Microeconomics I partie 7/9
Fiches de révisions
Fiches de révisions
Optimization solution 1
Write the lagrangian with logs
First order contition
Solve for lambda
Put into FOC 1 and FOC 2
Optimization according the utility function and the budget constraint
Tangency exceptions
Kinky tastes, corner solutions, multiple tangency
Consumers' choice: tangency condition, condition for optimization
MRS for perfect substitute: calculation
+ MRS for Cobb-Douglas
Definition of the MRS
Definition of MU
Sketch of the Cobb-Douglas function
Formula for the quasi-linear preferences
ũ = v (x1) + x2
x2 = k - v(x1)
Quasi-linear preferences
MRS independant from q of x2
Perfect complements: determine how much x2 per x1
u (x1,x2) = min {ax1. bx2} x2=(a/b)x1
2 teaspoon per cup of coffe
x1 = (1/2)x2 min {x1, (1/2)x2}
Perfect complements: utility function
u (x1,x2) = min {x1, x2} x2 = x1
perfect substitutes: when b<0
b is a bad
Perfect substitutes: when a = 0
Perfect substitutes: when a>0 and b>0
both goods
Perfect substitutes: utility function in general
mu(x1, x2) = ax1 + bx2
Perfect substitutes: price offer curve and demand curve
Price offer curve and demand curve with change of price
Ordinary goods VS Giffen goods
Change in prices
Quasi-linear preferences: income demand curve and Engel curve
Cobb-Douglas utility function: income offer curve and Engel curve
Perfect complements: Engel curve
Perfect complements: income offer curve
x1= m/ (p1+p2)
Perfect substitutes: One-to-one: with p < p : x1 = m/p1: Engel curve
Perfect substitutes: One-to-one: with p < p : x1 = m/p1: income offer curve
Engel curve
Income offer curve
Income expansion path
Inferior goods
(delta x1 / delta m) < 0, use of public transports, low quality food
Normal goods
(delta X, / delta m) > 0
Change in m: m'' < m <m'
Application to tax choice
If the government wants to raise R of revenue, should it use a quantity tax or an income tax?
Income tax better than quantity tax.
Solve with the lagrangian for good2
Cobb-Douglas optimal choices
Quasi-linear preferences
Establish the maximum utility
Express x2 from de budget constraint
Inject it into the utility function
Derive to get the FOC
Equalize the partial utility function according to the prices
Concave preferences
corner solution
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Lernkarten
Perfect substitutes: price offer curve and demand curve
Price offer curve and demand curve with change of price
Ordinary goods VS Giffen goods
Change in prices
Quasi-linear preferences: income demand curve and Engel curve
Cobb-Douglas utility function: income offer curve and Engel curve
Perfect complements: Engel curve
Perfect complements: income offer curve
x1= m/ (p1+p2)
Perfect substitutes: One-to-one: with p < p : x1 = m/p1: Engel curve
Perfect substitutes: One-to-one: with p < p : x1 = m/p1: income offer curve
Engel curve
Income offer curve
Income expansion path
Inferior goods
(delta x1 / delta m) < 0, use of public transports, low quality food
Normal goods
(delta X, / delta m) > 0
Change in m: m'' < m <m'
Application to tax choice
If the government wants to raise R of revenue, should it use a quantity tax or an income tax?
Income tax better than quantity tax.
Solve with the lagrangian for good2
Cobb-Douglas optimal choices
Quasi-linear preferences
Establish the maximum utility
Express x2 from de budget constraint
Inject it into the utility function
Derive to get the FOC
Equalize the partial utility function according to the prices
Concave preferences
corner solution
Consumers' demand: perfect complements
Exercice slide 18 lecture 5
Consumer demand: perfect substitutes
Assuming a one-to-one substitution
Solve for p1<p2, p1=p2 and p1>p2
Build a general case: (a/b)>(p1/p2) then x1* = m / p1 and x2* = 0
Optimization solution 3
Express x2 according to x1 with the budget constraint
Inject into the utility function
Derive the utility function to find de FOC
Optimization solution 2
Put MRS = -p1/p2
Inject the budget constraint
Optimization solution 1
Write the lagrangian with logs
First order contition
Solve for lambda
Put into FOC 1 and FOC 2
Optimization according the utility function and the budget constraint
Tangency exceptions
Kinky tastes, corner solutions, multiple tangency
Consumers' choice: tangency condition, condition for optimization
MRS for perfect substitute: calculation
+ MRS for Cobb-Douglas
Definition of the MRS
Definition of MU
Sketch of the Cobb-Douglas function
Formula for the quasi-linear preferences
ũ = v (x1) + x2
x2 = k - v(x1)
Quasi-linear preferences
MRS independant from q of x2
Perfect complements: determine how much x2 per x1
u (x1,x2) = min {ax1. bx2} x2=(a/b)x1
2 teaspoon per cup of coffe
x1 = (1/2)x2 min {x1, (1/2)x2}
