University:
Amherst College
Course:
ECON 301 | Advanced Microeconomics
Academic year:

2023
Views:

493

Pages:

2
Author:

Nathaly Davidson

Economics 58 The Cobb-Douglas Functions in Consumer Theory Utility Function: U ( x, y ) = x α y 1−α . Lagrangian: L = x α y 1−α + λ ( I − p x x − p y y ) FOCs: ∂L = αx α −1 y 1−α − λp x = 0 ∂x ∂L = (1 − α ) x α y −α − λp y = 0 ∂y ∂L = I − px x − p y y = 0 ∂λ x= Solving these yields: y= α Indirect Utility: V = x y Expenditure Function: αI px (1 − α ) I py 1−α = α α (1 − α )1−α I α px p 1−α y = KIp x−α p αy −1 ; K = α α (1 − α )1−α E = K −1 p αx p 1y−α V ln E = − ln K + α ln p x + (1 − α ) ln p y + ln V Numerical Examples: α = 0.5, x = .5I p x , y = .5I p y , V = .5I p x p y , E = 2Vp .x5 p .y5 Lump Sum Principle: I = 10, p x = 1, p y = 1, U = V = 5 Two ways to get to V = 6 1. Just give this person $2. 2. Subsidize x. How much? Use V = 6 = 5 p x = (5 6) ≈ 0.7 . Subsidize x at 0.3 per unit 2 p x ⋅ 1 . So, At this price x = 5 0.7 = 7.14 Cost of subsidy = 0.3 ⋅ 7.14 = 2.14 -- costs more than Income grant. Value of $1 in subsidy =2/2.14=0.93 Efficiency in Taxation: Govt. needs $2. Utility cost of income tax is 1; V falls from 5 to 4. If impose a per unit tax of t on x, price becomes p x = 1 + t . x = 5 /(1 + t ), x + tx = 5, but tx = 2 so x = 3, t = 2 3 V =5 5 3 = 3.87 . This per unit tax has an “excess burden” of 0.13 units of V. To put a dollar cost on this, note that $.26 would compensate for the 0.13 units of V lost. Hence, the excess burden is 13 percent of the tax collected. Exercise: Find excess burden of another dollar of tax. What is the marginal excess burden of the third dollar of tax?

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