Utility

Question 1

When can preferences be modelled as utility functions (what must preferences be)?

Answer 1

• Preferences must be complete, reflexive, transitive and continuous

Question 2

If x’ > x’’, what does this mean about utility?

Answer 2

• If x’ > x’’, U(x’) > U(x’’)
• This is also the case if x’ ~ x’’, meaning that U(x’) = U(x’’)

Question 3

What kind of concept is utility?

Answer 3

• Utility is ordinal, not cardinal
• This means that things can be ranked by position
• For example, if U(x) = 6 and U(y) = 2, then U(x) > U(y) BUT not x3

Question 4

If U(x) = (α,β), how can you calculate utility?

Answer 4

• To find utility, you must multiply the two levels (αβ)

Question 5

If bundles are on the same indifference curve, what does it mean with utility? What does this allow?

Answer 5

• They are equal
• Therefore, utility functions can be plotted on different indifference curves
• Higher indifference curves, the more preferred this is

Question 6

Explain the concept of monotonic transformations

Answer 6

• If U is a utility function and f is a strictly increasing function
• This means V = f(U) is also a utility function representing the same relation
• If w is an operation to U, then w preserves the same order as U and therefore represents the same preferences

Question 7

What are some examples of strange utility functions?

Answer 7

• Perfect Substitutes: V (x,y) = x + y
• Perfect Complements: w (x,y) = Min (x,y)
• Cobb-Douglas: U (x,y) = x^α * y^β where β and α are >0
• If β + α = 1, we can see the budget proportions
• Quasi-linear functions: U (x,y) = F(x) + x2 , this is relevant for production

Question 8

What is the Marginal Utility of a Commodity? What is the formula

Answer 8

• MU = the rate of change in total utility as quantities of commodities rise
• MUx = δU / δx

Question 9

What is the general equation for the utility functions? How does this give MRS?

Answer 9

• U (x,y) = k (constant)
• Totally differentiating gives:
  δU / δx *dX + δU / δy * dY = 0
• Rearranging gives:
  dY/dX = [δU / δx] / [δU / δy]
• We know that dY/dX is MRS, so MRS = [δU / δx] / [δU / δy]

Question 10

What is the effect of a monotonic transformation on the MRS?

Answer 10

• By creating another utility function representing the same preference relation, this means that MRS will not change