(1)### 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.

### 2014 2014

(2)### 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

(3)### 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

(4)### (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.

(5)### 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

(6)### 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 :

(7)### (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

(8)### 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} :

(9)### (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 + :

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