Microeconomics I

I complete: any two bundles can be compared
I reflexive: any bundle is at least as good as itself X X
I transitive: if X Y and Y Z then X Z
Often also ‘well-behaved’ (monotonic, convex)

Assume X and Y are on different
indifference curves and for
example X > Y .
X and Z are on the same
indifference curve so X = Z.
Y and Z are on the same
indifference curve so Y = Z.
By transitivity, X = Y which
contradicts X > Y .

The consumer is willing to substitute one good for another at a constant rate (measured by the slope of the indifference
curves)
Only the total amount of the two commodities in bundles determines their preference rank-order

Goods that are always consumed together in fixed proportions (measured by the slope of the ray through the origin)
Only the number of pairs of units of the two commodities determines the preference rank-order of bundles

Monotonicity: more is preferred to less

Convexity: the average is preferred to extremes

I complete: any two bundles can be compared
I reflexive: any bundle is at least as good as itself X >= X
I transitive: if X  >= Y and Y >= Z then X >= Z

Not the 3rd nor the 5th

mu(x1, x2) = ax1 + bx2

both goods

b is a bad

u (x1,x2) = min {x1, x2}    x2 = x1

x1 = (1/2)x2   min {x1, (1/2)x2}

u (x1,x2) = min {ax1. bx2}   x2=(a/b)x1

MRS independant from q of x2

ũ = v (x1) + x2 

x2 = k - v(x1)

Kinky tastes, corner solutions, multiple tangency

Write the lagrangian with logs

First order contition

Solve for lambda

Put into FOC 1 and FOC 2

Put MRS = -p1/p2

Inject the budget constraint

Express x2 according to x1 with the budget constraint

Inject into the utility function

Derive the utility function to find de FOC

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