Task.

Exercise 2 - Preferences II. We start with two definitions: homotheticity and quasilinearity (See Varian 6.3.)

1 ? Preferences are said to be homothetic if

(q A 1 , qA 2 ) ? (q B 1 , qB 2 ) implies (?qA 1 , ?qA 2 ) ? (?qB 1 , ?qB 2 )

for any positive constant ? > 0.

Graphically this means that marginal rates of substitution are constant along rays through the origin.

A utility function u(q1, q2) representing homothetic preferences (when it exists) satisfies

u(?qA 1 , ?qA 2 ) = ?u(q A 1 , qA 2 ).

? Preferences are said to be quasilinear (with respect to good 1 ? in this case, good 1 is called the numeraire.) if

(q A 1 , qA 2 ) ? (q B 1 , qB 2 ) implies (q A 1 + ?, qA 2 ) ? (q B 1 + ?, qB 2 )

for any (positive or negative) constant ?.

Graphically this means that marginal rates of substitution are constant along lines parallel to the q1-axis.

A utility function u(q1, q2) representing quasilinear preferences (when it exists) satisfies u(q1, q2) = q1 + f(q2), where f(.) is a function of q2.

For each of the preferences represented by the following utility functions:

1. (perfect substitutes) u(q1, q2) = aq1 + bq2,

2. (perfect complements) u(q1, q2) = min[aq1, bq2],

3. (Cobb-Douglas) u(q1, q2) = (q1) ? (q2) (1??) , ? ? (0, 1),

4. u(q1, q2) = q1 + b ln(q2),

answer the following questions (For a start, you may wish to set a = 1, b = 2 and ? = 1/3, although higher marks will be given for answers using generic a, b and ?. You are welcome to use the mathematica file from Canvas for support.):

(a) In the (q1, q2) space, draw the indifference map (i.e. sketch the indifference curves).

(b) Calculate the marginal rate of substitution.

(c) Can you make out, graphically, whether these preferences are homothetic? Quasilinear? Justify your answers.

(d) Now support your answer to (c) using the mathematical relationships used to define quasilinearity and homotheticity.

Bonus question: are the preferences represented by u(q1, q2) = a ln(q1) + b ln(q2) CobbDouglas?.

Exercise 3.

3. The balance of payments of a nation is an accounting statement of the international economic transactions, over some period of time (usually a year), of the residents (individuals, business firms, government units) of the country. Entries into the balance of payments are either credits (exports) or debits (imports). (a) In terms of the balance of payments, what does it mean to say that a country is experiencing a "capital outflow?" Specifically, what is "moving" between countries when a country experiences a capital outflow? What is being credited and what is being debited? Carefully explain. (b) Explain what is meant by the current account, the capital account, and the official settlements account. In what sense does the balance of payments always balance? In what sense does a deficit in the "basic balance" present a problem for a country that operates under a system of fixed exchange rates (as under a gold standard, for example)? Under what conditions would a "basic balance" deficit (sometimes referred to as a "balance of payments" deficit) not present a problem? Carefully explain.Quasi-Linear Preferences Example #4: Quasi-linear Preferences (Non-Linear Utility) . Preferences are said to be quasi-linear when they are represented by utility functions of the form u(x1, X2) = V(x1) + x2 . Treating the second good as a numeraire (money), we see that MRS(x1, X2) = V'(x1), i.e., the MRS is independent of wealth. . This is a good approximation for discussing goods that the consumer spends only a small fraction of their income on (e.g., milk, bread; but not cars, homes). .Quasi - Linear Preferences Example # 4 : Quasi -linear Preferences ( Non Linear Utility) . Preferences are said to be quasi Linear when they are represented by utility functions of the form U ( x , x . ] = V ( X , ] + X z * simply the other A plus good Here we are given some function of one good However, the second good in this case is interpreted as a nameraine, It is money. The utility from each unit of money is simply one unit of blility & the utility of the other good ( v(KI) ) depends on how much there is So by treating the second good as money, we can see that the MRS is just the derivative of this function V . - M R S ( X, X L ) = V ( x], He the MRSis independent of wealth As derivative with respect lo Ke is Simply 1 - D I unit of utility for I vail of mancy Jo MRS being Independent of wealth simply means it depends simply on how much of the good you are buying & Is actually a good approximation for Many of the purchases that are done. Lo Le a good approximation for discussing goods that the comeumer only spends a small practice of their income or Che Bread mille but not