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EC109 > Consumer Theory: Basics & Optimisation > Flashcards

Consumer Theory: Basics & Optimisation Flashcards

1
Q

Budget constraint depends on (2)

A

1) prices

2) exogenous income

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2
Q

Points below the BL =

A

Affordable - part of feasible set

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3
Q

Points on BL =

A

Just affordable - constraint is binding

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4
Q

Intercepts of BL

A

M/Px and M/Py

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5
Q

Slope of BL

A

(-) Px/Py - ratio of prices. The rate at which the market substitutes good X for Y.

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6
Q

Impact of change in price on BL

A

Change in price = BL pivots around one intercept= change in slope.

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7
Q

How does change in income impact BL?

A

Shifts BL = no change in slope, both intercepts change

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8
Q

3 proprieties of preferences

A

1) completeness - can always rank bundles
2) transitivity: if X>Y and Y>Z, then X>Z
3) continuous - tiny changes in bubbles don’t affect preference ordering

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9
Q

2 properties of well behaved preferences

A

1) monotonicity - more is better

2) convexity - averages > extremes

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10
Q

Checking for monotonicity

A

dU/dX1 and dU/dX2 > 0

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11
Q

Checking for convexity

A

Indifference curves should be convex to the origin.
We should have diminishing MRS.
MRS should decrease in absolute value as X1 rises.

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12
Q

Indifference curves show

A

All combinations of X1 and X2 that generate the same level of utility. Utility is constant along a given IC.

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13
Q

6 properties of ICs

A
  1. Higher IC = more utility - monotonicity
  2. Continuous
  3. Convex to origin
  4. Downwards sloping
  5. ICs cannot cross
  6. ICs cannot be > 1 bundle thick
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14
Q

Why must ICs be downwards sloping?

A

Otherwise contradict monotonicity

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15
Q

Why cannot ICs cross?

A

If they cross, transitivity and monotonicity cannot both hold.

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16
Q

Why cannot ICs be > 1 bundle thick?

A

Contradicts monotonicity as if two points on the same IC, utility must be the same, but if one to the north east of the other should be associated with higher utility as more of both goods should be better.

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17
Q

The slope of an IC =

A

MRS at a particular point

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18
Q

MRS shows

A

The rate at which a consumer is willing to substitute good Y for X based on their individual preferences

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19
Q

MRS formula

A

MRS = (-) MU1/MU2 - ratio of marginal utilities of each good. These are the partial derivatives of the utility function.

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20
Q

Why does MRS diminish for convex ICs?

A

When we don’t have much X1, MU1 is high = high MRS = steep
When we get more of X1, MU1 falls - we become less willing to sacrifice lots of units of X2 as X2 becomes scarce and we have lots of X1.

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21
Q

Formula for perfect substitutes

A

U(X, Y) = aX + bY

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22
Q

Formula for perfect compliments

A

U(X, Y) = MIN {aX, bY}

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23
Q

Formula for cobb Douglas

A

U(X, Y) = X^a Y^(1 - a)

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24
Q

Monotonic transformations must be…

A

POSITIVE transformations - this means ICs have same shape AND we have monotonicity. dU/dX and dU/dy >0

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25
Q

ICs and MRS for perfect substitutes

A 
MRS = a/b = constant along an IC - no diminishing MRS
ICs = straight lines
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26
Q

ICs and MRS for perfect compliments

A

MRS = infinity on vertical section; zero on horizontal; undefined at kink.
ICs are L shaped.

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27
Q

Does monotonicity hold for perfect compliments?

A

It holds from one IC to another one, but not along a given IC.
If we increase X1, utility is unchanged. Utility only increases if we increase both X1 and X2 in their certain ratio.

28
Q

When do we get symmetric ICs for cobb Douglas around 45 degree line?

A

When the exponents are equal.

29
Q

MRS for cobb Douglas

A

Negative a/(1 - a) X2/X1

30
Q

Formula for quasilinear tastes - which good is linear? Which is independent of income?

A

U(X, Y) = V(X) + Y

  • X is independent of income
  • Y is LINEAR good
31
Q

MRS for quasilinear and relationship between ICs

A

MRS = V’(X) / 1 = V’(X)

Each IC is vertically shifted version - MRS is constant along vertical lines. MRS only changes if X changes.

32
Q

MRS for non convex preferences

A

We have increasing MRS as we move done the IC.

33
Q

Shape of ICs if X= a bad and Y= a good.

A

Upwards sloping - if X rises, Y must rise to compensate for the disutility in order to remain along the same IC.

34
Q

Shape of ICs if X=a normal good and Y=neutral good

A

ICs are vertical lines. Change in the neutral good = no impact on utility. Utility only changes and so ICs only shift if X changes.

35
Q

MRS for neutral good on Y axis and normal good on X.

A

MRS = MUx/Muy = infinity / 0 = infinity.

36
Q

What is non-satiation?

A

We have an overall best bundle “point of bliss”
More is not always better - we can have too little and too much
ICs closer to bliss point = higher utility

37
Q

What is revealed preferences?

A

Relying on consumer behaviour to determine preferences - we don’t have a utility function. A consumer’s choices of bundles over others reveals their preferences.

38
Q

If (X1, X2) is chosen over (Y1, Y2), it must be that…

A

P1X1 + P2X2 = M
P1Y1 + P2Y2 < equal M
So P1X1 + P2X2 > equal P1Y1 + P2Y2
Therefore, (X1, X2) is directly revealed preferred

39
Q

What is WARP? Explain with an example.

A

Weak axiom reveal preference
Directly revealed preferences
E.g. Bundle (X1, X2) bought at prices (P1, P2) when (Y1, Y2) also affordable. Bundle Y bought at Q prices so bundle X must be more expensive at Q. If X > Y at prices P and Y not > X at Q then this satisfies WARP.

40
Q

What is SARP? Explain with an example.

A

Strong axiom revealed preference = directly & indirectly revealed by transitivity
If market behaviour if not transitive, SARP violated.

41
Q

Homothetic preferences =

A

Consumers preferences only depend on the ratio of good X to Y. If income increases, the ratio doesn’t change. So as we move onto higher ICs, MRS unchanged = MRS constant along any ray from the origin.

42
Q

Homothetic preferences in maths

A

If (X1, X2) ~ (Y1, Y2), then (tX1, tX2) ~ (tY1, tY2)

  • If we increase X and Y by same proportion t, equality is maintained. Two bundles on same IC1 now on same higher IC2 after being scaled up.
43
Q

3 examples of utility functions with homothetic preferences

A

Cobb-Douglas
Perfect substitutes
Perfect compliments
- Higher ICs are parallel shifts

44
Q

1 example of non- homothetic utility function

A

Quasi-linear - MRS constant along vertical rays, not a ray from the origin.

45
Q

What does elasticity of substitution measure? And give formula.

A

Elasticity of sub = % change in (X2/X1) / % change in MRS

  • measures degree of substitutability
  • how responsive a bundle of goods along an IC is to changes in MRS (perhaps due to change in price)
46
Q

Elasticity of sub for perfect substitutes

A

Infinity

- % change in MRS = 0 as it is constant along an IC

47
Q

Elasticity of sub for perfect compliments

A

Zero

- no substitutability

48
Q

Elasticity of sub for cobb Douglas

A

= 1

49
Q

Relation between curvature of ICs and elasticity of sub

A

Less curved = the more X2 has to fall and X1 has to rise for a 1% change in MRS.

So less curved = higher elasticity of substitution

50
Q

What is constant elasticity of substitution?

A

A formula that captures all values of sigma for different utility functions. All utility functions taking this form have sigma equal at all bundles.

51
Q

CES formula & what values can p take?

A

(aX1^-p + (1 - a) X2^-p) ^-1/p

-1 < equal p < equal infinity

52
Q

Elasticity of substitution from CES function

A

Elasticity of sub = 1 / (1 + p)

53
Q

What value of p for CES tends towards perfect compliments?

A

P –> infinity means elasticity of sub –> 0

= perfect compliments

54
Q

What value of p for CES tends towards perfect substitutes?

A

P –> - 1 means sigma –> infinity

= perfect substitutes

55
Q

What value of p for CES tends towards cobb Douglas?

A

P = 0 means sigma = 1

= cobb Douglas

56
Q

Tangency condition for utility max. When does it apply?

A

Applies to interior solutions.

BL slope = slope of IC
P1/P2 = MRS = MU1/MU2

57
Q

The solution to utility max yields …

A

Marshallian demands: X1* (p1, p2, M) & X2* (p1, p2, M)

58
Q

What is the degree of homogeneity of demand w.r.t. Prices and income?

A

K = 0

  • if prices double and income doubles, real income unchanged = demand for X unchanged.
59
Q

Is the tangency condition necessary? It is a sufficient condition?

A
  • it is necessary condition
  • not sufficient if preferences are not convex - if we have a cubic function that is convex and concave, FOCs may give minimums = check second order condition.
60
Q

The Langrange multiplier in utility max =

A

Lambda = marginal utility of income

dU/dM

61
Q

How do we find the optimal bundle for corner solutions?

A

Compare MUx/Px to MUy/Py

If MUx/Px > MUy/Py - X generates higher marginal utility per pound spent.
X* = M/Px and Y* = 0

62
Q

Optimum for perfect substitutes

A

Always a corner solution

63
Q

Optimum for perfect compliments

A

Optimise at the kink: aX = bY

Substitute this into the budget constraint.

64
Q

Optimum for cobb Douglas

A

Use Lagrangian - always an interior solution
X* = alpha M/ Px
Y* = (1 - alpha) M/ Py

65
Q

Optimum for quasilinear

A 
Either an interior or corner solution 
- Find interior using Lagrangian
- Sub X* into budget constraint to find Y*
- If Px X* > M then this gives Y* < 0
So we have a corner solution
66
Q

Optimum for concave ICs

A

Corner solution as extremes > averages

Compare MUx/Px to MUy/Py

EC109 (5 decks)

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