Graphs Details Math Examples Using data Tax example. Decision. Intermediate Micro. Lecture 5. Chapter 5 of Varian

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1 Decision Intermediate Micro Lecture 5 Chapter 5 of Varian

2 Decision-making Now have tools to model decision-making Set of options At-least-as-good sets Mathematical tools to calculate exact answer Problem to solve: max x1,x 2 u(x 1, x 2 ) s.t.p 1 x 1 + x 2 = m

3 The idea Budget set Set of options Given by m, p 1, Weakly-preferred set Set of bundles at least as good as (x 1, x 2 ) Which (x 1, x 2 ) to use?

4 The idea Choose (x1, x 2 ) such that 1. (x1, x 2 ) is affordable (within budget set) 2. No affordable (y 1, y 2 ) is preferred (within strictly-preferred set) Define this best choice of (x1, x 2 ) as the optimal choice.

5 The idea (x 1, x 2 ) On budget line and indifference curve The two lines are tangent Otherwise, preferred and feasible alternatives exist

6 Optimal choice: the specifics If preferences are monotone Optimal choice is on budget line If preferences are strictly convex and budget set is well behaved There is only one optimal choice Interior optimum: An optimal choice where a positive amount of every good is consumed The budget line will be tangent to the indifference curve Boundary optimum: An optimal choice where zero is consumed for at least one good The budget line need not be tangent to the indifference curve

7 Finding an solution with math How we will usually find a solution: Method 1 1. Find slope of budget line = p 1 2. Find MRS in terms of x 1 and x st equation: MRS = p nd equation: m = p 1 x 1 + x 2 5. Solve for x 1 and x 2 Example: Cobb-Douglas utility u(x 1, x 2 ) = x x = 2x 1 + x 2

8 Finding an solution with math How we will usually find a solution: Method 2 1. Solve out x 2 in the budget eqn: x 2 = m p 1 x 1 2. Substitute out x 2 from utility function 3. Rename new utility function: u(x 1, m p 1 x 1 ) = v(x 1 ) 4. Take derivative: d dx 1 v(x 1 ) = 0 5. Solve for x 1 6. x 2 = m p 1 x 1 Example: Cobb-Douglas utility u(x 1, x 2 ) = x x = 2x 1 + x 2

9 Lagrange multiplier method - slide 1 How we will usually find a solution: Method 3 1. Compose Lagrangian function: L(x 1, x 2, λ) = u(x 1, x 2 ) + λ[m (p 1 x 1 + x 2 )] Objective function: Function you want to maximize, in this case u(x 1, x 2 ) Constraint: Equation that limits possible values of x, in this case m (p 1 x 1 + x 2 ) = 0 Lagrange multiplier: variable that shows the cost of the constraint, in this case, λ. You do not need to know this definition, just include λ in the lagrangian, then solve it out.

10 Lagrange multiplier method - slide 2 How we will usually find a solution: Method 3 L(x 1, x 2, λ) = u(x 1, x 2 ) + λ[m (p 1 x 1 + x 2 )] 2. Take partial derivatives: L x 1 = u L x 2 L x 1 λp 1 = 0 = u x 2 λ = 0 λ = m (p 1x1 + x2 ) = 0 First order conditions: Set of equations from taking partial derivatives of a Lagrangian. They characterize the solution to the maximization problem. 3. Solve for λ, x 1, x 2 Example: Cobb-Douglas utility u(x 1, x 2 ) = x x = 2x 1 + x 2

11 Necessary and sufficient conditions Solution method in last slide gives all solutions where optimum is interior, and utility function is continuously differentiable (no kinks), and budget line is continuously differentiable Last slide: set of necessary conditions for a specific type of interior optimum Necessary conditions: If x is a solution, it satisfies the necessary conditions Sufficient conditions: If x satisfies the sufficient conditions, it is a solution

12 Other candidate solutions Consumption bundles to consider Bundles where MRS = p 1 and m = p 1 x 1 + x 2 Bundles where the indifference curve has a kink and m = p 1 x 1 + x 2 The endpoints of the budget line Exhaustive method: Calculate u(x 1, x 2 ) at all such points Better method: Learn some shortcuts

13 Demand functions We can solve for x as a function of p and m ie for unspecified values Demand function for good 1: x 1 (p 1,, m) Demand function for good 2: x 2 (p 1,, m)

14 Cobb-Douglas u(x 1, x 2 ) = x a 1 x b 2 u( ) continuous u(x 1, x 2 ) minimized when x 1 = 0 or x 2 = 0 Usual budget line: m = p 1 x 1 + x 2 Optimum given by: MRS = p 1 and budget line

15 Cobb-Douglas u(x 1, x 2 ) = x a 1 x b 2 m = p 1 x 1 + x 2 x 1 (p 1,, m) = x 2 (p 1,, m) = Spend and a b a+b a+b a m a+b p 1 b m a+b m on good 1, m on good 2

16 Perfect complements u(x 1, x 2 ) = min{ax 1, bx 2 } Indifference curves have kinks u(x 1, x 2 ) minimized when x 1 = 0 or x 2 = 0 Usual budget line: m = p 1 x 1 + x 2 Optimum is at kink: ax 1 = bx 2 and budget line

17 Perfect complements u(x 1, x 2 ) = min{ax 1, bx 2 } m = p 1 x 1 + x 2 x 1 (p 1,, m) = bm x 2 (p 1,, m) = bp 1 +a am bp 1 +a

18 Nonconvex preferences Example: u(x 1, x 2 ) = ax bx 2 2 u(x 1, x 2 ) may be maximized when x 1 = 0 or x 2 = 0 Usual budget line: m = p 1 x 1 + x 2 Need to check candidate solutions May be able to rule solutions out by graphing indifference curves

19 Nonconvex preferences u(x 1, x 2 ) = ax bx 2 2 m = p 1 x 1 + x 2 (x 1 (p 1,, m), x 2 (p 1,, m)) = (0, m b ) if < p 1 a (0, m ) or ( m b p 1, 0) if = p 1 a ( m b p 1, 0) if > p 1 a

20 Perfect substitutes Example: u(x 1, x 2 ) = ax 1 + bx 2 m = p 1 x 1 + x 2 MRS = a b 3 cases for (x1, x 2 ) p 1. 1 < MRS: (x1, x 2 ) = ( m p 1, 0) p 2. 1 > MRS: (x1, x 2 ) = (0, m ) p 3. 1 = MRS:...

21 Perfect substitutes Example: u(x 1, x 2 ) = ax 1 + bx 2 m = p 1 x 1 + x 2 Case 3: p 1 = MRS : What is optimal x? All bundles on the budget line!

22 Neutrals/bads Suppose good 1 is a neutral or a bad. How much of good 1 will you buy? None! x = (0, m )

23 Neutrals/bads Suppose good 1 is a neutral or a bad. How much of good 1 will you buy? None! x = (0, m )

24 Discrete goods Suppose good 1 can only be an integer. Can consume (0, m ), (1, m p 1 ), (2, m 2p 1 ),... Calculate u(x) at each of these points Find best combination

25 Quasilinear utility u(x 1, x 2 ) = v(x 1 ) + x 2 u( ) continuous Usual budget line: m = p 1 x 1 + x 2 Optimum given by: MRS = p 1 and budget line (except for very small m)

26 Quasilinear utility u(x 1, x 2 ) = v(x 1 ) + x 2 m = p 1 x 1 + x 2 MRS = v (x 1 ) 1 v ( x 1 ) = p 1, for some x 1 x { 1 (p 1,, m) = } x1 if m > p 1 x 1 m p 1 if m p 1 x 1 x { 2 (p 1,, m) = m p1 x 1 if m > p 1 x 1 0 if m p 1 x 1 x1 does not depend on m, except at very low m }

27 Estimating utility functions Real world - Don t know utility functions Do observe p, m, x If households always spend same fraction of income on x 1, x 2 Cobb-Douglas utility If x 1 changes with price but not income Quasilinear utility Otherwise, may need more complicated utility functions

28 Raising revenue with a quantity tax The government wants to raise $R Charges per-unit tax t on good 1 t until tx 1 = R Label choice x

29 Raising revenue with an income tax The government wants to raise $R Collects $R from consumer Income tax, or per-head tax Budget line includes optimum from per-unit tax

30 Which tax? At x Per-unit tax budget line tangent to IC Income tax budget line crosses IC Can reach higher IC with income tax! Both taxes raise $R, but income tax is better for consumers

31 Which tax? At x Per-unit tax budget line tangent to IC Income tax budget line crosses IC Can reach higher IC with income tax! Both taxes raise $R, but income tax is better for consumers

32 Which tax? - Caveats 1. This analysis may break down if consumers are not all the same Income tax more harmful for those who don t consume x1 Maybe we want to discourage x 1, or encourage x 2 2. We ve only discussed the demand side of the market Response of suppliers may lessen impact 3. Consumers may work less when faced with income tax 4. Per-head tax can be inhumane if consumers differ a lot in pre-tax income