Utility, Demand Functions, Elasticity

Question 1

What is indirect utility?

Answer 1

max utility = U[x₁*(p₁, ..,p_n, I), x₂*(p₁, …, p_n, I),…]

= V(p₁, p₂, …, p_n, I).

v = indirect utility

Optimal utility depends indirectly on price of goods and income.

Question 2

What does the expenditure function show and what is its relationship to indirect utility?

Answer 2

Expenditure function shows minimal expenditures necessary to achieve a given utility level for a particular set of prices. minimal expenditures = E(p_1,p₂, …,p₃, U).

Expenditure funtion and indirect utility function are inverse functions of one another.

e.g. V(p_x, p_y, I) = I / [2p_x^0.5p_y^0.5] , where I is income.

E(P_xP_y, U) = 2P_x^0.5P_y^0.5U, where U is utility

(*Don’t inverse the constraint).

Question 3

What are the axioms of rational choice?

Answer 3

1. Completeness –> All preference orderings are known.
2. Transivity –> If A < B and B < C, then A < C
3. Continuity –> If A is preferred to B, then situations suitably close to A must be preferred to B
4. Non-satiation –> More is better

Question 4

Explain uniqueness of utility measures.

Answer 4

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.

Question 5

What is the marginal rate of substitution?

Answer 5

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

Question 6

A convex indiffernce curve says what about consumer preferences?

Answer 6

Consumers prefer a balance of goods rather than a lot of one good one little of another.

Question 7

Name four common utility functions.

Answer 7

1. Cobb-Douglas: U(x,y) = x^ay^1-a
   
   1. IC is convex
2. Perfect substitutes: U(x,y) = ax + by
   
   1. IC is straight diaganol line.
   2. a = alpha; b = beta
3. Perfect complements: U(x,y) = min(ax,by)
   
   1. L shaped IC
   2. a = alpha; b = beta
4. CES Utility: Constant elasticity of substitution

Question 8

Are Cobb-Douglas, perfect substitute, perfect complements, and CES utilities homothetic?

Answer 8

Yes: The MRS depends only on the ratio of the amounts of two goods, not on absolute quantities of the goods.

1. Cobb-Douglas: MRS = a/b * y/x
   
   1. a = alpha; b = beta
2. perfect substitutes: MRS is the same at every point.
3. perfect complements:
   
   1. MRS is infinity for y/x > a/b
   2. MRS is undefined when y/x = a/b
   3. MRS is 0 when y/x < a/b

Question 9

What is an example of a utility function with non-homothetic preferences?

Answer 9

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.

Question 10

Outline steps for maximizing utility given a budget constraint (BC).

Answer 10

1. FOC: Point of tangency between BC and IC
   
   1. Slope of BC = Slope of IC
   2. P_x/P_y = -dy/dx = MRS. ==> Substitution of x for y.
2. 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.

Question 11

Rule of thumb for maximizing utility when have corner solutions?

Answer 11

1. If slope of the BC is flatter than slope of ICs, the optimal point is on horizontal axis.
2. If slope of the BC is steeper than the slope of the ICs, the optimal point is on the vertical axis.

Question 12

Outline steps to maximize utility w/more than 2 goods s.t. BC (interior solutions).

Answer 12

1. Set-up Lagrangian expression
2. Set-up partial derivatives of L.
   
   1. Set equal to 0 and solve.
3. SOC for maximum: Assumption of strict quai-concavity (diminishing MRS) is sufficient to ensure solution would be true max.

Question 13

Interpret the Lagrange multiplier (lambda) in utility maximization.

Answer 13

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

Question 14

How do FOC for maximization of utility with a corner solution differ from max with an interior solution?

Answer 14

Rater than partial derivatives of Lagrangian expression being set = 0, they are set < 0.

Question 15

Given a demand function:

Q_D = 2P^-0.25

What is the inverse demand function?

Answer 15

Inverse Demand Function:

P = 16Q_D^-4

Question 16

How can the type of good (substitute, complement, normal/inferior) be determined from the demand function?

Answer 16

• Substitute: f(P_x2) > 0
• Complement: f(P_x2) < 0
• Normal Good: f(Inc) > 0
• Inferior Good: f(Inc) < 0

**Above are all partial derivatives of the function:

Q_D1 = f(P_x1,P_x2,I)

Question 17

Formula for MRS (Indifference Curves)

Answer 17

MRS = -dy/dx -for all- [U(x,y)=k]

MRS = -dy/dx = MU_x / MUy

Question 18

What is the formula for the marginal rate of transformation (Budget Constraint)?

Interpret an example of MRT.

Answer 18

MRT = | (I/P_y)/(I/P_x) | = | P_x/P_y | = | slope |

[Straight lines are “absolute value]

If MRT = 2, then 2 units of y will trade for one unit of x.

Question 19

How can convexity of ICs be shown?

Answer 19

For two goods: Calculate the MRS = MU_x/MU_y

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.

Question 20

What is the lump sum principle?

Answer 20

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.

Question 21

What are 3 properties of expenditure functions?

Answer 21

1. 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.
2. Nondecreasing in prices: f(P_i) > 0 for every good i
   
   1. the expenditure function report minimum expense necessary to reach U*, so an increase in price for any good must increase this minimum.
3. Concave in prices

Question 22

TRUE/FALSE: Demand functions that are homogeneous of degree 0 (doubling of prices leaves Q_D unchanged) are a direct result of utility maximization assumption.

Answer 22

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.

Question 23

What are the two analytically different effects that come into play when a good’s price changes?

Answer 23

1. 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.
2. 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.

Question 24

Outline steps demonstrating income and substitution effects graphically when the price of one good changes.

Answer 24

1. Graph BC₁
2. Indicate where MRS₁ = Slope of BC₁
   
   1. Tangency between IC₁ and BC₁
3. Graph BC₂ showing the change in axis-intercept for the good whose price changed.
4. Indicate where MRS₂ = Slope of BC₂
   
   1. i.e. draw the new IC w/tangency to new BC
5. On IC₁ sketch in BC₃ that is parrallel to BC₂
   
   1. Note: BC₃ is not a real BC.
6. The distance between “tangecy of IC₁ and BC₁” and “tangency of IC₁ and BC₃” along the effected good’s axis is the substitution effect.
7. The distance between “tangency of IC₁ and BC₃” and “tangency of IC₂ and BC₂” is the income effect.

Question 25

How does interpretation of the income and substitution effects differ by type of good (i.e. normal or inferior)?

Answer 25

• For a normal good, the income and substitution effects are reinforcing (move in the same direction).
• For an inferior good, the IE and SE move in opposite directions so the effect of change in price is indeterminate. For a price decrease: SE tends to decrease Q demanded while income effect tends to increase Q demanded.

Graph of price increase of inferior good:

Question 26

What is a giffen good?

Answer 26

A mythological good for which a price increase increases Q_D of that good. –> If the income effect is strong enough, the change in price and resulting change in Q_D could move in the same direction.

Supposed event: Price of potatoes rose in Ireland and Q_D potatoes also rose. Supposed explanation. Potatoes were staples so people needed to buy potatoes reducing real income. People could no longer afford other foods and so had to use remaining income on buying more potatoes.

Question 27

What causes shifts in demand curve?

Answer 27

1. Income
2. Prices of other goods
3. Preferences

Question 28

What are the assumptions inherent in a Marshallian (uncompensated) demand function?

Answer 28

Nominal income (I-bar) and prices of other goods are held constant.

x* = x(p_x,p_y-bar,I-bar)

Question 29

What is a Hicksian (compensated) demand curve? How does it differ from Marshallian demand curve?

Answer 29

Hicksian (compensated) demand curve holds real income (U-bar) and prices of other goods constant whereas Marshallian demand holds nominal income (I-bar) constant.

• U-bar = constant utility or real income
• I-bar = constant income or nominal income.

x^c= x^c(p_x,p_y,U).

Effects of change in price on purchasing power are “compensated” to constrain individual to remain on the same IC (level of U) –> So reactions to price changes include only substitution effects.

• Price of x increases –> Nominal income increased to keep person on same IC
• Price of x decreases –> Nominal income decreased to keep person on same IC

Question 30

What is Shephard’s lemma?

Answer 30

Application of the envelope function which shows that a consumer’s compensated demand function x^c can be derived from the partial derivation of the expenditure function:

E(p_x,p_y,U): f(p_x) = partial differentiation of the Lagrangian expression w.r.t. p_x = x^c(p_x,p_y,U). Because…

1. Both function (E and x^c) depend on the same variables Px, Py, U
2. Differerentiating a minimized function so any price change will be met by series of adjustments in Q_d to continue to minimize expenditures.
3. Price change of a good will effect expenditures roughly in proportion of Q_d of that good.

Question 31

What is an insight and a benefit of the compensated (Hicksian) demand curve?

Answer 31

Benefit: No ambiguity resulting from when income and substitution effects work against one another as with uncompensated demand curve.

Insight: slope of Hicksian demand curve is negative.

Question 32

Graphically, how to Marshallian and Hicksian demand curves differ?

Answer 32

• At higher prices:
  
  • Q_d higher under Hicksian than Marshallian demand functions because nominal income is increased to keep consumer at same level of U (under Hicksian)
• At lower prices:
  
  • Q_d lower under Hicksian because nominal income is decreased to keep consumer at same level of U.
• For a normal good, Marshallian demand is more responsive to price changes than Hicksian demand.

Question 33

Demonstrate the Slutsky equation.

Answer 33

Question 34

What are the formulas for elasticity of Marshallian demand?

Answer 34

Given function: x(P_x, P_y, I)

• Price elasticity of demand:
  
  • e_x,Px = f(P_x) * P_x/x , where f(P_x) is the partial w.r.t. P_x and x = demand function
• Income elasticity of demand:
  
  • e_x,I = f(I) * I/x
• Cross-price elasticity of demand:
  
  • e_x,Py = f(P_y) * P_y/x

Question 35

When will compensated and uncompensated demand elasticities be similar?

Answer 35

1. Share of income (s_x) of good x is small
2. Income elasticity for good x (e_x,I) is small

Question 36

Define unit elastic, elastic, and inelastic.

Answer 36

• Unit elastic: % change in Q_d = % change in P
  
  • e_x,Px = -1 —-> 1% increase in price results in 1% descrease in Q_d
• Elastic: % change in Q_d > % change in P
  
  • e_x,Px < -1 —–> 1% increase in price results in 3% decrease in Q_d
• Inelastic: % change in Q_d < % change in P
  
  • e_x,Px > -1 —-> 1% increase in price results in 0.3% decrease in Q_d

Question 37

What is compensating variation?

Answer 37

Amount of compensation required to keep someone on the same IC after a price increase. It is a concept in welfare economics.

CV = E (P_x¹, P_y, U_o) - E(P_x⁰, P_y, U_o)

Compensating surplus (variation) assumes that an individual has the right to an initial level of public goods, Q, for proposed decreases in Q, but does not have the right to an increased level of Q. Put another way, compensating surplus is the amount an individual must give in order to receive an increase in Q (WTP) or the amount the individual must receive to accept a decrease in Q (WTA); thus, in both instances, leaving the individual at their initial utility level.

Question 38

How is compensating variation determined graphically?

Answer 38

1. Draw expenditure functions at original price
   
   1. E(P_x⁰, P_y, U_o)
2. Draw new expenditure function reflecting price change as if it were BC and person was moving to new indifference curve
3. Construct new expenditure function by drawing a line parallel line to BC from #2 above that is tangent to U_o. Extend function to y-axis.
   
   1. E(P_x¹, P_y, U_o)
4. CV = difference between curves #2 and #3 above.

CV can also be determined by integrating the x^c between the old and new prices of x.

Question 39

What is consumer surplus?

Answer 39

Graphically it is the area below the demand curve above equilibrium price. It represents the $ consumers would be willing to pay, but don’t have to.

Question 40

What are determinants of own price elasticity of demand?

Answer 40

1. Availability of substitutes which in turn is impacted by time horizon under consideration, i.e. the longer the time horizon the more substitutes will be available.
2. Degree to which good is a luxury. Elasticity will be greater for luxury goods.
3. Market definition. As good is defined more specifically, demand becomes more elastic.
4. Budget share (s). The greater s_x, the more elastic Q_d for that good will be.

Question 41

Re-write the Slutsky equation to reflect cross-price effects on demand.

Answer 41

See Notes.

Question 42

What is equivalent compensation (EV)?

Answer 42

The amount that an individual would be willing to give up (or be paid) to prevent prices from changing.

Equivalent surplus (variation) assumes that an individual has the right to an increased level of Q and so must be compensated in order to remain indifferent between the current and higher levels of Q. Put another way, equivalent surplus is the amount of compensation an individual must receive in order to forego increased levels of Q (WTA) or the amount an individual must give in order prevent a decrease in Q (WTP); thus, in both instances, leaving the individual at the new utility level.

Question 43

How do compensating and equivalent variations/surplus work?

Answer 43

Compensating and equivalent welfare measures are measures of the amount of compensation required to make an individual indifferent to an exogenous change. Compensating and equivalent welfare measures differ with respect to their reference points and so with respect to the initial assignment of property rights. Compensating/equivalent variation refers to price changes while compensating/equivalent surplus refers to quantity changes. The discussion below will be in terms of surplus (quantity changes).