CCT2 Utility Functions and Marshallian Demand Flashcards
What is the equation to show where the indifference curve is tangential to the budget constraint?
MRSxy=px/py
What do we mean by ordinal utility?
The ranking (or ordering) of bundles based on consumer preferences.
Does ordinal utility relate to how much more utility one bundle provides than another?
No, it only focuses on which bundle is preferred and has no mention of by how much.
What happens to the budget line when the price of x falls?
The budget line will rotate anti clockwise.
When the price of x falls, which effects cause demand for x to increase?
Both the substitution effect and the income effect combine to increase the demand for x.
Define the income effect.
The income effect is the change in the consumption of goods based on income ie in general ↑income → ↑ consumption.
Define the substitution effect
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.
What do Marshallian demand curves show?
Marshallian demand curves simply show the relationship between the price of a good and the quantity demanded of it.
How could we derive the Marshallian demand for good x?
Associate the quantities of x with its price and draw a line through the points.
Why would we derive the functional form of the Marshallian demand curve?
In order to solve a utility maximising problem subject to a budget constraint.
Define a monotonic transformation.
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.
What does v’(u)>0. imply?
That on a graph the function v(u) is upward sloping.
If v(u)=5x^2 then is v(u) is a (positive) monotonic transformation of u(x)? Why/Why not?
Yes as v’(u) = 10x>0
What is true of utility functions which are (positive) monotonic transformations of other utility functions?
Both utility functions have the exact same Marshallian demand curve.
Is the concept of diminishing marginal utility relevant to understanding downward sloping demand curves?
No, as (positive) monotonic transformations do not necessarily preserve properties like diminishing marginal utility but produces the same demand curves.
Under which circumstances is a utility function said to be homothetic ?
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.
Under homothetic utility functions, what is true along any ray from the origin?
- The marginal rate of substitution between two goods is the same
- The slope of the indifference curve is the same
What are the 5 different types of utility function?
- Cobb-Douglas
- Perfect Substitution
- Perfect Complements
- Quasi-linear
- Stone-Geary
What is the functional form for a Cobb-Douglas utility function?
U(x,y)=x^a*y^b
What kind of indifference curves do a Cobb-Douglas utility function give us?
A Cobb-Douglas utility function gives us nice, smooth and convex indifference curves.
Do Cobb-Douglas utility functions satisfy homotheticity?
Yes
List the steps to finding the Marshallian demand in a Cobb-Douglas utility function, and what is significant about this solution?
- Maximise the Lagrangean to find the first order conditions
- Then solve simultaneously
- The solution is a tangency
What is the functional form for a Perfect Substitutes utility function and what is different about this?
- U(x,y) = ax+ by
* The function is linear
What kind of indifference curves do a Perfect Substitutes utility function give us and what is a consequence of this?
- The indifference curves are straight lines
* Therefore the MRS is constant along an indifference curve.
How do we find the optimum solution under a Perfect Substitutes utility function and is this a tangency?
We find the point where the highest indifference curve meets the budget constraint, and no, this is not a tangency
Do Perfect Substitute utility functions satisfy homotheticity?
Yes
What shape are indifference curves under a Perfect Complements utility function and what does this say about the MRS along the indifference curves?
- 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.
What is the functional form for a Perfect Complements utility function?
U(x,y) = min{ax,by}
Do Perfect Complements utility functions satisfy homotheticity?
Yes
How do we work out the optimising solution for a Perfect Complements utility function, and is this a tangency solution?
- 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
What happens to one of the goods under a Quasi-linear utility function? What does this mean?
- 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.
What is the functional form for a Quasi-Linear utility function?
• U(x,y) = lnx + y
or
• U(x,y) = lny + x
What is important to note about indifference curves under a Quasi-Linear utility function?
They are vertically or horizontally parallel depending on their functional form
Does a Quasi-Linear utility function have a tangency solution? Is it homothetic?
- It has a tangency solution
* It isn’t homothetic
Outline the steps towards solving for the optimising levels of x and y in a Quasi-Linear utility function.
- Form a Lagrangean
- Differentiate with respect to x, y and λ
- Solve simultaneously
How do you solve optimisation problems under Stone-Geary
You will not be asked this in an exam, don’t worry
What is the functional form for a Stone-Geary utility function and what does this mean?
- 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
What is Cobb Douglas in relation to Stone-Geary?
Cobb-Douglas is a special case of Stone-Geary, where both α and β =0.
Does a Stone-Geary utility function have a tangency solution? Is it homothetic?
- It has a tangency solution
* It isn’t homothetic in general as the origin is in a new place
