Utility, Demand Functions, Elasticity Flashcards
What is indirect utility?
max utility = U[x1*(p1, ..,pn, I), x2*(p1, …, pn, I),…]
= V(p1, p2, …, pn, I).
v = indirect utility
Optimal utility depends indirectly on price of goods and income.
What does the expenditure function show and what is its relationship to indirect utility?
Expenditure function shows minimal expenditures necessary to achieve a given utility level for a particular set of prices. minimal expenditures = E(p1,p2, …,p3, U).
Expenditure funtion and indirect utility function are inverse functions of one another.
e.g. V(px, py, I) = I / [2px0.5py0.5] , where I is income.
E(PxPy, U) = 2Px0.5Py0.5U, where U is utility
(*Don’t inverse the constraint).
What are the axioms of rational choice?
- Completeness –> All preference orderings are known.
- Transivity –> If A < B and B < C, then A < C
- Continuity –> If A is preferred to B, then situations suitably close to A must be preferred to B
- Non-satiation –> More is better
Explain uniqueness of utility measures.
Notion of utility is defined only up to an order-preserving (monotonic) transformation. One can say A > B, but not A is 1.5 times greater than B.
Implies not possible to compare utilities of different people.
What is the marginal rate of substitution?
The rate at which good x can be traded for good y. It is negative of the slope of the indifference curve (IC) at some specific point.
MRS = -dy/dx = MUx/MUY
A convex indiffernce curve says what about consumer preferences?
Consumers prefer a balance of goods rather than a lot of one good one little of another.
Name four common utility functions.
- Cobb-Douglas: U(x,y) = xay1-a
- IC is convex
- Perfect substitutes: U(x,y) = ax + by
- IC is straight diaganol line.
- a = alpha; b = beta
- Perfect complements: U(x,y) = min(ax,by)
- L shaped IC
- a = alpha; b = beta
- CES Utility: Constant elasticity of substitution
Are Cobb-Douglas, perfect substitute, perfect complements, and CES utilities homothetic?
Yes: The MRS depends only on the ratio of the amounts of two goods, not on absolute quantities of the goods.
- Cobb-Douglas: MRS = a/b * y/x
- a = alpha; b = beta
- perfect substitutes: MRS is the same at every point.
- perfect complements:
- MRS is infinity for y/x > a/b
- MRS is undefined when y/x = a/b
- MRS is 0 when y/x < a/b
What is an example of a utility function with non-homothetic preferences?
y is a neutral, i.e. straight horizontal line extending from y-axis. MRS = y so MRS diminishes as y diminishes, BUT is independent of x ‘cuz x has constant MU.
Willingness to give up y to get one more x depends only on how much y have.
Outline steps for maximizing utility given a budget constraint (BC).
- FOC: Point of tangency between BC and IC
- Slope of BC = Slope of IC
- Px/Py = -dy/dx = MRS. ==> Substitution of x for y.
- SOC: In order for necessary condition to also be sufficient one assumes that the MRS is diminshing, i.e. the utility function is strictly quasi-concave.
Rule of thumb for maximizing utility when have corner solutions?
- If slope of the BC is flatter than slope of ICs, the optimal point is on horizontal axis.
- If slope of the BC is steeper than the slope of the ICs, the optimal point is on the vertical axis.
Outline steps to maximize utility w/more than 2 goods s.t. BC (interior solutions).
- Set-up Lagrangian expression
- Set-up partial derivatives of L.
- Set equal to 0 and solve.
- SOC for maximum: Assumption of strict quai-concavity (diminishing MRS) is sufficient to ensure solution would be true max.
Interpret the Lagrange multiplier (lambda) in utility maximization.
At the optimum point, each good purchased should have an idential MB-to-MC ratio, i.e. an extra dollar should yield the same “additional utility” no matter which good it’s spent on.
Lambda can be regarded as MU of an extra $1 of consumption expenditure, i.e. MU of Income
How do FOC for maximization of utility with a corner solution differ from max with an interior solution?
Rater than partial derivatives of Lagrangian expression being set = 0, they are set < 0.
Given a demand function:
QD = 2P-0.25
What is the inverse demand function?
Inverse Demand Function:
P = 16QD-4
How can the type of good (substitute, complement, normal/inferior) be determined from the demand function?
- Substitute: f(Px2) > 0
- Complement: f(Px2) < 0
- Normal Good: f(Inc) > 0
- Inferior Good: f(Inc) < 0
**Above are all partial derivatives of the function:
QD1 = f(Px1,Px2,I)
Formula for MRS (Indifference Curves)
MRS = -dy/dx -for all- [U(x,y)=k]
MRS = -dy/dx = MUx / MUy
What is the formula for the marginal rate of transformation (Budget Constraint)?
Interpret an example of MRT.
MRT = | (I/Py)/(I/Px) | = | Px/Py | = | slope |
[Straight lines are “absolute value]
If MRT = 2, then 2 units of y will trade for one unit of x.
How can convexity of ICs be shown?
For two goods: Calculate the MRS = MUx/MUy
If as x increases and y decreases/increases, the MRS decreases, then convex.
If as x increases and y decreases, the MRS increases, then NOT convex and function is NOT quasi-concave.
What is the lump sum principle?
Taxes or subsidies on general purchasing power are “superior” than taxes or subsidies on specific goods. Taxes/subsidies on specific goods change a person’s purchasing power & distorts their choices.

What are 3 properties of expenditure functions?
- Homogeneity to degree 1: BC is linear in prices so any proportional increase in both prices and income will permit purchase of same bundle as before.
- Nondecreasing in prices: f(Pi) > 0 for every good i
- the expenditure function report minimum expense necessary to reach U*, so an increase in price for any good must increase this minimum.
- Concave in prices
TRUE/FALSE: Demand functions that are homogeneous of degree 0 (doubling of prices leaves QD unchanged) are a direct result of utility maximization assumption.
TRUE –>
- Demand functions derived from utility max. will be homogeneous.
- Demand functions that are not homogeneous cannot reflect utility max UNLESS prices enter directly into utility function as in a good w/snob appeal.
What are the two analytically different effects that come into play when a good’s price changes?
- Substitution effect: Even if person stays on the same IC, consumption patterns would be allocated so as to equate the MRS w/ the new price ratio.
- Income effect: A price change changes an individual’s “real” income –> person can’t stay on initial IC and must move to a new one.
When the price of one good changes, the axis-intercept for that good will change (of the BC). Thus, for the MRS to = slope of the BC, the MRS will change.
Outline steps demonstrating income and substitution effects graphically when the price of one good changes.
- Graph BC1
- Indicate where MRS1 = Slope of BC1
- Tangency between IC1 and BC1
- Graph BC2 showing the change in axis-intercept for the good whose price changed.
- Indicate where MRS2 = Slope of BC2
- i.e. draw the new IC w/tangency to new BC
- On IC1 sketch in BC3 that is parrallel to BC2
- Note: BC3 is not a real BC.
- The distance between “tangecy of IC1 and BC1” and “tangency of IC1 and BC3” along the effected good’s axis is the substitution effect.
- The distance between “tangency of IC1 and BC3” and “tangency of IC2 and BC2” is the income effect.

