraphicallySlide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Effects of an Increase in IncomeEffects of an Increase in IncomeEffects of an Increase in IncomeConsumer Theory and Income ElasticitiesIncome-Consumption Curve and Income ElasticitiesIncome-Consumption Curve and Income ElasticitiesEffects of a Price IncreaseIncome and Substitution EffectsSlide Number 23Slide Number 24Slide Number 25Slide Number 26Slide Number 27Slide Number 28Slide Number 29Compensated Demand CurveCompensated Demand CurveCompensated Demand CurveRevealed PreferenceSlide Number 34Slide Number 35Slide Number 36Slide Number 37Slide Number 38Slide Number 39Slide Number 40Slide Number 415. Demand • Deriving Demand Curves • Effects of an Increase in Income • Effects of a Price Increase • Revealed Preference 1Deriving Demand Curves • If we hold people’s tastes, their incomes, and the prices of other goods constant, a change in the price of a good will cause a movement along the demand curve. • We saw this in Chapter 2: 2Deriving Demand Curves • In Chapter 3, we used calculus to maximize consumer utility subject to a budget constraint. • This amounts to solving for the consumer’s system of demand functions for the goods. • Example: q1 = pizza and q2 = burritos • Demand functions express these quantities in terms of the prices of both goods and income: • Given a specific utility function, we can find closed-form solutions for the demand functions. 3Example: Deriving Demand Curves • Cobb-Douglas utility function: • Budget constraint: • The demand functions that result from this constrained optimization problem are: • With Cobb-Douglas, quantity demanded of each good is a function of only the good’s own-price and income. 11 2 12(, )aaUq q qq−=11 2 2pq pq Y+=4Deriving Demand Curves • The graph below shows the demand curve for q1, which we plot by holding Y fixed and varying p1. 5Example: Suppose U = x + y. Income, I, is spent only on x and y. Prices are px and py. What are the price consumption functions (curves)? Answers: • When px < py, X* =I/px, Y*=0 • When px > py, X*=0, Y*=I/py • When px = py, any (X*, Y*) satisfying X* + Y* = I/ py. 6X PX 0 PY I/PY I/PX Demand curve for X / when [0, / ] when 0 when x xyy xyxyIp p px Ip p ppp7Deriving Demand Curves Graphically • Allowing the price of the good on the x-axis to fall, the budget constraint rotates out and shows how the optimal quantity of the x-axis good purchased increases. • This traces out points along the demand curve. 8Algebraically, we can solve for the individual’s demand using the following equations: 1. pxx + pyy = I 2. MUx/px = MUy/py … at a tangency. (If this never holds, a corner point may be substituted where x = 0 or y = 0) Interior solution case 9Example: Finding a Demand Curve with an Interior Solution Suppose that U(x,y) = xy. The prices of x and y are px and py, respectively, and income = I. We have: pxx + pyy = I (1) y/px = x/py (2) Substituting the second condition into the budget constraint, we have: pxx + py(px/py)x = I or…x = I/2px Substituting x = I/2px into (2), we have Y=PxX/Py=I/2Py 10Example: Finding a Demand Curve with a Corner Solution Suppose U = xy + 10x. MUx=y + 10 and MUy = x. All the other notation is as in the last example. The indifference curves are convex and intersect the x axis. X Y 11We have: pxx + pyy = I (1) (y+10)/x = px/py (2) (2) => X = Py (y+10)/Px (3) Substituting equation (3) into the budget line, we have the demand curve for y: y = (I-10py)/(2py) • If I > 10py, demand for y is positive (we have interior solution) 12• What if I < 10py? We have a corner point because: At y = 0, it is the case that: MUx/px = (y+10)/ px = 10/ px MUy/py = x/py = (I/px)/py I < 10py implies 10/px >(I/px)/py i.e., MUx/px > MUy/py Hence y=0 will be the optimal choice. 13X Y I>10Py I<10Py A B A: interior solution B: corner solution 14Effects of an Increase in Income • An increase in an individual’s income, holding tastes and prices constant, causes a shift of the demand curve. • An increase in income causes an increase in demand (e.g. a parallel shift away from the origin) if the good is a normal good and a decrease in demand (e.g. parallel shift toward the origin) if the good is inferior. • A change in income prompts the consumer to choose a new optimal bundle. • The result of the change in income and the new utility maximizing choice can be depicted three different ways. 15Effects of an Increase in Income 16Effects of an Increase in Income • The result of the change in income and the new utility maximizing choice can be depicted three different ways. 1. Income-consumption curve: using the consumer utility maximization diagram, traces out a line connecting optimal consumption bundles. 2. Shifts in demand curve: using demand diagram, show how quantity demanded increases as the price of the good stays constant. 3. Engle curve: with income on the vertical axis, show the positive relationship between income and quantity demanded. 17Consumer Theory and Income Elasticities • Recall the formula for income elasticity of demand from Chapter 2: • Normal goods, those goods that we buy more of when our income increases, have a positive income elasticity. • Luxury goods are normal goods with an income elasticity greater than 1. • Necessity goods are normal goods with an income elasticity between 0 and 1. • Inferior goods, those goods that we buy less of when our income increases, have a negative income elasticity. 18Income-Consumption Curve and Income Elasticities • The shape of the income-consumption curve for two goods tells us the sign of their income elasticities. 19Income-Consumption Curve and Income Elasticities • The shape of the income-consumption and Engle curves can change in ways that indicate goods can be both inferior and normal depending on an individual’s income level. 20Effects of a Price Increase • Holding tastes, other prices, and income constant, an increase in the price of a good has two effects on an individual’s demand: 1. Substitution effect: the change in quantity