MULTIPLE -CHOICE QUESTIONS

 Imagine that 4 units of exes and 3 units of whys cost the same as 3 units of exes and 5 units of whys, and your income m allows you to buy exactly 4 units of exes and 3 units of whys. Then, the budget line is    given by the equation

(a) x + y = 7.

(b) x/2+ y = 5.

(c)  2x + y = 11.

(d) x + y = 8.            (e) None of the above.

(HINT: As a first step, write the budget equations with the information given and express unit prices as a function of m.)

DETAILS :

4px +3py     =   3px +5py  ! px  = 2py

4px +3py   =   m ! 8py +3py = m

m                  2m

 Kana thinks that 2 units of good 1 is always a perfect substitute for 3 units of good 2. Which of the fol-

lowing utility functions represents Kana’s preferences?

(a)  U(x1 ,x2 ) = 3x1 +2x2 +1000

(b)  U(x1 ,x2 ) = (2x1 +3x2 )2

(c)  U(x1 ,x2 ) = min{3x1 , 2x2 }

(d)  U(x1 ,x2 ) = x1(2)x2(3)

(e)  U(x1 ,x2 ) = p3x1 +2x2

DETAILS : According to Kana the two goods are perfect substitutes and the utility of consuming 2 units of good 1 must be equal to the utility of consuming 3 units of good 2.

 Georgina consumes only grapefruits and pineapples. Her utility function is U(x,y) = x2  · y8 , where x is the number of grapefruits consumed and y is the number of pineapples consumed. Georgina’s income is $105, and the prices of grapefruits and pineapples are $1 and $3, respectively. How many grapefruits will she consume?

(a)  10.5

(b)  7

(c) 63

(d)  21

None of the above.

DETAILS : Georgina is going to spend  of her income on grapefruits, given that her preferences can  be represented by a Cobb-Douglas type utility function. With those $21 is able to purchase 21 units of grapefruits.

1

 Ambrose’s indifference curves can be described by the following equation: x2  = constant − 6x1(2) , where a larger constant corresponds to a higher indifference curve. If good 1 is drawn on the horizontal axis     and good 2 on the vertical axis, what is the slope of Ambrose’s indifference curve when his consumption bundle is (9, 5)?

(a)  −

(b)  −1

(c)  −

(d)  −2 (e)  −6

1                                                                                                                                1

DETAILS : From the above equation we get that u(x1 ,x2 ) = 6x1(2)  + x2 . This means that MU1  = 3x1(−) 2 ,

1

MU2  = 1, and MRS = 

 The only information we have about Goldie is that she chooses bundle (6, 6) when prices are (6, 3) and she chooses bundle (10, 0) when prices are (5, 5). From this information, we can conclude that ...

(a)  ... bundle (6, 6) is revealed preferred to (10, 0) by Goldie and there is no evidence that she violates

WARP.

(b)  ... neither bundle is revealed preferred to the other by Goldie.

(c)  ... Goldie violates WARP.

(d)  ... bundle (10, 0) is revealed preferred to (6, 6) by Goldie and she violates WARP.

(e)  ... the bundle (10, 0) is revealed preferred to (6, 6) and there is no evidence that she violates WARP.

(HINT: Represent Goldie’s budget constraints and the bundles that she chooses in a graph.)

DETAILS : When prices are (6, 3), the cost of bundle (6, 6) is 54 and the cost of bundle (10, 0) is 60. When prices are (5, 5), the cost of bundle (6, 6) is 60 and the cost of bundle (10, 0) is 50. Note that when bundle (6, 6) was chosen, (10, 0) was not affordable, and vice versa.

 If the demand curve is a downward-sloping straight line, then ...

(a)  ... the price elasticity of demand is constant all along the demand curve.           (b)  ... the price elasticity of demand is zero at the peak of the sellers’ total revenue.

(c)  ... the price elasticity of demand is larger than one when the marginal cost is sufficiently low.

(d)  ... the price elasticity of demand changes from zero to minus infinity as the price rises.

None of the above.

(HINT: Consider the following demand function: D(p) = a − b · p, where a,b > 0.)

DETAILS: Consider D(p) = a − b · p, and note that ✏p(D)  =  ·  = −  .

 Miss Muffet insists on consuming 3 units of whey per 1 unit of curds. If the price of curds is $5, the     price of whey is $3, and Miss Muffet’s income is m, then Miss Muffet’s demand function for curds can be written as

(a) (b)

(c)

(d)

(e)

m

18 .

m

20 .

3m

18 .

3m

20 .

None of the above.

(HINT: Note that whey and curds are perfect complements for Miss Muffet.)

DETAILS : Given that the two goods are perfect complement, Miss Muffet would want to consume 3       times as much whey as curds: w = 3 · c. Her budget constraint is 5c +3w = m. From the two equations, we get that c =  .

8.  Which of the following statements is NOT correct as a description of the impact of quantity taxes (such as a sales tax or an excise tax)?

(a) It does not matter if the tax is levied on sellers or on buyers.

(b) Tax incidence depends on both the price elasticity of demand and the price elasticity of supply.

(c) The fraction of a quantity tax paid by buyers rises as supply becomes more price elastic.

(d) The fraction of a quantity tax paid by buyers rises as demand becomes more price elastic.

(e) Imposing a quantity tax may not necessarily reduce both consumer surplus and producer surplus.

9.  A firm has the production function f(x1 ,x2 ) = 1.2 · (x1(0) .5 + x2(0) .5 ) whenever x1  > 0 and x2  > 0. When the amounts of both inputs are positive, this firm has ...

(a)  ... increasing returns to scale.

(b)  ... decreasing returns to scale.

(c)  ... constant returns to scale.

(d)  ... increasing returns to scale if x1 + x2  > 1 and decreasing to returns to scale otherwise.

(e)  ... increasing returns to scale if output is less than 1 and decreasing to returns to scale if output is greater than 1.

DETAILS: Take t > 1 and consider f(t · x1 ,t · x2 ) = 1.2 · [(t · x1 )0.5 +(t · x2(0) .5 )] = t0 5. · 1.2 · (x1(0) .5 + x2(0) .5 ). Then we can conclude that f(t · x1 ,t · x2 ) < t · f(x1 ,x2 ).

10.  When the total cost function is C(y) = 99 + (y − 1)2 , how much output should the firm produce in order to minimize the average total cost?

(a) 0

(b)  1

(c)  2

(d)  10

11

DETAILS : At the minimum point of ATC(y), we have that ATC(y)  =  MC(y). It means that  + (yy(−)1)2   = 2(y − 1) ! 99 + y2 − 2y +1 = 2y2 − 2y ! y2  = 100.

OPEN QUESTIONS

11.  Carol’s preferences are represented by the utility function

U(x1 ,x2 ) = min{2x1 + x2 ,x1 +2x2 } =

If Carol is initially consuming 5 units of x1 and 7 units of x2 , then what is the largest number of x2 that she would be willing to give up in return for an additional 4 units of x1 ?

Carol’s initial bundle is (5, 7), while the proposed one can be written as (5 + 4, 7 − ∆x2 ), where ∆x2 is the largest number of x2 that she would be willing to give up in return for an additional 39 units of x1 .

The two bundles must give the same utility level to Carol, so we can write that

u(5, 7)   =   u(5 + 4, 7 − ∆x2 ).

min{2 · 5+7, 5+2 · 7}   =   min{2 · 9+7 − ∆x2 , 9+2 · (7 − ∆x2 )} min{17, 19}   =   min{25 − ∆x2 , 23 − 2 · ∆x2 }

17   =   23 − 2 · ∆x2

2 · ∆x2     =   6

∆x2     =   3

Charlie’s utility function is xA  · xB , where xA is the number of apples and xB is the number of bananas that he consumes. Both the price of apples and the price of bananas used to be $2, and his income used to be $40. The price of apples has recently increased to $6, while the price of bananas has remained $2

and Charlie’s income has remained $40.

Compute the Slutsky substitution effect on Charlie’s apple consumption.

Given his utility function, we know that Charlie is going to spend the same amount on apples as on ba- nanas. For that reason, before the price change he consumes 10 units of apples and 10 units of bananas. After the price change he consumes  units of apples and 10 units of bananas.

Note that Charlie would need an additional ∆m = 10 · ∆pA  = 40 units of income in order to be able to afford the original bundle after the price change.

If he had an income of $80, Charlie would consume  units of apples and 20 units of bananas. For that reason, the Slutsky substitution effect on Charlie’s apple consumption is  − 10 = −  ⇡ −3.33.

Casper consumes cocoa and cheese. Cocoa is sold in an unusual way. There is only one supplier, and     the more cocoa you buy from him, the higher the price you have to pay per unit. In fact y units of cocoa will cost Casper y2 dollars. Cheese is sold in the usual way at a price of 2 dollars per unit. Casper’s in-   come is 20 dollars and his utility function is U(x,y) = x+2y, where x is his consumption of cheese and y is his consumption of cocoa.

(a) Sketch Casper’s budget set and shade it in.

(b) Sketch some of his indifference curves and mark the point that he chooses.

(c) Calculate the amount of cheese and the amount of cocoa that Casper demands at these prices and this income.

Casper’s budget constraint can be written as 2x + y2    20. From that, we get that y  p20 − 2x.

Note that Casper’s indifference curves are straight lines with a slope of MRS = −  .

Even if the indifference curves are straight lines, the utility-maximizing bundle can be characterized by a tangency condition, because the budget “line” is strictly concave.

The slope of the frontier of the budget set is: p20 − 2x =  = −  .

The slope of the indifference curves is MRS = −  .

Setting the to to be equal gives x⇤ = 8. Substituting this back to the budget constraint (2x⇤ + y ⇤2  = 20) y ⇤ = 2.

Suppose that a firm’s production function is f(L,M)  =  4LM , where L is the number of units of labor and M is the number of machines used.

(a) Find the firm’s conditional factor demand functions.

(b) If the cost of labor is $40 per unit and the cost of machines is $40 per unit, then how much is the total cost of producing y units of output?

We need to solve the firm’s cost-minimization problem, i.e.

minpL  · L + pM  · M

L,M

subject to y = 4L M

Given that the production function is a Cobb-Douglas type function, the solution is characterized by a

tangency condition: |TRS| =  !  =  .

From there, we get that M = L ·  . By substituting this expression into the production function and  rearranging it for y, we have one of the firm’s conditional factor demand functions: L⇤ =  · q  . The other one is then M ⇤ =  · q  .

With the help of the previous results, we can write the firm’s cost function as c(y) = pL · L⇤ +pM · M ⇤ = 2 ·  · pwM  · wL  =  · p40 · 40 = 20y.