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  MRS and Special utility functions

Marginal Rate of Substitution (MRS):

171_marginal rate of substitution.jpg

U = U(x, y). Totally differentiating to get d U = MUxdx + MUydy = 0 (why?).

217_MRS formula.jpg

cf) Can you depict an indifference curve with increasing MRSx, y ?

Special Utility Functions:

2359_special utility function.jpg

Cobb-Douglas Utility Function (by Charles Cobb and Paul Douglas 1928)

U = Axα yβ , where A, α , and β are positive constants.

Cobb-Douglass efficacy function has three properties that make it of interest in the study of consumer choice.

i) MU’s are positive Check it out.
ii) Since MU’s are all positive the apathy curves will be downward sloping.
iii) It also exhibits a diminishing MRS

Quasi-Linear Utility Function (imperfect substitution, No income effect on x) It can describe preferences for a consumer who purchases the same amount of a commodity regardless of his income. Ex) toothpaste and coffee U(x, y) = v(x) + by , where v(x) is a function that increases in x and b is positive constant. The indifference curves are parallel so for any value of y the slopes of I.C. will be the similar.

U = √x + y

Budget Constraint:

Px x + Py y ≤ M From this constraint we can derive the budget line (or price line) to visualize in 2-D space.

484_budget constraint.jpg

where - Px/Py is slope and M/Py is vertical intercept.

955_budget constraint.jpg

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