CCT2 Utility Functions and Marshallian Demand

Question 1

What is the equation to show where the indifference curve is tangential to the budget constraint?

Answer 1

MRSxy=px/py

Question 2

What do we mean by ordinal utility?

Answer 2

The ranking (or ordering) of bundles based on consumer preferences.

Question 3

Does ordinal utility relate to how much more utility one bundle provides than another?

Answer 3

No, it only focuses on which bundle is preferred and has no mention of by how much.

Question 4

What happens to the budget line when the price of x falls?

Answer 4

The budget line will rotate anti clockwise.

Question 5

When the price of x falls, which effects cause demand for x to increase?

Answer 5

Both the substitution effect and the income effect combine to increase the demand for x.

Question 6

Define the income effect.

Answer 6

The income effect is the change in the consumption of goods based on income ie in general ↑income → ↑ consumption.

Question 7

Define the substitution effect

Answer 7

The substitution effect is the decrease in sales for a product that can be attributed to consumers switching to cheaper alternatives when its price rises- ie ↑price of beef → ↑ consumption of chicken.

Question 8

What do Marshallian demand curves show?

Answer 8

Marshallian demand curves simply show the relationship between the price of a good and the quantity demanded of it.

Question 9

How could we derive the Marshallian demand for good x?

Answer 9

Associate the quantities of x with its price and draw a line through the points.

Question 10

Why would we derive the functional form of the Marshallian demand curve?

Answer 10

In order to solve a utility maximising problem subject to a budget constraint.

Question 11

Define a monotonic transformation.

Answer 11

Let u(x) be a function. Then if we transform the function into a new function v(u), then this is said to be a (positive) monotonic transformation of u(x) if the derivative v’(u)>0.

Question 12

What does v’(u)>0. imply?

Answer 12

That on a graph the function v(u) is upward sloping.

Question 13

If v(u)=5x^2 then is v(u) is a (positive) monotonic transformation of u(x)? Why/Why not?

Answer 13

Yes as v’(u) = 10x>0

Question 14

What is true of utility functions which are (positive) monotonic transformations of other utility functions?

Answer 14

Both utility functions have the exact same Marshallian demand curve.

Question 15

Is the concept of diminishing marginal utility relevant to understanding downward sloping demand curves?

Answer 15

No, as (positive) monotonic transformations do not necessarily preserve properties like diminishing marginal utility but produces the same demand curves.

Question 16

Under which circumstances is a utility function said to be homothetic ?

Answer 16

If increasing income and keeping the price ratio the same (aka an outwards shift in the budget constraint) results in the goods being consumed in the same proportion.

Question 17

Under homothetic utility functions, what is true along any ray from the origin?

Answer 17

• The marginal rate of substitution between two goods is the same
• The slope of the indifference curve is the same

Question 18

What are the 5 different types of utility function?

Answer 18

• Cobb-Douglas
• Perfect Substitution
• Perfect Complements
• Quasi-linear
• Stone-Geary

Question 19

What is the functional form for a Cobb-Douglas utility function?

Answer 19

U(x,y)=x^a*y^b

Question 20

What kind of indifference curves do a Cobb-Douglas utility function give us?

Answer 20

A Cobb-Douglas utility function gives us nice, smooth and convex indifference curves.

Question 21

Do Cobb-Douglas utility functions satisfy homotheticity?

Answer 21

Yes

Question 22

List the steps to finding the Marshallian demand in a Cobb-Douglas utility function, and what is significant about this solution?

Answer 22

• Maximise the Lagrangean to find the first order conditions
• Then solve simultaneously
• The solution is a tangency

Question 23

What is the functional form for a Perfect Substitutes utility function and what is different about this?

Answer 23

• U(x,y) = ax+ by

* The function is linear

Question 24

What kind of indifference curves do a Perfect Substitutes utility function give us and what is a consequence of this?

Answer 24

• The indifference curves are straight lines

* Therefore the MRS is constant along an indifference curve.

Question 25

How do we find the optimum solution under a Perfect Substitutes utility function and is this a tangency?

Answer 25

We find the point where the highest indifference curve meets the budget constraint, and no, this is not a tangency

Question 26

Do Perfect Substitute utility functions satisfy homotheticity?

Answer 26

Yes

Question 27

What shape are indifference curves under a Perfect Complements utility function and what does this say about the MRS along the indifference curves?

Answer 27

• The indifference curves are L-shaped, with lines parallel to the axes.
• For this reason, the MRS is either zero or infinite along the indifference curves.

Question 28

What is the functional form for a Perfect Complements utility function?

Answer 28

U(x,y) = min⁡{ax,by}

Question 29

Do Perfect Complements utility functions satisfy homotheticity?

Answer 29

Yes

Question 30

How do we work out the optimising solution for a Perfect Complements utility function, and is this a tangency solution?

Answer 30

• Work out the equation of the line from the origin which passes through the corner of the indifference curve
• Work out the equation of the budget constraint
• Solve simultaneously
• This is not a tangency solution

Question 31

What happens to one of the goods under a Quasi-linear utility function? What does this mean?

Answer 31

• One of the goods will not have an income effect (the Marshallian demand will not be a function of m)
• When there is a shift in the budget constraint, the tangency happens vertically/horizontally above/next to the previous budget constraint, there is no change in x/y.

Question 32

What is the functional form for a Quasi-Linear utility function?

Answer 32

• U(x,y) = lnx + y
or
• U(x,y) = lny + x

Question 33

What is important to note about indifference curves under a Quasi-Linear utility function?

Answer 33

They are vertically or horizontally parallel depending on their functional form

Question 34

Does a Quasi-Linear utility function have a tangency solution? Is it homothetic?

Answer 34

• It has a tangency solution

* It isn’t homothetic

Question 35

Outline the steps towards solving for the optimising levels of x and y in a Quasi-Linear utility function.

Answer 35

• Form a Lagrangean
• Differentiate with respect to x, y and λ
• Solve simultaneously

Question 36

How do you solve optimisation problems under Stone-Geary

Answer 36

You will not be asked this in an exam, don’t worry

Question 37

What is the functional form for a Stone-Geary utility function and what does this mean?

Answer 37

• U(x,y)=(x-α)^a (y-β)^b

* You need at least a minimum amount of x and y greater than α and β before you can even start gaining utility

Question 38

What is Cobb Douglas in relation to Stone-Geary?

Answer 38

Cobb-Douglas is a special case of Stone-Geary, where both α and β =0.

Question 39

Does a Stone-Geary utility function have a tangency solution? Is it homothetic?

Answer 39

• It has a tangency solution

* It isn’t homothetic in general as the origin is in a new place