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Esame di Microeconomia avanzata (16 aprile 2004)

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Esame di Microeconomia avanzata (16 aprile 2004) ...

    Advanced Microeconomics I

    thExamination Set (July 6, 2007)

    (Paolo Bertoletti)

Time: 90 minutes. (Gli studenti di laurea specialistica rispondano solo alle prime tre domande,

    avendo a disposizione 75 minuti)

I. An individual with income m has preferences over two goods, consumption c and the

    bequest to her children b. Her preferences are described by the following utility function (for

    b ? ? > 0):

     U(c,b)?lnc?ln(b??)Given p = p and p = 1, and assuming an interior solution (which conditions are required for cb

    it?): a) derive the Marshallian demands and the indirect utility function; b) discuss if the above

    preferences are homothetic; c) define s = 1 (pc)/m the propensity to save out of income: what is the relation between s and m? Does ? play any role?

    II. Show graphically the case of an “inferior” input: why under homotheticity no input can be

    inferior? Why an “inferior” input must also be “regressive”? III. Prove that, for a rational consumer: 1) if she is interested only in two commodities they must

    be net substitutes for her (under convexity of the preferences); 2) a Giffen commodity

    cannot be “normal”; 3) not all commodities can be inferior. IV. Consider the case of Leontief preferences defined over two commodities (perfect

    complements): in the case of two commodities, illustrate graphically the equivalent and the

    compensating variation if only a price changes at the time.

    thAdvanced Microeconomics I - Examination Set (July 6, 2007) - SOLUTIONS

    (Paolo Bertoletti)

    I An individual with income m has preferences over two goods, consumption c and the bequest to her children b. Her preferences are described by the following utility function (for b ? ? > 0):

     U(c,b)?lnc?ln(b??)

    Given p = p and p = 1, and assuming an interior solution (which conditions are required for it?): a) cb

    derive the Marshallian demands and the indirect utility function; b) discuss if the above preferences

    are homothetic; c) define s = 1 (pc)/m the propensity to save out of income: what is the relation

    between s and m? Does ? play any role?

    A) The MRS (in absolute value) is given by (b - ?)/c, so the condition for an interior solution is that

    pc = b - ? and the Marshallian demands result: c(p, m) = (m - ?)/2p and b(p, m) = (m + ?)/2 (this clearly requires m > ?). B) Since the MRS is not constant along a ray in the (c, b) space (or, equivalently, since the ratio b(p, m)/c(p, m) does depend on m), preferences are not homothetic. C) 2By direct substitution, s = ? + ?/(2m), thus s/m < 0 and s/m? < 0. ?????

II Show graphically the case of an “inferior” input: why under homotheticity no input can be

    inferior? Why an “inferior” input must also be “regressive”?

    In the following picture input 1 is inferior.

    x2

    ? =w/wtg12

    y’

    y> y ?

    y

    ? x1 xx1 1

    ~xUnder homotheticity, the cost function can be written c(w, y) = g(w)h(y), with (w, y) = g(w)h(y), iiby Shephard’s lemma. Accordingly, all inputs have the same elasticity wrt the level of output,

    h’(y)y/h(y), which is also the elasticity of cost wrt to y and thus cannot be negative. I.e., all inputs are normal.

    ~x(p, w) ? (w, y(p, w)), where y(p, w) is the supply function, it follows that: By the identity xii

    ~?x?x?yii?, ?p?y?pand since the supply function is increasing wrt to the output price p, it turns out that the

    unconditional demand of input i is decreasing wrt p iff the conditional demand of the same input decreases wrt y. I.e., and input is regressive iff is inferior.

III Prove that, for a rational consumer: 1) if she is interested only in two commodities they must be

    net substitutes for her (under convexity of the preferences); 2) a Giffen commodity cannot be

    “normal”; 3) not all commodities can be inferior.

    ?h/?p?h/?p1) By differentiating the identity u(h(p, u), h(p, u)) ? u wrt ot p, u(h)= - u(h), 121121121

    ?h/?pwhere u > 0 by monotonicity and ? 0 by concavity of the expenditure function. Thus i11

    ?h/?p?h/?p = ? 0 and inputs are net substitutes. 2) By Slutsky’s equation: 2112

    ?x?h?xiii??x. i?p?p?mii

    ?h/?pSince ? 0, the lhs can be positive only if the second term on the rhs is positive, which ii

    ?x/?mrequires < 0: i.e., commodity i needs to be inferior. 3) By differentiating the budget i

    constraint wrt m one gets

    n?xi?p1 ?i?m?1i

    ?x/?mwhich would be false if < 0, i = 1, …, n. i

IV Consider the case of Leontief preferences defined over two commodities (perfect complements):

    in the case of two commodities, illustrate graphically the equivalent and the compensating variation

    if only a price changes at the time.

    ’ < p, p = 1: I examine the case of p112

    x2

    m + EV

    m

     u

    u

    x1 m/pm/p1 1’

     x2

    m

     m - CV u

    u

    x1 m/pm/p1 1’

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