### Universidad Carlos III de Madrid June 2014 Microeconomics

### Name: Group:

### 1 2 3 4 5 Grade

### You have 2 hours and 45 minutes to answer all the questions. The maximum grade for each question is in parentheses. You should manage your time accordingly.

### 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. Which of the following axioms are not satis…ed by the Pareto preferences _{P} (de…ned for (x; y) ; (x ^{0} ; y ^{0} ) 2 R ^{2} + as (x; y) P (x ^{0} ; y ^{0} ) if x x ^{0} and y y ^{0} ):

### A:1 (completeness) A:2 (transitivity) A:3 (monotonicity) A:4 (continuity).

### 1.2. If a consumer’s preferences are represented by the utility function u(x; y) = 2 minfx; 2yg; her income is I = 12 and the prices are p _{x} = p _{y} = 1; then her optimal consumption bundle is:

### (6; 6) (8; 4) (0; 12) (12; 0).

### 1.3. If x is an inferior good, then the signs of the substitution (SE), income (IE) and total (T E) e¤ects of an increase of its price p x are:

### SE < 0; IE > 0; T E < 0 SE < 0; IE > 0; T E: inderterminate SE < 0; IE < 0; T E < 0 SE > 0; IE > 0; T E: inderterminate.

### (QUESTION CANCELLED IN THIS ENGLISH VERSION.)

### 1.4. If a consumer’s preferences are represented by the utility function u(x; y) = 2x + y; her income is I = 4 and the prices are p _{x} = p _{y} = 1; then the equivalent variation of a sales tax on 1 euro per unit of good x is:

### 0 1 2 4:

### 1.5. Identify the expected utility and risk premium of the lottery l that pays x = (0; 1; 4) with probabilities p = ( ^{1} _{4} ; ^{1} _{2} ; ^{1} _{4} ); for an individual with preferences represented by the Bernoulli utility function u(x) = p x:

### Eu(l) = 1; RP (l) = 1

### 4 Eu(l) = 1

### 2 ; RP (l) = 1 4 Eu(l) = 1; RP (l) = 1

### 2 Eu(l) = 1

### 2 ; RP (l) = 1 2 :

### 1.6. A …rm that produces a good using labor (L) and capital (K) according to the production function F (L; K) = minf2L; p

### Kg has:

### increasing returns to scale constant returns to scale economies of scale a convex cost function.

### 1.7. If a …rm’s average cost function is CM _{e} (Q) = p

### Q, then:

### its marginal cost is decreasing

### its average cost is below it marginal cost it has constant returns to scale

### its total cost function is concave.

### 1.8. The demand of a good is D(P ) = maxf70 10P; 0g and the cost function generated by the single technology available to produce the good is C(Q) = Q ^{3} 6Q ^{2} + 10Q: Then in the long run competitive equilibrium the price p and the number of …rms n are:

### p = 3; n = 60 p = 1; n = 60 p = 1; n = 20 p = 3; n = 20.

### 1.9. A landowner that has Q = 600 hectare of land monopolizes the rental market for farm land in a region in which the demand is D(P ) = maxf800 2P; 0g: (P is given in thousand of euros per hectare.) Land has no alternative use to farming (that is, the landowner’s opportunity cost of renting the land for farm use is zero). Then the farming area in the region and the rents of the landowner (her pro…t, in million of euros) are:

### Q = 600; = 0 Q = 400; = 80 Q = 600; = 150 Q = 400; = 60:

### 2. A consumer’s preferences for food (x) and clothes (y) are represented by the utility function u(x; y) = 2x + ln y. The prices are p _{x} and p _{y} euros per unit, respectively, and the consumer’s income is I euros.

### (a) (10 points) Calculate the consumer’s ordinary demands, x(p x ; p y ; I) and y(p x ; p y ; I). Rep- resent the consumer’s budget set and calculate her optimal consumption bundle if her income is I = 4 and prices are (p _{x} ; p _{y} ) = (4; 1):

### Solution: Let us calculate the consumer’s M RS:

### M RS(x; y) = 2

### 1 y

### = 2y:

### An interior solution solves the system of equations

### 2y = p _{x} p y

### xp x + yp y = I:

### Solving:

### x = I p x

### 1

### 2 ; y = p _{x} 2p y

### > 0:

### In order for x > 0 the inequality I > ^{p} _{2}

^{x}

### must hold. Otherwise x = 0 and y = _{p} ^{I}

y

### . Hence the ordinary demands are:

### x(p _{x} ; p _{y} ; I) = ( I

### p

x### 1

### 2 if I ^{p} _{2}

^{x}

### 0 if I < ^{p} _{2}

^{x}

### , y(p _{x} ; p _{y} ; I) = ( _{p}

x

### 2p

y### if I ^{p} _{2}

^{x}

### I

### p

y### if I < ^{p} _{2}

^{x}

### .

### The optimal bundle at prices (p x ; p y ) = (4; 1) and income I = 4 is (x ; y ) = ( ^{1} _{2} ; 2), and the

### consumer’s utility is u _{0} = 1 + ln 2:

### (b) (10 points) Calculate the consumer’s true price index CP I assuming that her income is I = 4; the prices in the base period are (p _{x} ; p _{y} ) = (4; 1), and the prices in the current period are (p ^{0} _{x} ; p ^{0} _{y} ) = (4; 2): (Use the approximation ln 2 0:7:) Calculate as well the Laspeyres consumer price index CP I L . Explain why these two price indices di¤er.

### Solution. The cheapest bundle that allows to maintain the utility level u _{0} at price (4; 2) is the solution to the system of equations

### 2^ y = p ^{0} _{x} p ^{0} _{y} = 2 2^ x + ln ^ y = 1 + ln 2:

### Solving:

### (^ x; ^ y) = ( 1

### 2 (1 + ln 2) ; 1) (0; 85; 1):

### Hence

### IP C = 0; 85 (4) + 1 (2)

### 4 = 1:35:

### On the other hand

### IP C _{L} = 0:5 (4) + 2 (2)

### 4 = 1:5:

### The Laspeyes consumer price index does not take into account the substitution e¤ ect, and there-

### fore is always greater then the true consumer price index.

### 3. Ana is a student whose welfare depends on her average grade m 2 R ^{+} and her consumption c 2 R + : (Assume that consumption is measured in euros, so that p _{c} = 1:) Her preferences are represented by the utility function u(h; c) = ln m + ln c: Ana has H = 15 hours that she can allocate to study and/or supply as labor. Ana’s average grade m is determined by the number of hours she studies e according to the formula m = ^{2} _{3} e: The hourly wage is w 0 euros, and Ana has no other income (besides her labor income).

### (a) (10 points) Describe Ana’s budget constraint and graph her budget set in the plane (m; c).

### Calculate the number of hours she studies and works as a function of w. Assuming that w = 4, calculate Ana’s optimal average grade and consumption (m ; c ) bundle, and represent it in the graph.

### Solution. Ana’s labor income (her only income) is

### wl = w(H e)

### = w(H 3

### 2 m) (since m = 2 3 e) Hence her budget constraint is

### c w(H 3

### 2 m);

### Substituting H = 15 and moving m to the left hand side of the inequality we can write Ana’s budget constraint as

### 3

### 2 wm + c 15w:

### Observe that the opportunity cost of an increase of a point in the average grade ^{3} _{2} w euros in consumption. (That is, the e¤ ective price of m is p _{m} = ^{3} _{2} w.) Of course, c 0; m 0:

### Since Ana’s RM S(h; c) = c=m; an interior solution to her utility maximization problem must solve the system of equations

### c

### m = 3

### 2 w 3

### 2 wm + c = 15w:

### Solving

### m(w) = 5 c(w) = 15

### 2 w:

### For w = 4, we have (m ; c ) = (5; 30):

### (b) (10 points) Assume now that a program is introduced that rewards students with an average grade m = 7 or above with M = 10 euros. Assuming that w = 4; graph Ana’s new budget set and determine the impact of this change on her average grade and consumption.

### Solution. Ana’s new budget set is described in the graph below: for m < 7; Ana’s budget constraint does not change, while for m 7 her new budget constraint is

### 3

### 2 wm + c 15w + 10:

### In particular, the bundle (m; c) = (7; 28) is now feasible. Moreover,

### u(5; 30) = ln 5 + ln 30 = ln 150 < ln 7 + ln 28 = ln 196 = u(7; 28):

### Hence the optimal bundle in (a), (m ; c ) = (5; 30); is not longer optimal. Since RM S(7; 28) = 4 > 3

### 2 w = 6;

### that is, the value of an additional unit of average grade m is below its cost. Hence the bundle

### (m ; c ) = (7; 28) is optimal.

### 4. (10 points) Jorge must renew his car insurance policy. His company o¤ers him to alternative policies that cover the repair costs of any accident he may have: one policy involves a deductible of 200 euros per accident for a premium of 400 euros; the other one involves no deductible and a premium of 600 euros. Jorge believes that the probability of having 0; 1 and 2 accidents over the year are ^{1} _{4} , ^{1} _{2} and ^{1} _{4} , respectively. Also he knows that the repair costs of any accident are always above 200 euros. If Jorge is risk averse, which policy will he subscribe, the one with the deductible or the one with no deductible? (You can answer this question knowing that Jorge is risk averse, and without knowing precisely his preferences.) Assume now that Jorge has an annual wealth of 2200 euros, that his preferences are described by the Bernoulli utility function u(x) = p x; where x is his income net of the insurance premium and repair costs. Assuming that these two insurance policies are the only ones available, describe the lottery that Jorge faces when he has perfect information and chooses optimally his insurance policy. (It should be understood that this lottery describes Jorge’s prospects before he receives the information.) Would Jorge be willing to pay 100 euros for this information?

### Solution. Let us denote by l _{D} and l _{N D} the lotteries representing the insurance with and without deductible, and let u be Jorge’s Bernoulli utility functions. The expectations of these lotteries are

### E(l _{D} ) = 400 + 1

### 4 (0) + 1

### 2 ( 200) + 1

### 4 (2) ( 200) = 600 = E(l _{N D} ):

### Moreover, since l _{N D} involves paying with certainty 600 euros, then Eu(l _{N D} ) = u(E(l _{N D} )): As Jorge is risk averse and the lottery l _{N D} is non-degenerated, then u(E(l _{D} )) > Eu(l _{D} ): Therefore

### Eu(l _{N D} ) = u(E(l _{N D} )) = u(E(l _{D} )) > Eu(l _{D} );

### that is, Jorge prefers the insurance with no deductible.

### Now, since the Bernoulli utility function u(x) = p

### x represents the preferences of a risk averse individual, then with these preferences Jorge’s optimal insurance is the one with no deductible, and Jorge’s expected utility with no information is

### Eu(l SF ) = p

### 2200 600 = 40:

### If Jorge knew with certainty that he is going to have two accidents over the year, then he would subscribe the insurance with no deductible, whereas if he knew with certainty that he is going to have no accidents over the year, then he would subscribe the insurance with deductible. Hence, with perfect information and assuming that Jorge pays 100 euros for this information, then the lottery he faces is l P I that pays x IP = (2200 400 100; 2200 600 100; 2200 600 100) = (1700; 1500; 1500) with probabilities p _{IP} = ( ^{1} _{4} ; ^{1} _{2} ; ^{1} _{4} ): The expected utility of this lottery is

### Eu(l _{IP} ) = 1 4

### p 1700 + 1 2

### p 1500 + 1 4

### p 1500 = 39; 55 < 40 = Eu(l _{SF} ):

### Thus, Jorge would not be willing to pay 100 euros in order to obtain perfect information.

### 5. A market is monopolized by a …rm that produces the good using only labor. Its production function is Q = f (L) = p

### L, where L is the number of workers employed by the …rm. The

### …rm is a price-taker in the labor market, in which the wage is w = 1: The market demand is D(P ) = max f120 P; 0g : (Price and wage are given in euros per unit.)

### (a) (15 points) Calculate the total, average, and marginal cost functions, as well as the monopoly equilibrium price and quantity. What is the dead-weight loss of the monopoly relative to the equilibrium in which the …rm behaves as a price-taker?

### Solution. The total cost function:

### Q = p

### L ) L(w; Q) = Q ^{2} Therefore,

### C(w; Q) = wQ ^{2} and

### C(1; Q) = C(Q) = Q ^{2} :

### The marginal and average cost functins, respectively, M C(Q) = 2Q and

### AC(Q) = Q:

### Monopoly Equilibrium: Solving the equation

### IM a(Q) = CM a(Q);

### i.e.,

### 120 2Q = 2Q;

### get Q ^{M} = 30 y P ^{M} = 120 30 = 90: The pro…t of the monopoly is

### M = 90 (30) (30) ^{2} = 1800:

### If the …rm behaves as price-taker the equilibrium condition is P = M C(Q): That is, its supply function would be S(P ) = P=2: The market equilibrium price would be the solution of the equation

### 120 P = P=2;

### (b) (5 points) The …rm may adopt a new technology, which would allow it to produce the good according to the production function Q = f (L) = 2 p

### L: If adopting this new technology requires an investment of T euros, for which values of T would the …rm adopt this technology?

### Solution. With the new technology the cost function ~ C(Q) = ^{Q} _{4}

^{2}