Lecture 3: the Foundations of Demand Functions
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Lecture 3: The Foundations of Demand Functions
Are Demand Curves downward sloping?
The Basic Premises Assume - consumers have preferences and are consistent
If a consumer is confronted with two market baskets, containing different commodities, they can Rank the baskets. Exhibit consistent or transitive Preferences. More is better than less.
The Utility Function
Utility Function represents consumers preferences by assigns a numeric value to various combina- tions of goods
Properties - a two commodity utility function for apples and bananas - U(A,B)
Example: U = (AB)1/2
Utility from Different Baskets Choice Apples Bananas Utils A 4 1 2 B 2 2 2 C 3 3 3
Basket C is preferred to baskets A and B The consumer is indifferent between choices A and B.
The units in which utility is measured are irrelevant, other than for ordering.
Suppose second utility function W = U3. Thus,
W = U3 =(AB)3/2
Utility from same baskets, different utility function Choice Apples Bananas Utils A 4 1 8 B 2 2 8 C 3 3 27
Numbers differ, but the conclusions do not.
Do not attach any special significance to magnitude of these numbers, only their rank order. Indifference Curves - a plot of different combinations of goods that represent equal utility
- any combination of the goods lying along the same indifference curve indicates that the con- sumer is indifferent to them.
Note two properties: Indifference curves for two ‘goods’ are downward sloping Indifference curves do not intersect.
Indifference curves for apples and bananas
Indifference curves simply connect points representing different combina- tions that give equal utility. If you get fewer apples more bananas are re- quired to give equal utility. There are multiple indifference curves, corre- sponding - as one moves further from the origin - to higher utility.
Exercise - Consider the following statements and translate them into indifference curves.
“I could care less whether you have Coors or Budweiser, so long as it is beer.”
“I can’t stand either gin or vermouth, but martinis in the proper proportion turn me on.”
“Apples and Bananas are good substitutes. Double the number of Apples and halve the num- ber of Bananas, and I’m just as well off.”
The Mechanics of Maximization - used to show how consumers the combinations they purchase
- Indifference curves are independent of income, however, total expenditures cannot exceed total income. Mathematically,
paA + pbB = Y NOTE: the slope of this budget line is the relative price of A in terms of B (or vice versa).
Consumer wants to maximize utility subject to their budget constraint
Mathematically, max U(A,B) A,B
s.t. paA + pbB = Y The Solution The optimal combination occurs where the indifference curve is tangent to the budget line.
o The tangency condition means that the slope of the indifference curve is exactly equal to the slope of the budget line (or, if you wish, the relative price).
Utility Maximization
Utility maximization requires being on the highest indifference curve tan- gent to the budget line. Maximization is, after all, always constrained by budgets. The point (A*,B*) represents the best that can be done given the budget constraint. A Restatement in Terms of Marginal Utility Consider an individual with a utility function
U = AB with pa = 0.50, pb = 0.10, and Y = 100. - If he consumes 60 apples and 700 bananas, his utility is (60)(700) = 42,000. - the marginal utili- ty of an additional apple would be 700 units - the MUb = 60
- the marginal utility per dollar for apples is MUa/pa = 700/$0.50 = 1,400. - the marginal utility per dollar for bananas is MUb/pb = 60/$0.10 = 600. Which one will this person purchase next?
Total and marginal utility for different combinations of apples and bananas
A B U MUa MUb MUa/pa MUb/pb 60 700 42000 700 60 1400 600 70 650 45500 650 70 1300 700 80 600 48000 600 80 1200 800 90 550 49500 550 90 1100 900 100 500 50000 500 100 1000 1000 110 450 49500 450 110 900 1100
Utility Maximization occurs where the marginal utility per dollar are equal.
MUa/pa = MUb/pb
Diminishing MRS Restated: the ratio of marginal utilities must be equal to the ratio of prices
MUa / MUb = pa /pb
- pa / pb is the slope of the budget line.
- MUa / MUb is the slope of the indifference curve, the marginal rate of substitution marginal rate of substitution - the rate at which consumers are willing to substitute one good for another. A particular Indifference Curve
Apples Bananas 60 700 61 688.5 62 677.4 63 666.7 64 656.5
- diminishing marginal rate of substitution.
The case of diminishing MRS leads to an interior solution; that is, the minimum cost of purchasing a given level of utility involves both goods.
The case of increasing MRS means that you will buy only one good.
Cost of Different Points on the Indifference Curve with Dimin- ishing MRS
Apples Bananas MRS Cost
60 700.0 $100.00 61 688.5 11.5 $99.35 62 677.4 11.1 $98.74 63 666.7 10.8 $98.17 64 656.3 10.4 $97.63
90 466.7 5.2 $91.67 91 461.5 5.1 $91.65 92 456.5 5.0 $91.65 93 451.6 4.9 $91.66 Cost of Same Indifference Curve with a second utility function, with increasing MRS Apples Bananas Cost 60 700.0 $100.00 61 677.2 $98.22 62 655.6 $96.56 63 634.9 $94.99 64 615.2 $93.52 65 596.4 $92.14 66 578.5 $90.85
With a constant MRS and we find that we will purchase either one or the other, but not both. Which one depends on relative prices.
Composite Goods – multiple choices Consider a consumer choosing between Apples, Bananas, Oranges, and Grapes.
- consumer maximizes a utility function U(A,B,O,G) subject to a budget constraint that
pAA + pBB + pOO + pGG = I,
- to graphically analyze - assume that the choice is between consumption of Apples and “all other goods” or a composite good. Thus if we let
X = pBB + pOO + pGG,
And the utility function becomes U(A,X) and the budget constraint is
pAA + pxX = I, where the price of x is set at $1, or unitary.
drawback is if there are simultaneous changes in the price of apples and bananas or a change in the price of bananas influences the demand for apples
An Application to Grants in Kind An economic proposition: that gifts in cash are better than gifts in kind. Why a Gift in Cash is Preferable to a Gift in Kind
Initially the consumer is at (Xo, Ao). She is in indifference curve Io. She
now receives a gift of A1 apples with a cash value of X1. The extra apples place her on indifference curve I1. But a gift of X1 would have placed her in
indifference curve I2
Some Normative Problems The example of food stamps.
- consider one who has a weekly income of $400 and is initially on indifference curve Io. Suppose that he is spending $200 a week on food and $200 on other commodities.
- suppose they are offered the chance to buy $300 of food stamps for $150 in cash.
budget line becomes kinked and the optimum point does not satisfy MRS = pf/pog An Application to Food Stamps
Food stamps are a gift in kind. As this example shows, introducing food stamps gives a boost in utility, but not as much as if this were given as a gift in cash. The right to buy food stamps not as valuable as the extra $150 in cash.
So, do economists give Christmas gifts?