1.2 Utility functions and Preferences

Question 1

How to denote preferences

Answer 1

a >~ b : weakly prefers a to b

a > b: strictly prefers

A ~ B: indiffernt

Question 2

Assumptions of consumer preferences

Answer 2

1. Completeness - consumers rank goods using the 3 preference denotions
2. Transitivity - consumer rankings logically consistent in the sense that if a>b then b>c
3. Non satiation - when all else equal, more of a good is better than less

Question 3

What are indifference curves and their properties?

Answer 3

Shows preference of 2 different goods (kinda like PPF)
Show the rate consumer trades a good for another

This is also known as the Marginal Rate of Substitutions

Properties:
- Bundles far from origin prefered
- All bundles on a curve
- ICs cannot cross
- Cant slope up
- Cant be thick

Question 4

How do you calculate the total differential of an indifference curve?

Answer 4

How much function changes if we simultaneously change all variables by a small amount:

The marginal change in f(x,y) if x moves by dx and y by dy:

df(x,y) = (partial derivative w.r.t x *dx) + (partial derivative w.r.t. y * dy)

Question 5

What are utility functions and how are they compared?

Answer 5

Assignment of numerical values to each bundle making them a function of U(x)

Compared with >, < and =

Question 6

What are the 2 types of utility function

Answer 6

Ordinal - rankings, not utility levels so if U(x) = 2 x U(y) does not mean they like x twice as much
- Do not allow interpersonal comparisons of utility

Cardinal: assigns an exact value to the bundles and so if U(x) is 2 x U(y) then implies likes it twice as much

Question 7

How do indifference curves relate to utility?

Answer 7

U with a line over it is the utility function. It is a function of q1 and q2

Marginal utility: change in total utility from consuming an extra unit of a good

U1 = partial derivative w.r.t q1

U2 = partial derivative w.r.t q2

(SHOWN ON PAGE 3)

If we move down the graph, we give up q2 to gain q1:
Losing q2 (dq2) decreases total utility by (pd w.r.t q2)*dq2

Gaining q1 (dq1) increases utility by (pd w.r.t. q1)*dq1

Question 8

What are the properties of utility and MRS along a utility curves IC?

What is the relationship between utility and the MRS?

Answer 8

Along the curve, U is constant and equal to:

(pd wrt q1)dq1 + (pd wrt q2)dq2

^ = utility x change in q

The MRS:

dq2/dq1 = -(pd wrt q1)/(pd wrt q2) = -U1/U2

Note the slope is equal to the negative ratio of marginal utilities

Question 9

What are properties result from the IC being convex?

Answer 9

MRS is diminishing

-U1/U2 gets smaller as we move down the curve, so MRS falls

When q1 rises each unit becomes worth less, so U1 falls

When q2 falls each unit worth more, so U2 goes up

Question 10

What are some of the common utility functions?

Answer 10

(Page 4)

Perfect substitutes

Perfect complements

Imperfect substitutes (most common, two famous ones):

Cobb-Douglas
Quasilinear