Problem 1.

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?

Suppose a representative consumer has preference over consumption (c) and leisure (1). His indifference curve is described by the following function: u(c, I) = 3 In(c) + 2 In(!). His endowment over time is given by h = 1. His wage rate is 2 in terms of consumption goods. Suppose he has 2 units of dividend income from firm and 1 unit of lump-sum tax from the government. (2) Compute this consumer's optimal choice over consumption, leisure and working time. (15 points) Suppose a representative consumer has preference over consumption (c) and leisure (). His indif- ference curve is described by the following function: u(c, l) = 31n(c) + 21n(1). His endowment over time is given by h = 1. His wage rate is 2 in terms of consumption goods. Suppose he has 2 units of dividend income from firm and 1 unit of lump-sum tax from the govern- ment. (2) Compute this consumer's optimal choice over consumption, leisure and working time. (15 points) (3) Now suppose there is an increase in the lump sum tax. Diagram the effects on consumer's optimal consumption and labor supply. Explain your results. (15 points) Note: under the above utility function, du/al MRSl,c= du/ac (3) Now suppose there is an increase in the lump sum tax. Diagram the effects on consumer's optimal consumption and labor supply. Explain your results. (15 points) Note: under the above utility function, MRSI,c = au/81 au/acVery Berry Strawberry Farm produces strawberries. Strawberries are produced according to: Number of Workers Tons of Strawberries Produced 0 0 12 2 22 3 30 36 5 50 According the table above, the production function of the farm exhibits O A. diminishing total product O B. diminishing marginal product O C. increasing marginal product O D. constant marginal product O E. Increasing total product OF. None of the aboveSpring 20lo, Towson University n For the demand and supply curve P- 10l- 30 ap - 3040, the per demand at the equilibrium point in: A) me - 2/0 m) 60 - 5/3 0) 60 - 3/2 D) 60 - 2/3 7. A consumer faced with the utility function U - 2X 4 373 consumes 10 units of and 5 units of good Y. What is his marginal utility of good X at this consumption A) NOY =2 B) MUY = 10- C) MUY = 3 D) MUY = 5 8. The demand function for basketball tickets is Q = 45000 - 509P and perfectly inelastic @ = 30000. The market clearing price is therefore: 4) 30 3) 35 45 50 For the utility function U = 5X + 2Y, the marginal rate of substit IRS = X/Y (RS = Y/X RS = 5X/2Y RS = 5/2 the demand and supply curves P = 100 - 3Q and P eiling of $25 creates: lus of 20 units he of 25 units