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  • Microeconomics 3

    Economics Programme, University of Copenhagen

    Spring semester 2006

    Week 10

    Lars Peter sterdal

  • Todays programme

    Production economies.

    Theme 3: Market imperfections. Introduction

    Public goods.

  • General equilibrium in production economies

    We now consider a more general (and perhaps more relevant) the model.

    We go from an exchange economy to an economy with production.

    Modelling issues:

    Production technology

    Firm behavior

    Ownership

  • Production technology

    m rms

    k goods (a good can be consumed by consumers, it can be an input factor toproduction, or both)

    A net output vector for rm j is a vector yj = (y1j ; :::; ykj ).

    Negative entries: net input (ex., steel, crude rubber, labor, ...)

    Positive entries: net output.(ex., cars)

  • Yj is the the production possibilities set for rm j.

    Example: One input, one output

  • Y

    Input

    Output

  • Production function (one output only):

    f(x) = fmax y j (y;x) 2 Y g

  • Transformation function (multiple outputs):

    We say that y in Y is e cient if there is no y0 2 Y such that y0 y, y0 6= y.

    A transformation function is a function T such that T (y) 0 for all y 2 Yand T (y) = 0 if and only if y is e cient:

  • Our notation is very convenient:

    If p is a price vector, then pyj is prot associated with production plan yj.

    Firm behavior:

    Assume that each rm solves

    maxpyj

    subject to yj 2 Yj

  • Remark 1: This problem has generally no solution under increasing returnsto scale (why? ).

    Remark 2: This problem has generally not a unique solution under constantreturns to scale (why? )

  • Y

    Input

    Output

    Isoprofit curves

    Profitmax

  • Solving this prot-maximization problem gives us net supply yj(p) for rm j.

    For m rm

    y(p) =mXj=1

    yj(p)

    is aggregate net supply.

  • Y =Pmj=1 Yj is the aggregate production possibilities set.

    i.e. y 2 Y if and only if y 2 Pmj=1 yj, where yj 2 Yj.NB: No production externalities in this model! (why? )

    Theorem: The following is equivalent:

    i) Aggregate production y maximizes aggregate prot,

    ii) Each rms production yj maximizes its individual prot.

    Intuitively, this theorem is rather clear given our assumption that there are noproduction externalities. (make sure you understand this intuition).

  • We now look at labor supply and prot distribution.

    Labor supply (input to production) is modelled as follows:

    A consumer has L units of time available (ex. L = 24):

    Divide between labor l and leisure L = L l.

  • Example:

    The consumer cares about leisure L and a consumption good c.

    w: wage rate.

    p: price of consumption good.

    c: endowment of consumption good.

  • Consumers problem:

    maxu(c; L)

    subject to pc = pc+ w(L L); 0 L L; 0 c.

    (we can rewrite budget constraint: pc+ wL = pc+ wL).

  • Distribution of prots

    Consumers own the rms.

    We assume that ownerships are historically given and xed (thus consumerscannot buy and sell stocks in this model).

    Tij is consumer is share of the prots of rm j.

    (remember that we have n consumers and m rms)

    Pni=1 Tij = 1, j = 1; :::;m.

  • Consumer is total prot income:

    Pni=1 Tijpyj(p)

    and his/her budget constraint is:

    pxi = p!i +Pni=1 Tijpyj(p)

    Consumer is demand: xi(p):

  • Aggregate demand: x(p) =Pni=1 xi(p)

    Aggregate excess demand: z(p) = x(p) y(p)| {z }remember:input negative!

    !.

    Remark about notation: Varian writesX(p) instead of x(p), and he sometimeswrites Y(p) instead of y(p) (you can write it the way you prefer as long asyou make sure the meaning is clear).

  • Walraslaw holds again: pz(p) = 0 for all p.

    The reason is the same: If each consumer satises his/her budget constraintwith equality, then the economy as a whole also satises the aggregate budgetconstraint with equality.

    Walrasian Equilibrium (production economy): (x;y;p) is a W.E. if z(p) 0.

    i.e., if supply demand on all markets, when consumers maximize utility givenp and rms maximize prots given p.

  • Theorem: Existence of Walras Equilibrium (Arrow-Debreu 1954):

    An equilibrium exists if the following is satised (see next two slides):

  • For each consumer i:

    1. The consumption set is closed, convex, and bounded from below (ex. Rk+).

    2. There is no satiation consumption bundle.

    3. Preferences are continuous. That is, the sets fxijxi x0ig and fxijx0i xig are closed for each x0i.

    4. The initial endowment is in the interior of the consumption set.

    5. xi i x0i implies txi + (1 t)x0i x0i for any 0 < t < 1. (convexitycondition).

  • Firms:

    6. For each rm j; 0 2 Yj (always possible to produce nothing).

    7. Y =Pmj=1 Yj closed, convex (guarantees continuity of rms aggregate

    net supply correspondence)

    8. Y \ (Y ) f0g:That is, the only vector y 2 Y for which y 2 Y isy =f0g (irreversible production).

    9. Y R+. (free disposal)

  • The welfare theorems can be generalized:

    First theorem of welfare economics: If (x;y;p) is a W.E. then (x;y) isweakly Pareto e cient:

    Proof: Somewhat similar to that for an exchange economy. Read yourself.

  • Second theorem of welfare economics: Suppose that (x;y) is a feasi-ble Pareto e cient allocation in which each consumer holds strictly positiveamounts of each good, and where preferences are convex, continuous, andstrongly monotonic. Suppose further that Yj is convex for j = 1; :::;m. Thenthere exists some vector p 0; and a suitable reallocation of initial wealth(i.e. a reallocation of prot shares and endowments), such that (x;y;p) isa W.E.

    (skip the proof)

  • Robinson Crusoe economy.

    Suppose that unit price of labor w and unit price of consumption good is 1.

    Initial endowment: (0; L)

    Illustrate in a "Koopmans diagram":

  • Labor (input)

    Leisure

    Consumption

    L

    Real equlibrium profit

    Equilibriumconsumption

    Income

    Slope=-w/1=-w

  • Theme 3: Market imperfections.

    Market imperfections: Deviations form the assumptions of perfect competition(as in Varian ch. 17 & 18).

    3.a. Public goods. Varian ch. 23

    3.b. Externalities. Varian ch. 24

    3.c. Externalities in production economies. PNS

  • Public goods

    Until now, we have assumed that goods are ordinary private consumption goods:

    In particular, they are excludable (people can be excluded from consuming it)and rival (one persons consumption reduces the amount available to others).

    Public goods are non-excludable and non-rival.

  • Classical examples:

    Streetlights (non-excludable, non-rival)

    Lighthouse (non-excludable, non-rival)

    Clean air (non-excludable, non-rival)

    Military defence (non-excludable, non-rival)

    Basic research (non-excludable, non-rival)

  • Some good are non-rival but excludable ("club good"):

    TV broadcast (non-rival, excludable if coded or requires payment of licencefee)

    Music and other digital goods that can be downloaded on the web.

  • Some good are rival but non-excludable:

    Highways (non-excludable unless charging tolls to road users, rival)

    Libraries (non-excludable unless charging fees, rival)

    Beaches (usually non-excludable, rival)

  • There are many in-between cases. Whether a good is rival/non-rival andexcludable/non-excludable is often a matter of interpretation/judgement. Strictlyspeaking, we need an explicit model (as at next slide) for an exact investigation.

    Note that if a good is provided by the public, it doesnt have to be a publicgood:

    Some good of often provided by the public, but still ordinary private in nature(ex. many kinds of health care goods)

  • As we will see, competitive markets may not be good for allocating publicgoods.

    First, we will establish conditions that can tell us under what circumstances itis e cient to provide a public good.

  • E cient provision of a discrete public good

    Model with two agents and two goods.

    xi consumption of private good (think of it as money spent on private con-sumption)

    wi initial endowment of private good.

    G public good (non-excludable, non-rival)

    gi agent is contribution to public good.

  • xi = wi gi

    ui(G; xi) (assume strictly increasing in xi)

    G =

    (1; if g1 + g2 c0; g1 + g2 < c.

  • If there is (g1; g2); g1 + g2 c,

    such that

    u1(1; w1 g1) > u1(0; w1)

    and

    u2(1; w2 g2) > u2(0; w2):

    Then providing the public good (with payments (g1; g2)) is Pareto e cient.

  • Let ri be is reservation price; that is, is maximum willingsness-to-pay forthe public good.

    ui(1; wi ri) = ui(0; wi)

    If ui(1; wi gi) > ui(0; wi) for both i, we have:

    ui(1; wigi) > ui(0; wi) = ui(1; wiri). And since ui is strictly increasingin its second argument:

    wi gi > wi ri for both i.

  • This implies:

    ri > gi:

    Therefore:

    r1 + r2 > g1 + g2 c.

    That is: the sum of reservation prices is greater than c:

  • Conversely, suppose that we have r1 + r2 > c.

    For "i > 0, dene gi = ri "i:

    Then we can select "1; "2 su ciently small such that r1 + r2 > g1 + g2 > c:Thus:

    ui(1; wi gi) > ui(0; wi)

    and providing public goods with payments g1; g2 is a Pareto