Chapter 4 UTILITY MAXIMIZATION AND CHOICE 1
Our Consumption Choices Suppose that each month we have a stipend of $1250. What can we buy with this money? 2
What can we buy with this money? Pay the rent, 750 $. Food at WholeFoods, 300 $ Clothing 50 $ Gasoline, 40 $ Movies and Popcorn, 60 $ CD s, 50 $ Tennis racket? Cable TV? 3
What can we buy with this money? Pay the rent, 700 $. Food at Ralph s, 200 $ Clothing 40 $ Gasoline, 80 $ Movies and Popcorn, 60 $ CD s, 20 $ Tennis racket 150 $ Cable TV 50 $ 4
Which is Better? If we like healthy food and our own bedroom, we will choose the first option If we like crowded apartments and we absolutely need the Wilson raquet, we will choose the second option The actual choice depend on our TASTE, on our PREFERENCES 5
Our Consumption Choices Our decisions can be divided in two parts We have to determine all available choices, given our income: BUDGET CONSTRAINT Of all the available choices we choose the one that we prefer: PREFERENCES and INDIFFERENCE CURVES 6
UTILITY MAXIMIZATION PROBLEM 7
There are N goods. Basic Model Consumer has income m. Consumer faces linear prices {p 1,p 2,,p N }. Preferences satisfy completeness, transitivity and continuity. Preferences usually satisfy monotonicity and convexity. Consumer s problem: Choose {x 1,x 2,,x N } to maximize utility u(x 1,x 2,,x N ) subject to budget constraint and x i 0. 8
Budget Constraints If an individual has m dollars to allocate between good x 1 and good x 2 p 1 x 1 + p 2 x 2 m Quantity of x 2 m p 2 If all income is spent on x 2, this is the amount of x 2 that can be purchased The individual can afford to choose only combinations of x 1 and x 2 in the shaded triangle Slope=-p 1 /p 2 If all income is spent on x 1, this is the amount of x 1 that can be purchased m p 1 Quantity of x 1 9
Budget Constraints Suppose income increases Then budget line shifts out Suppose p 1 increases Then budget line pivots around upper-left corner, shifting inward. 10
Nonlinear Budget Constraints Example: Quantity Discounts. m=30 p 2 =30 p 1 =2 for x 1 <10 p 1 =1 for x 1 >10 11
Consumer s Problem (2 Goods) Consumer chooses {x 1,x 2 } to solve: Max u(x 1,x 2 ) s.t. p 1 x 1 +p 2 x 2 m Solution x* i (p 1,p 2,m) is Marshallian Demand. Results: 1. Demand is homogenous of degree zero: x* i (p 1,p 2,m)= x* i (kp 1,kp 2,km) for k>0 Idea: Inflation does not affect demand. 2. If utility is monotone then the budget binds. 12
UMP: The Picture We can add the individual s preferences to show the utility-maximization process Quantity of x 2 A B C The individual can do better than point A by reallocating his budget The individual cannot have point C because income is not large enough U 1 U 2 U 3 Point B is the point of utility maximization Quantity of x 1 13
Soln 1: Graphical Method Utility is maximized where the indifference curve is tangent to the budget constraint Equate bang-per-buck, MU 1 /p 1 = MU 2 /p 2 Quantity of x 2 slope of budget constraint dx slope of indifferen ce curve dx 2 1 U constant U x1 p p 1 2 U x2 B At optimum: p p 1 2 U x1 U x2 U 2 Quantity of x 1 14
Soln2: Substitution Method Rearrange budget constraint: x 2 = (m-p 1 x 1 )/p 2 Turn into single variable problem: Max u(x 1,(m-p 1 x 1 )/p 2 ) FOC(x 1 ): MU 1 + MU 2 (-p 1 /p 2 ) = 0. Rearrange: MU 1 /MU 2 = p 1 /p 2 Example: u(x 1,x 2 ) =x 1 x 2 15
Soln3: Lagrange Method Maxmize the Lagrangian: L = u(x 1,x 2 ) + [m-p 1 x 1 -p 2 x 2 ] First order conditions: MU 1 - p 1 = 0 MU 2 - p 2 = 0 Rearranging: MU 1 /MU 2 = p 1 /p 2 Example: u(x 1,x 2 ) =x 1 x 2 16
LIMITATIONS OF FIRST ORDER APPROACH 17
1. Second-Order Conditions The tangency rule is necessary but not sufficient It is sufficient if preferences are convex. That is, MRS is decreasing in x 1 If preferences not convex, then we must check second-order conditions to ensure that we are at a maximum 18
1. Second Order Conditions Example in which the tangency condition is satisfied but we are not at the optimal bundle. Quantity of x 2 B A = local min B = global max C = local max A C Quantity of x 1 19
1. Second-Order Conditions Example: u(x 1,x 2 ) = x 12 + x 2 2 FOCs from Lagrangian imply that x 1 /x 2 = p 1 /p 2 But SOC is not satisfied: L 11 = L 22 = 2 > 0 Actual solution: x 1 = 0 if p 1 >p 2 x 2 = 0 if p 1 <p 2 20
2. Corner Solutions In some situations, individuals preferences may be such that they can maximize utility by choosing to consume only one of the goods Quantity of y U 1 U 2 U 3 At point A, the indifference curve is not tangent to the budget constraint Utility is maximized at point A A Quantity of x 21
2. Corner Solutions Boundary solution will occur if, for all (x 1,x 2 ) MRS>p 1 /p 2 or MRS<p 1 /p 2 That is bang-per-buck from one good is always bigger than from the other good. Formally, we can introduce Lagrange multipliers for boundaries. If preferences convex, then solve for optimal (x 1,x 2 ). If find x 1 <0, then set x 1 *=0. 22
2. Corner Solutions Example: u(x1,x2) = x 1 + x 2 MRS = - / Price slope = -p 1 /p 2 Which is bigger? 23
3. Non-differentiable ICs Suppose preferences convex. At kink MRS jumps down. Solution occurs at kink if MRS - p 1 /p 2 MRS + 24
3. Non-differentiable ICs Perfect complements: U(x 1,x 2 ) = min (x 1, x 2 ) Utility not differentiable at kink. Solution occurs at: x 1 = x 2 25
GENERAL THEORY 26
The n-good Case The individual s objective is to maximize utility = U(x 1,x 2,,x n ) subject to the budget constraint m = p 1 x 1 + p 2 x 2 + + p n x n Set up the Lagrangian: L = U(x 1,x 2,,x n ) + (m - p 1 x 1 - p 2 x 2 - - p n x n ) 27
The n-good Case First-order conditions for an interior maximum: L/x 1 = U/x 1 - p 1 = 0 L/x 2 = U/x 2 - p 2 = 0 L/x n = U/x n - p n = 0 L/ = m - p 1 x 1 - p 2 x 2 - - p n x n = 0 28
Implications of First-Order Conditions For any two goods, U U / / x x i j p p i j This implies that at the optimal allocation of income MRS ( x i for x j ) p p i j 29
Indirect Utility Function Substituting optimal consumption in the utility function we find the indirect utility function V(p 1,p 2,,p n,m) = U(x* 1 (p 1,,p n,m),,x* n (p 1,,p n,m)) This measures how the utility of an individual changes when prices/income varies Enables us to determine the effect of government policies on utility of an individual. 30
Indirect Utility: Properties 1. v(p 1,..,p n,m) is homogenous of deg 0: v(p 1,..,p n,m) = v(kp 1,..,kp n,km) Idea: Inflation does not affect utility. 2. v(p 1,..,p n,m) is increasing in income and decreasing in prices. 3. Roy s identity: V p i xi( p1,...,, m) V m If p 1 rises by $1, then income falls by x 1 $1. Agent also changes demand, but effect small p N 31
EXPENDITURE MINIMISATION PROBLEM 32
Expenditure Minimization We can find the optimal decisions of our consumer using a different approach. We can minimize her/his expenditure subject to a minimum level of utility that the consumer must obtain. This is important to separate income and substitution effects. 33
Basic Model There are N goods. Consumer has utility target u. Consumer faces linear prices {p 1,p 2,,p N }. Preferences satisfy completeness, transitivity and continuity. Preferences usually satisfy monotonicity and convexity. Consumer s problem: Choose {x 1,x 2,,x N } to minimise expenditure p 1 x 1 + +p N x N subject to u(x 1,x 2,,x N ) u and x i 0. 34
Consumer s Problem (2 Goods) Consumer chooses {x 1,x 2 } to solve: Min p 1 x 1 +p 2 x 2 s.t. u(x 1,x 2 ) u Solution h* i (p 1,p 2,u) is Hicksian Demand. Expenditure function e(p 1,p 2,u) = min [p 1 x 1 +p 2 x 2 ] s.t. u(x 1,x 2 ) u = p 1 h* 1 (p 1,p 2,u) +p 2 h* 2 (p 1,p 2,u) Problem is dual of UMP 35
EMP: The Picture Point A is the solution to the dual problem Quantity of x 2 Expenditure level E 2 provides just enough to reach U Expenditure level E 3 will allow the individual to reach U but is not the minimal expenditure required to do so A Expenditure level E 1 is too small to achieve U U Quantity of x 1 36
Graphical Solution Slope of Indifference Curve = -MRS Slope of Iso-expenditure curves = -p 1 /p 2 At optimum p p 1 2 MRS U x 1 U x 2 Idea: equate bang-per-buck If p 2 =2p 1 then MU 2 =2MU 1 at optimum. 37
Lagrangian Solution Minimize the Lagrangian: L = p 1 x 1 +p 2 x 2 + [u-u(x 1,x 2 )] First order conditions: p 1 - MU 1 = 0 p 2 - MU 2 = 0 Rearranging: MU 1 /MU 2 = p 1 /p 2 Example: u(x 1,x 2 ) =x 1 x 2 38
EMP & UMP UMP: Maximize utility given income m Demand is (x 1,x 2 ) Utility v(p 1,p 2,m) EMP: Minimize spending given target u Suppose choose target u = v(p 1,p 2,m). Then (h 1,h 2 ) = (x 1,x 2 ) Then e(p 1,p 2,u) = m Practically useful: Inverting Expenditure Fn Fixing prices, e(v(m)) = m, so v(m)=e -1 (m) If u=x 1 x 2 then e=2(up 1 p 2 ) 1/2. Inverting, v(m)=m 2 /(4p 1 p 2 ) 39
Expenditure Function: Properties 1. e(p 1,p 2,u) is homogenous of degree 1 in (p 1,p 2 ) If prices double constraint unchanged, so need double expenditure. 2. e(p 1,p 2,u) is increasing in (p 1,p 2,u) 3. Shepard s Lemma: e( p1, p2, u) hi( p1, p2, u) pi If p 1 rises by p, then e(.) rises by p h 1 (.) Demand also changes, but effect second order. 4. e(p 1,p 2,u) is concave in (p 1,p 2 ) 40
Expenditure Fn: Concavity and Shepard s Lemma At p* 1, the person spends e(p* 1, )=p* 1 x* 1 + + p* n x* n e(p 1, ) e sub e(p 1, ) If he continues to buy the same set of goods as p* 1 changes, his expenditure function would be e sub e(p* 1, ) p* 1 p 1 But the consumer can do better by reallocating consumption to goods that are less expensive. Actual expenditures will be less than e sub 41
42 Hicksian Demand: Properties 1. h(p 1,p 2,u) is homogenous of degree 0 in (p 1,p 2 ) If prices double constraint unchanged, so demand unchanged. 2. Symmetry of cross derivatives Uses Shepard s Lemma 3. Law of demand Uses Shepard s Lemma and concavity of e(.) 2 1 2 1 1 2 1 2 h p e p p e p p h p 0 1 1 1 1 e p p h p