W12 MIC 07 Fancy Budget Constraints

W12 MIC 07 Fancy Budget Constraints - FANCY BUDGET...

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1 Northwestern University ECON 310-1: Microeconomic Theory Profs Jim Hornsten & Ron Braeutigam Winter 2012 FANCY BUDGET CONSTRAINTS (B&B Section 4.3) Notes for Lecture #07 Two Types of Optima The Composite Good Model Vouchers & Grants Member Discounts Quantity Discounts Buy One, Get One Free Rationing A Menu of Calling Plans Intertemporal Choice Two Toolkits x y
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2 Northwestern University ECON 310-1: Microeconomic Theory Profs Jim Hornsten & Ron Braeutigam Winter 2012 Consumer Choice: An Interior Solution CASE : Nice Cobb- Douglas Utility In general, max x , y U [ x , y ] subject to I = xP x + yP y Here, max x , y xy subject to 24 = 2 x + 2 y " # MRS x , y = # P x P y " MU x MU y = y x = 2 2 " x = y " 24 = 2 x + 2( x ) = 4 x " x * = 6 = y * The tangency condition and budget constraint give us 2 equations in 2 unknowns. This is a nice, smooth indifference curve with diminishing MRS, mixed with a linear budget constraint (with a constant slope), so we expect an interior solution (i.e., x * > 0 and y * > 0). The BL and IC will be tangent when their slopes are equal, or equivalently, the marginal utility per dollar spent is the same for the two goods: " Slope [ IC ] = " Slope [ BL ] # MU x MU y = P x P y # MU x P x = MU y P y With Leontief prefs, we cd get an interior solution without tangency. x BL IC 2 x * y * IC 1 y
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3 Northwestern University ECON 310-1: Microeconomic Theory Profs Jim Hornsten & Ron Braeutigam Winter 2012 Consumer Choice: Two Corner Solutions First, try max x , y x ( y + 12) subject to 24 = 2 x + 2 y " # MRS x , y = # P x P y " MU x MU y = y + 12 x = 2 2 " x = y + 12 " 24 = 2( y + 12) + 2 y = 4 y + 24 " y * = 0, x * = 12 The tangency condition and budget constraint still work RESULT : A corner solution with tangency! Now try max x , y x ( y + 12) subject to 24 = 2 x + 4 y Doubling P y flattens the budget line... # MRS x , y = # P x P y " MU x MU y = y + 12 x = 2 4 " x = 2 y + 24 " 24 = 2(2 y + 24) + 4 y = 8 y + 48 " y = # 3. Huh?! Can' t have y < 0. We' ve traded in as much y as possible. Use bang for the buck : MU x P x = y + 12 2 > MU y P y = x 4 If spend all $24 on 12 units of x, get 0 + 12 2 > 12 4 , which confirms our decision to load up on only x. At this corner solution, we observe IC is steeper : its slope is more negative! Slope [ IC ] " Slope [ BL ] # $ ( Slope [ IC ]) % $ ( Slope [ BL ]) # $ $ MU x MU y & ( ( ) * + + % $ $ P x P y & ( ( ) * + + # MU x MU y % P x P y # MU x P x % MU y P y so marginal utility per dollar spent (or "bang for the buck") is higher for x than for y , and consumer wants to give up some y to get more x ... but runs out of y . x y BL IC 2 x * y * =0 IC 1
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4 Northwestern University ECON 310-1: Microeconomic Theory Profs Jim Hornsten & Ron Braeutigam Winter 2012 PERFECT SUBSTITUTES & PERFECT COMPETITION Consider U[x, y] = x + y , which has indiff curves with slope -1 Simple budget line xP x + yP y I , which has slope -P x /P y Suppose goods x and y are identical products sold by rival firms With indivisible goods and income is $4, find optimal choice(s) when P x = $1 and P y = $2 when P x = $2 and P y = $1 when P x = P y = $1
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5 PERFECT SUBSTITUTES & PERFECT COMPETITION Consider U[x, y] = x + y , which has indiff curves with slope -1 Simple budget line xP x + yP y I , which has slope -P x /P y Suppose goods x and y are identical products sold by rival firms
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