You have 2 hours and 45 minutes to answer all the questions.

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Universidad Carlos III de Madrid May 2016 Microeconomics

Name: Group:

1 2 3 4 5 Grade

You have 2 hours and 45 minutes to answer all the questions.

1. Multiple Choice Questions. (Mark your choice with an “x.” You get 2 points if your answer is correct, -0.66 points if it is incorrect, and zero points if you do not answer.)

1.1. An individual’s preferences over consumption bundles (x; y) 2 R 2 + are complete and transitive (axioms A:1 and A:2). If the individual considers the consumption of good x to be detrimental to her welfare and the consumption of good y to be bene…cial, then his indi¤erence curves

cross are increasing

are concave have area.

1.2. A consumer with a monetary income I = 4 is considering buying the bundle (x; y) = (4; 0) at prices p x = p y = 1: If her marginal rate of substitution at this bundle is M RS (4; 0) = 1=2, then the consumer should

buy more x and less y buy more x and more y buy more y and less x buy the bundle (4; 0).

1.3. Assume that x is an inferior good. If p x decreases, then

the demand of x decreases the demand of x may increase or decrease the demand of y increases the demand of x increases.

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1.5. If the preferences of the consumer in question 1.4 are represented by the utility function u(x; y) = 2x + y, then her true CPI is:

3 4

3 2

4

3 1.

1.6. The preferences of a consumer over lotteries are represented by the Bernoulli utility function u(x) = p x. Identify the expected utility and the risk premium of the the lottery l = (x; p) which pays x = (0; 4; 16) with probabilities p = ( 1 4 ; 1 2 ; 1 4 ):

Eu(l) = 2; RP (l) = 2 Eu(l) = 4; RP (l) = 2 Eu(l) = 2; RP (l) = 2 Eu(l) = 4; RP (l) = 2:

1.7. A …rm that produces a good from labor (L) and capital (K) according to the production function F (L; K) = minf2L; Kg has:

decreasing returns to scale constant returns to scale economies of scale a concave total cost function.

1.8. If a competitive …rm produces a positive output, then the market price is equal to its marginal cost and larger than its average cost

equal to its marginal cost and larger than its average variable cost equal to its average cost and larger than its marginal cost

equal to its average variable cost and no smaller than its marginal cost.

1.9. In a long run competitive equilibrium of a market

all …rms produce the same quantity the price equals the …rms’average cost all …rms use the same technology

the price is smaller …rms are in the market.

1.10. If a monopoly produces a good with zero cost, then the monopoly’s Lerner index is

0 1

2 2 1

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2. A consumer’s preferences over clothing (x) and food (y) are represented by the utility function u(x; y) = x + 2py. The price of clothing is p x = 1 euro per unit, the price of food is p y = p euros per unit and the consumer’s income is I euros.

(a) (10 points) Calculate her demand for food y(p; I) for I 1

p and for I < 1 p : Solution: Since

M RS(x; y) = 1

2 2py

= p y;

an interior solution to the consumer’s problem solves the system:

p y = 1 p x + py = I:

Solving the system we get

y(p; I) = 1

p 2 ; x(p; I) = I 1 p :

Since x cannot be negative, when the consumer’s income is less that one euro ( I < 1

p ), the she consumes x = 0 and y = I

p ; that is, if I < 1

p (and therefore the consumer cannot by 1

p 2 units of food ), then she buys the maximum amount of food feasible, y = I

p , and no clothing.

(b) (5 points) Represent graphically the consumer’s budget set for p = 1

2 and I = 1; and calculate her optimal bundle and utility level.

Solution: The budget line for (p; I) = ( 1 2 ; 1) is

x + y 2 = 1:

Since I = 1 < 2 = 1

p ; the optimal consumption bundle is (x ; y ) = (0; 2); the consumer’s utility utility is u(0; 2) = 2 p

2: The graph below illustrates these calculations.

1 2

y 3

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(c) (10 points) With the data of part (b), calculate the income and substitution e¤ects over the demand for food of a unit tax of 50 cents on the price of food. Calculate the tax revenue.

Solution: The tax increases the price of food to p 0 = 1: In order to calculate the substitution e¤ ect ( SE) we solve the system:

p y B = 1 x + 2 p y = 2 p

2:

Hence y B = 1; and the substitution e¤ ect is

SE = y B y = 1 2 = 1:

To calculate the income e¤ ect ( IE)we …rst calculate the total e¤ ect ( T E):

T E = y(p 0 ) y(p) = 1 2 = 1:

Hence

IE = T E SE = 1 ( 1) = 0:

The tax revenue is

T = 1

2 y(p 0 ) = 1

2 euros.

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3. (15 points) Consider an individual about to retire with preferences for leisure (h; measured in hours) and consumption (c; measured in euros) represented by the utility function u(h; c) = hc, whose salary is 15 euros/hour, and has a total of 140 hours per month available to use as leisure or supply as labour. He is entitled to a monthly pension 1200 euros, but if he continues working, this pension would be reduced by t(15l); where l is the number of hours he works and t 2 [0; 1=2].

Write down the individual’s budget constrain, draw his budget set and calculate his labor supply l(t). For which values of t will he choose to retire altogether?

Solution: For t 2 [0; 1=2] the consumer’s budget constrain is:

0 c 1200 + 15(1 t) (140 h) ; 0 h 140:

His budget set is graphed below.

h c

140 1200

Since the consumer’s M RS(h; c) = c=h; an interior solution to his maximization problem is a solution to the system

c

h = 15 (1 t)

c = 1200 + 15 (140 h) (1 t) : Solving this system we obtain the consumer’s leisure demand

h(t) = 70 + 40 1 t : Hence his labor supply is

l(t) = 70 40 1 t : A solution to the equation

l(t) = 70 40

1 t = 0

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4. Consider a competitive market where 10 …rms produce a good with the same production function, F (L; K) = p

L + 3K: Prices of labour and capital are w = 1 and r = 4 euros per unit.

(a) (10 points) Compute the conditional demands for labour and capital, the total, average and marginal cost functions, and the supply function of each …rm. (Hint: draw an isoquant.)

We can see that the isoquants are linear and parallel, as shown by marginal rate of technical substitution,

M RT S(L; K) = 1 3 ; which is constant. With the prices w = 1 and r = 4, we have

M RT S(L; K) = 1 3 > 1

4 = w r :

Hence producing using only labor as input is cheaper (capital can be replaced by labor and save cost). Therefore, the conditional input demands are

L(Q) = Q 2 K(Q) = 0:

The total cost function is

C(Q) = Q 2 : The marginal and average cost functions are

M C(Q) = 2Q; AC(Q) = Q:

Note C( Q) = 2 Q 2 > Q 2 for all > 1; which implies that …rms have diseconomies of scale.

Also, since

M C(Q) = 2Q > AC(Q) = Q;

the closing down condition holds for all P 0:

Therefore, the …rms’s supply function is the solution to the equation P = M C(Q) = 2Q:

Hence

Q S i = S i (P ) = P

2 :

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(b) (10 points) The market demand is D(p) = max f100 5p; 0g : Compute the market equilibrium, and evaluate the impact of a unit tax T = 2 on the equilibrium output and price, and on the consumer and producer surplus.

Since there are 10 …rms in the market, the market supply is X 10

i=1

Q S i = 10S i (P ) = (10) P 2 = 5P:

The competitive equilibrium price P is the solution to the equation D(P ) = S(P ):

Assume that P 20; the equation becomes 100 5P = 5P . Therefore, P = 10( 20) and Q = 50:

Let P be the consumer price, and let ^ P = P 2 be the price received by producers. Then the market supply is

S( ^ P ) = 5 ^ P ;

which is calculated in part (b), we can re-write the function of P (for P 2) as S(P ) = 5 (P 2) :

The equilibrium price under tax P T can be derived from the following 100 5P = 5(P 2):

Hence, P T = 11: The equilibrium quantity Q T = 100 5 (11) = 45: The price for producers is P ^ T = 11 2 = 9:

In the equilibrium without tax, the consumer and producer surplus are

CS = 1

2 (20 10) 50 = 250 P S = 1

2 (10) 50 = 250:

With the tax the consumer and surplus are CS T = 1

2 (20 11) 45 = 202; 5 P S T = 1

2 (9) 45 = 202; 5:

Therefore, the tax reduces both consumer surplus and producer surplus by 47; 5 euros each. The

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5. The owner of sport events TV broadcasting rights monopolizes the market of a country in which the number of households (in thousands) connected by adsl that demand the service is D A (p) = maxf200 p; 0g; and the number of households connect by cable that demand the service is D C (p) = maxf300 p; 0g: The monopoly’s marginal cost is zero.

(a) (10 points) Calculate the equilibrium price, the number of households of each type viewing sports events on TV, the …rms pro…ts, and the consumer surplus in the absence of price discrimi- nation.

Solution: In order to calculate the aggregate demand, we note that for prices p > 200 euros only the demand of households connected by cable is positive. Hence if the monopoly sets a price greater than 200 euros only these households will have access to viewing sport events, and the demand is 300 p = q; that is, P (q) = 300 q: At prices below 200 also the household connected by adsl will have a positive demand, and the total demand is q = (200 p) + (300 p) ; that is, P (q) = 250 q

2 : Therefore the aggregate (inverse) demand is

P (q) = 8 >

<

> :

300 q if 0 q > 100 250 q

2 if 100 q < 500

0 if q 500:

The monopoly’s revenue function is R(q) = P (q)q; and its marginal revenue (R 0 (q)) is

R 0 (q) 8 <

:

300 2q if 0 q > 100 250 q if 100 q < 500

0 if q 500:

The monopoly equilibrium is obtained solving the condition R 0 (q) = M C(q): Assuming that q < 100; we have the equation

300 2q = 0;

whose solution is q = 150 > 100: Hence in equilibrium q > 100; so that the condition R 0 (q) = M C(q) yields the equation

250 q = 0;

that is, q = 250:

Thus, the monopoly equilibrium is q M = 250 and p M = 125: The number of households con- nected by cable who have access to viewing sport events is 175 thousand and that of households connected by adsl is 75 thousand. The monopoly’s pro…t is

= 125 (250) = 31 250:

The consumer surpluses are

CS A = 1

2 (75) 2 ; CS C = 1

2 (175) 2 :

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(b) (10 points) Determine the equilibrium prices and quantities under third degree price discrimi- nation, and evaluate its e¤ect over the monopoly’s pro…ts, the surplus of households of either type, the consumer surplus, and the total surplus.

Solution: The monopoly’s problem is

q

A

max ;q

C

0 I A (q A ) + I C (q C ) C (q A + q C ) = P A (q A )q A + P C (q C )q C ; where

P E (q) = 300 q if 0 q > 300

0 if q 300; and P H (q) = 200 q if 0 q > 200

0 if q 200:

Equilibrium is obtained by solving the system:

300 2q A = 0 200 2q C = 0:

That is, q A = p A = 100; q C = p C = 150:

The monopoly pro…t is

d = 100 2 + 150 2 = 32:500;

which is greater than without price discrimination.

The consumer surpluses are

CS A d = 1

2 100 2 ; CS C d = 1 2 150 2 :

Hence with price discrimination the households connected by adsl are better of, but the households connected by cable are worse o¤ . The total consumer surplus is smaller than without price discrim- ination,

CS d = 1

2 100 2 + 1

2 150 2 < 1

2 75 2 + 1

2 175 2 = CS:

Also the total surplus is smaller with price discrimination than without price discrimination, CS d + d = 3

2 100 2 + 3

2 150 2 < 1

2 75 2 + 1

2 175 2 + 125 (250) = CS + :

Figure

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