Utility Maximization and Choice Chapter 4 Utility Maximization and Choice

Consumer Behavior ● Theory of consumer behavior Description of how consumers allocate incomes among different goods and services to maximize their well-being. Consumer behavior is best understood in these distinct steps: Consumer Preferences Utility Budget Constraints Consumer Choices

Key Definitions Budget Set: The set of baskets that are affordable Budget Constraint: The set of baskets that the consumer may purchase given the limits of the available income. Budget Line: The set of baskets that one can purchase when spending all available income.

The Budget Constraint PXX + PYY ≤ I Assume only two goods available: X and Y Price of good x: Px ; Price of good y: Py Income: I Total expenditure on basket (X,Y): PxX + PyY The Basket is Affordable if total expenditure does not exceed total Income: PXX + PYY ≤ I

The Budget Constraint pxx + pyy  I Assume that an individual has I amount of income to allocate between good x and good y pxx + pyy  I The individual can afford to choose only combinations of x and y in the shaded triangle Quantity of y If all income is spent on y, this is the amount of y that can be purchased If all income is spent on x, this is the amount of x that can be purchased Quantity of x

A Budget Constraint Example Two goods available: X and Y I = $10 Px = $1 Py = $2 All income spent on X → I/Px : units of X bought All income spent on Y → I/Py : units of Y bought Budget Line 1: 1X + 2Y = 10 Or Y = 5 – X/2 Slope of Budget Line = -Px/Py = -1/2

• B • • A Budget Constraint Example Y Budget line = BL1 I/PY= 5 A -PX/PY = -1/2 B • • C X I/PX = 10

Budget Constraint Shifting of Budget line: 1) Change in income 2) Change in price of good X 3) Change in price of good Y 4) Change in the price of both good X and Y.

Budget Constraint Location of budget line shows what the income level is. Increase in Income will shift the budget line to the right. More of each product becomes affordable Decrease in Income will shift the budget line to the left. less of each product becomes affordable

A Budget Constraint Example Y Shift of a budget line I = $12 PX = $1 PY = $2 Y = 6 - X/2 …. BL2 If income rises, the budget line shifts parallel to the right (shifts out) If income falls, the budget line shifts parallel to the left (shifts in) 6 5 BL2 BL1 10 12 X

A Budget Constraint Example Y Rotation of a budget line If the price of X rises, the budget line gets steeper and the horizontal intercept shifts in If the price of X falls, the budget line gets flatter and the horizontal intercept shifts out I = $10 PX = $1 PY = $3 Y = 3.33 - X/3 …. BL2 6 BL1 5 3.33 BL2 10 X

Consumer Choice Assume:  Only non-negative quantities "Rational” choice: The consumer chooses the basket that maximizes his satisfaction given the constraint that his budget imposes. Consumer’s Problem: Max U(X,Y) Subject to: PxX + PyY < I

Constrained Consumer Choice Consumers maximize their well-being (utility) subject to their budget constraint. The highest indifference curve attainable given the budget is the consumer’s optimal bundle. When the optimal bundle occurs at a point of tangency between the indifference curve and budget line, this is called an interior solution / Interior optimum.

Interior Optimum MRSx,y = MUx/MUy = Px/Py Interior Optimum: The optimal consumption basket is at a point where the indifference curve is just tangent to the budget line. A tangent: to a function is a straight line that has the same slope as the function…therefore…. MRSx,y = MUx/MUy = Px/Py “The rate at which the consumer would be willing to exchange X for Y is the same as the rate at which they are exchanged in the marketplace.”

• • • Interior Consumer Optimum Y B Preference Direction Optimal Choice (interior solution) IC C • BL X

Interior Consumer Optimum Utility is maximized where the indifference curve is tangent to the budget constraint Quantity of y B U2 Quantity of x

Interior Consumer Optimum

Interior Consumer Optimum: Assumptions U (X,Y) = XY and MUx = Y while MUy = X I = $1,000 P = $50 and P = $200 Suppose Basket A contains (X=4, Y=4) Suppose Basket B contains (X=10, Y=2.5) Question: Is either basket the optimal choice for the consumer? Basket A: MRSx,y = MUx/MUy = Y/X = 4/4 = 1 Slope of budget line = -Px/Py = -1/4 Basket B: MRSx,y = MUx/MUy = Y/X = 1/4

• Interior Consumer Optimum Example Y 50X + 200Y = I U = 25 2.5 X 10 10 Chapter Four

Contained Optimization What are the equations that the optimal consumption basket must fulfill if we want to represent the consumer’s choice among three goods or more than three goods?

Optimization Principle To maximize utility, given a fixed amount of income, an individual will buy the goods and services: that exhaust total income for which the MRS is equal to the rate at which goods can be traded for one another in the marketplace

A Numerical Illustration Assume that the individual’s MRS = 1 willing to trade one unit of x for one unit of y Suppose the price of x = $2 and the price of y = $1 The individual can be made better off trade 1 unit of x for 2 units of y in the marketplace

Second Order Conditions for a Maximum The tangency rule is necessary but not sufficient unless we assume that MRS is diminishing if MRS is diminishing, then indifference curves are strictly convex if MRS is not diminishing, we must check second-order conditions to ensure that we are at a maximum

SOCs for a Maximum The tangency rule is only a necessary condition we need MRS to be diminishing Quantity of y U1 B U2 A There is a tangency at point A, but the individual can reach a higher level of utility at point B Quantity of x

Corner Solutions Individuals may maximize utility by choosing to consume only one of the goods At point A, the indifference curve is not tangent to the budget constraint Quantity of y U2 U1 U3 A Utility is maximized at point A Quantity of x

Consumer Maximization: Corner Solution At point e, the indifference curve is not tangent to the budget constraint Utility is maximized at point e , Quantity r e 25 Y I 3 I 2 Budget line I 1 50 X , Quantity

Example for quasilinear corner solution David is considering his purchases of food (x) and clothing (y). He has a utility function U(x,y) = xy + 10x. His income is I = 10. He faces a price of Px = $1 and a price of clothing Py = $2. What is David’s optimal basket.

Constrained Consumer Choice with Quasilinear Preferences If the relative price of one good is too high and preferences are quasilinear, the indifference curve will not be tangent to the budget line and the consumer’s optimal bundle occurs at a corner solution.

Constrained Consumer Choice with Perfect Substitutes With perfect substitutes, if the marginal rate of substitution does not equal the marginal rate of transformations, then the consumer’s optimal bundle occurs at a corner solution, bundle b.

Example for perfect substitutes Sara views chocolate and vanilla ice cream as perfect substitute. She likes both and is willing to trade one scoop of chocolate for two scoops of vanilla ice cream. Q. if the price of a scoop of chocolate ice cream is (Pc) is three times the price of vanilla (Pv), will Sara buy both types of ice cream? If not which will she buy?

Rational Choice and Marginal Utility 1) If MUx/Px > MUz/Pz, consume more of good x. 2) If MUy/Py > MUz/Pz, consume more of good y.

Constrained Consumer Choice with Perfect Complements The optimal bundle is on the budget line and at the right angle (i.e. vertex) of an indifference curve.

ℒ = U(x1,x2,…,xn) + (I - p1x1 - p2x2 -…- pnxn) The n-Good Case The individual’s objective is to maximize utility = U(x1,x2,…,xn) subject to the budget constraint I = p1x1 + p2x2 +…+ pnxn Set up the Lagrangian: ℒ = U(x1,x2,…,xn) + (I - p1x1 - p2x2 -…- pnxn)

The n-Good Case FOCs for an interior maximum: ℒ/x1 = U/x1 - p1 = 0 ℒ /x2 = U/x2 - p2 = 0 • ℒ /xn = U/xn - pn = 0 ℒ / = I - p1x1 - p2x2 - … - pnxn = 0

Interpreting the Lagrangian Multiplier  is the marginal utility of an extra dollar of consumption expenditure the marginal utility of income

Interpreting the Lagrangian Multiplier At the margin, the price of a good represents the consumer’s evaluation of the utility of the last unit consumed how much the consumer is willing to pay for the last unit

ℒ/xi = U/xi - pi  0 (i = 1,…,n) Corner Solutions A corner solution means that the first-order conditions must be modified: ℒ/xi = U/xi - pi  0 (i = 1,…,n) And If ℒ/xi = U/xi - pi  0, then xi = 0 This means that any good whose price exceeds its marginal value to the consumer will not be purchased

Cobb-Douglas Demand Functions With constant utility Cobb-Douglas utility function: U(x,y) = xy Setting up the Lagrangian: ℒ = xy + (I - pxx - pyy) Let us assume that  +  = 1 FOCs: ℒ/x = x-1y - px = 0 ℒ/y = xy-1 - py = 0 ℒ/ = I - pxx - pyy = 0

Cobb-Douglas Demand Functions First-order conditions imply: y/x = px/py Since  +  = 1: pyy = (/)pxx = [(1- )/]pxx Substituting into the budget constraint: I = pxx + [(1- )/]pxx = (1/)pxx

Cobb-Douglas Demand Functions Solving for x yields Solving for y yields The individual will allocate  percent of his income to good x and  percent of his income to good y

Example of Cobb – Douglas Utility Suppose a person has a utility function given as U = x0.8y0.2 Given that Px = 5 and Py = 3 and Income = 75. Calculate the amount of utility the consumer derives. Suppose there is 1% increase in income, what will be the changed utility.

CES Demand To illustrate cases in which budget shares are responsive to economic circumstances, 3 specific examples of CES Function. 1) Assume that  = 0.5 2) If  = -1 3) If  = -

CES Demand (y/x)0.5 = px/py 1) Assume that  = 0.5 U(x,y) = x0.5 + y0.5 Setting up the Lagrangian: ℒ = x0.5 + y0.5 + (I - pxx - pyy) FOCs: ℒ/x = 0.5x -0.5 - px = 0 ℒ/y = 0.5y -0.5 - py = 0 ℒ/ = I - pxx - pyy = 0 This means that (y/x)0.5 = px/py

CES Demand Substituting into the budget constraint, we can solve for the demand functions In these demand functions, the share of income spent on either x or y is not a constant depends on the ratio of the two prices The higher is the relative price of x, the smaller will be the share of income spent on x

U(x,y) = -x -1 - y -1 = -(1/x) – (1/y) CES Demand 2) If  = -1, U(x,y) = -x -1 - y -1 = -(1/x) – (1/y) Though Utility function sounds strange the marginal utilities are positive and diminishing. First-order conditions imply that y/x = (px/py)0.5 The demand functions are These demand functions are less price responsive. As Px ↑, the individual cuts back only modestly on good x, so the spending on x ↑.

I = pxx + pyy = pxx + py(x/4) CES Demand 3) If  = -, U(x,y) = Min(x,4y) The person will choose only combinations for which x = 4y This means that I = pxx + pyy = pxx + py(x/4) I = (px + 0.25py)x

CES Demand Hence, the demand functions are Share of the person’s budget to good x ↑as Px ↑. Why? X and y must be consumed in fixed proportion.

Indirect Utility Function It is often possible to manipulate first-order conditions to solve for optimal values of x1,x2,…,xn These optimal values will be • x*n = xn(p1,p2,…,pn,I) x*1 = x1(p1,p2,…,pn,I) x*2 = x2(p1,p2,…,pn,I)

Indirect Utility Function We can use the optimal values of the x’s to find the indirect utility function maximum utility = U(x*1,x*2,…,x*n) maximum utility = V(p1,p2,…,pn,I) The optimal level of utility will depend indirectly on prices and income

The Lump Sum Principle Taxes on an individual’s general purchasing power are superior to taxes on a specific good an income tax allows the individual to decide freely how to allocate remaining income a tax on a specific good will reduce an individual’s purchasing power and distort his choices

The Lump Sum Principle A tax on good x would shift the utility-maximizing choice from point A to point B B U2 Quantity of y A U1 Quantity of x

The Lump Sum Principle An income tax that collected the same amount would shift the budget constraint to I’ I’ Quantity of y Utility is maximized now at point C on U3 U3 C A B U1 U2 Quantity of x

The Lump Sum Principle 1) If the utility function is Cobb-Douglas with  =  = 0.5, we know that The indirect utility function is

The Lump Sum Principle If a tax of $1 was imposed on good x the individual will purchase x* = 2 indirect utility will fall from 2 to 1.41 An equal-revenue tax will reduce income to $6 indirect utility will fall from 2 to 1.5

The Lump Sum Principle 2) If the utility function is fixed proportions with U = Min(x,4y), we know that The indirect utility function is

The Lump Sum Principle If a tax of $1 was imposed on good x indirect utility will fall from 4 to 8/3 An equal-revenue tax will reduce income to $16/3 Since preferences are rigid, the tax on x does not distort choices

Expenditure Minimization Dual minimization problem for utility maximization allocate income to achieve a given level of utility with the minimal expenditure the goal and the constraint have been reversed

Expenditure Minimization Point A is the solution to the dual problem E2 provides just enough to reach U1 Quantity of y E3 E3 will allow the individual to reach U1 but is not the minimal expenditure required to do so E2 E1 E1 is too small to achieve U1 U1 Quantity of x E1, E2, E3, are different expenditure level

total expenditures = E = p1x1 + p2x2 +…+ pnxn Expenditure Minimization The individual’s problem is to choose x1,x2,…,xn to minimize total expenditures = E = p1x1 + p2x2 +…+ pnxn subject to the constraint utility = Ū = U(x1,x2,…,xn) The optimal amounts of x1,x2,…,xn will depend on the prices of the goods and the required utility level

minimal expenditures = E(p1,p2,…,pn,U) Expenditure Function The expenditure function shows the minimal expenditures necessary to achieve a given utility level for a particular set of prices minimal expenditures = E(p1,p2,…,pn,U) The expenditure function and the indirect utility function are related both depend on market prices but involve different constraints

Expenditure Minimization with Calculus Minimize expenditure, E, subject to the constraint of holding utility constant: The solution of this problem, the expenditure function, shows the minimum expenditure necessary to achieve a specified utility level for a given set of prices:

Duality The mirror image of the original (primal) constrained optimization problem is called the dual problem. Min PxX + PyY (X,Y) subject to: U(X,Y) = U* where: U* is a target level of utility. If U* is the level of utility that solves the primal problem, then an interior optimum, if it exists, of the dual problem also solves the primal problem.

• Optimal Choice Example: Expenditure Minimization Y Example: Expenditure Minimization • Optimal Choice (interior solution) U = U* Decreases in expenditure level PXX + PYY = E* X

Expenditure Function Two ways to compute expenditure function. 1) Cobb- Douglas case The indirect utility function in the two-good If we interchange the role of utility and income (expenditure), we will have the expenditure function E(px,py,U) = 2px0.5py0.5U

Expenditure Function 2) Fixed – Proportions case The indirect utility function is If we again switch the role of utility and expenditures, we will have the expenditure function E(px,py,U) = (px + 0.25py)U