I. Review of markets

    B. Elasticity

    Recall: Sometimes it is not enough to know the direction in which Qd or Qs changes as a result of some shock to an equilibrium.

    Sometimes we want to measure how much Qd or Qs changes--i.e., responsiveness

    Elasticity does this.

    Handout: Elasticity formulas
    Worksheet: Elasticity review (tip: recall the arc elasticity formula)

General formula for market elasticities (e):

           %DQ
    e = ---------
           %DX

where X=something which influences Q, such as price (P)

    Larger elasticity => greater responsiveness

    (1) Real world relevance: direction vs. size of change in Qd.

    (2) Percentages are the key

        Across goods: Asahi vs. Mazda

        Over time: $400 change in PC prices

There are many elasticities:

Own-price elasticity of demand:
           %DQd
    e = ----------
           %DP

e is negative, but we'll usually drop the minus sign.

Income elasticity of demand:
            %DQd
    e_I = ----------
            %DI

Cross-price elasticity of demand:
                %DQdx
    e_x,Py = -----------
                 %DPy

Elasticity of supply:
             %DQs
    e_S = -----------
              %DP

    1. CALCULATING ELASTICITY

    a. ARC ELASTICITY

    Arc elasticity (e_ARC): elasticity for a distinct pair of points (X1,Q1) and (X2,Q2):

                 DQ/avgQ
    e_ARC = --------------
                 DX/avgX

where
    DQ=(Q2-Q1)
    avgQ=(Q1+Q2)/2
    DX=(X2-X1)
    avgX=(X1+X2/2.

Ex: Own-price elasticity of demand:                  DQd/avgQd     eARC = -----------------                    DP/avgP    We use avgQ and avgP so we get the same percentage answers whether P rises or falls. Consider D with two points:      P  Qd   a  6   4   b  4   8 Worksheet: Elasticity review

    b. POINT ELASTICITY

    Point elasticity gives you a precise estimate of elasticity; as the name suggests, at a specific point

Ex: own-price point elasticity of demand:

	%DQd		DQd/Qd		DQd		P			P
e =	----------	=	-----------	=	------	·	----	=	dQd/dP ·	----
	%DP		DP/P		DP		Qd			Qd

where, at the point (Qd,P):
    dQd/dP = the derivative of D, and
    P/Qd = the ratio of P to Qd.

    point e = dQd/dP · P/Qd

    Since dq/dp is negative by law of demand, e < 0.
    By convention, we often drop the sign on own-price elasticity of demand, or talk only about its size (I do that, text doesn't).
    I don't care which you use

Example: linear demand for movie theater tickets:

    Qd = 4771 - 656 P
    dQd/dP = -656, drop (-) sign =>
    e = dQd/dP · P/Qd
       = 656 · P/Qd
    P = 5 =>
    Qd = 1491 =>
    e = 656 · 5/1491 = 2.2

   Geometry and calculations (Excel)

    To do on homework: Use the P and Qd information and the linear elasticity formula to prove that halfway down a linear demand curve: e = 1 (demand is unit elastic).

    So note that a linear demand curve contains all the possible elasticity values:
    (1) elasticity drops continuously from infinity to 0 as you move down the demand curve.
    (2) elasticity switches from elastic to inelastic exactly halfway down.

    Note: The simple formula here holds only for linear demand, not all demand curves.

    2. PRICE ELASTICITY (e) AND TOTAL EXPENDITURE (TE)

    TE = P·Q
    +P --> -Q (by law of demand)
    DTE = ?

    P and Q move in opposite directions when we move along D, so which effect wins?
    e tells us: e = %DQ/%DP

    Rules:
(1) |e| < 1: inelastic: P and TE move in the same direction
(2) |e| > 1: elastic: P and TE move in opposite directions
(3) |e| = 1: unit elastic: TE is constant as P changes

    ? Your advice to theater owner to raise revenue?

Start: Day 4

How does the geometry look?            DQd    P       A     e = ------ · ---- = ---             DP    Qd      B in the diagram to the right.     So in terms of geometry, elasticity corresponds to the ratio of area A to area B when price changes.     (1) Inelastic demand => B>A;     (2) Elastic demand => A>B; and     (3) Unit elastic demand => A=B.

    (One final note: Recall that for a linear demand curve, demand is elastic until halfwy down the curve; then it becomes inelastic. That means that lower prices keeps raising P·Q until the midpoint; after that, P·Q starts to fall again. So for a linear demand curve, TR and TE are maximized at the midpoint price and quantity.)

    3. APPLYING PRICE ELASTICITY

    Ex.1: U.S. sugar quotas

    Ex.: Tax incidence (Java)

Who really pays a tax? The answer is given by 'tax incidence' which breaks a tax into the percentages paid by buyers and sellers. How a tax gets divided depends on the elasticities of demand and supply.

For any good:
    Pb=Price buyers pay
    Ps=Price sellers receive
    A per-unit tax (t) raises Pb above Ps by the amount t:
        Pb - Ps = t => Pb = Ps +t

    Note that who pays a tax depends on the shapes of D and S.
    In general, the party which has the lower elasticity gets 'stuck' paying more of the tax
    Gasoline: demand is pretty inelastic
    Labor market: supply is pretty inelastic
    These results are true regardless of what any laws say.

    A per-unit subsidy (sub) raises Ps above Pb by the amount sub:
        Ps - Pb = sub => Ps = Pb + sub

    To draw it: shift S down by size of subsidy

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