II. CONSUMER DEMAND

    A. THE RATIONAL CONSUMER

    3. THE CONSUMER'S OPTIMUM

    Goal now: to combine the utility and constraint components of consumer theory to see how consumers reach an optimum, maximizing their utility given their income and price constraints.

    a. COMPARING VALUE AND COST

    ? What should you do?
    a. Consume more Y and less X.
    b. Consume less Y and more X.
    c. Continue consuming your present quantities.

    --the logic: bargain hunting
    (Worksheet: Choosing the best feasible bundle: comparing value and cost)

    ? What is your cheapest alternative?

    Shift your budget toward your cheapest alternative.

    b. GEOMETRY

Example: Food budget = $160     Health food (H=goodX): Ph = $2     Junk food (J=goodY): Pj = $1     ? BL equation?     BLo: J = 160 - 2H     Initial situation: Consuming on Uo.     ? How do points a-d compare in terms of desirability?     ? In terms of affordability?     Suppose that presently you are on your BL and       MRS=3     ? Which point in the diagram, a or c, would reflect that, and how did you decide?

    ? What does the slope of Uo tell you?

    ? What does the slope of BL tell you?

    ? How does relative value compare to relative price in this case?

    Linking back to bargain hunting:
    X the better bargain => PyDY > PxDX => DY/DX (=MRS) > Px/Py

    ? How would you adjust your budget to raise your utility above Uo, and how did you decide?

    ? How would you decide where to stop?

    End result:
    Tangency Condition (T.C.) for an optimum:

    Bargain hunting situation at your optimum:

    All equally preferred options cost the same.

    Rules to maximize your satisfaction.:
        (1) Be on BL => use all your income
        (2) Allocate income such that: relative value = relative cost

    --buy more if value > cost
    --buy less if value < cost

    Other ways to consider the value-cost comparison:
    1. Since you're willing to trade $3 of junk food for $2 of health food and be indifferent about it, you can be better off by getting more health food.
    2. When you consider consuming more X and less Y, you are essentially using good Y as a "currency" for buying good X. If you're willing to spend more Y for another X than you have to, then do it.
    If you are willing to spend less Y than you have to for X, then stop spending so much on X.

    General rule for an optimum:
    Allocate your income so that marginal value = marginal cost

    If you're not doing that, you can always improve your well-being by making a consumption adjustment

    c. MATHEMATICS

    (1) Generalizing to many goods

    Through calculus we can demonstrate how the geometry generalizes to our world of many items to choose from.

    We start by rearranging the tangency condition:

    In this form we can tack on as many items as we like.
    The result is...

    The equi-marginal principle:

    => Allocate your income to get the same amount of extra satisfaction per $ spent on any item.

    No item offers any better value per $ than any other item

    Example: Utils of pleasure from your last $ spent on...
        Food: 9 / Shelter: 9 / Clothing: 12
    ? What should you do?

    (2) Solving consumer optimization problems

    Why do this?
    It's artificial - U is not quantifiable in practice.
    But it explicitly forces you to utilize the 2 key optimality relationships: the tangency condition and the budget constraint.

    Given U = U(X,Y), I, px, py
    Goal: find X and Y which maximize U

    2 unknowns: X and Y
    2 equations:

    (1) Tangency Condition (T.C.):
            MRS = Px/Py, where MRS = MUx/MUy

    (2) Budget Constraint (B.C.): I = Px X + Py Y

    Worksheet
    Worksheet results
    Spreadsheet (Excel)

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