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Final exam 200 points

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ECON 311 Spring Quarter, 2009 NAME:_______________________

Prof. Hamilton


200 points
1. (20 points). When workers from other parts of the country relocate to San Luis Obispo, they typically find the cost of living to be much higher than it is in the cities where they move from. Suppose a worker from Fresno considers relocating to San Luis Obispo. The worker derives utility from 2 goods: housing (X) and recreational goods (Y).

(a) (5 points) Provide a diagram that shows the workers optimal consumption of X and Y in Fresno for given prices (PX and PY) and income (I).

(b) (5 points) Show on your diagram the optimal consumption of goods X and Y in San Luis Obispo when the workers income and the price of recreational goods is identical in Fresno and San Luis Obispo, but housing costs twice as much in San Luis Obispo than in Fresno.

(c) (10 points) According to your diagram, are housing (X) and recreational goods (Y) substitutes or complements? Explain in terms of the income effect and substitution effect why the worker’s consumption of recreational goods can go up or down after the worker moves from Fresno to San Luis Obispo.

Since demand for recreational goods rises in San Luis Obispo when the worker is faced with a higher price of housing, housing (X) and recreational goods (Y) are substitutes. The reason is that the worker substitutes away from housing and towards recreational goods when moving to San Luis Obispo and the substitution effect (SE) is larger than the income effect (IE).

2. (20 points). Suppose that the average household in San Luis Obispo consumes 800 gallons of gasoline per year. A 20-cent gasoline tax is introduced, coupled with a $160 annual tax rebate per household. Will the household be better off, worse off, or equally well off under the new program? Provide a graph to support your argument.

The household cannot be any worse off. Household income is adjusted upwards by the tax rebate so that the household can afford to buy 800 gallons of gasoline at the same cost as before. It turns out that the household will be better off, and the key to seeing this is to think of substitution effects. Gasoline is now relatively more expensive than other goods the household can buy, so the household will substitute away from buying gasoline and into consumption of other goods.
To see this, consider the following diagram. In the graph, X is gasoline and Y represents other goods. Notice that relative prices change after the gas tax is imposed. The household’s old budget line was B1, their original bundle was point A (800 gallons of gasoline and the remaining income spent on other goods). The household’s original utility level was U1. The new budget line (B2) –the dotted line-- is adjusted to allow the household to continue buying bundle A (800 gallons of gasoline and the same level of other goods), so it also goes through point A. Notice that relative prices have changed, so the slope of B2 is different than the slope of B1. Therefore, the tangency between MRS and the (new) price ratio no longer occurs at point A, and the household will adjust by shifting consumption towards the (relatively) lower priced good Y, hence obtaining the higher indifference curve U2 –the dashed curve.


4. (30 points). Suppose the variable x represents the output of a firm that takes price p as given. The firm has a cost function C(x) and seeks to maximize profit.

(a) Use words and a diagram to discuss the solution to this problem. Will the firm increase or decrease its output when the price goes up? Explain. Show the firm’s profit level on your diagram.

(10 points for describing equilibrium supply as p=MC; 10 points for showing AC and profit)

Profit:  = px – c(x) => FOC: p – c = 0 (or P=MC)

SOC: -c < 0 (or MC increases in x)

The firm produces all units for which MC < P and then ceases production when P = MC. Because MC is rising with output (by the second order condition), MC must be beneath price for small levels of output (it is sometimes said “MC cross price from below”). If the initial price is p1, then the profit maximizing output level is x1*. If the price rises to p2, then the profit maximizing output level rises to x2*. Output must increase as the price goes up by the second-order condition: MC rises with output, so P=MC at a higher price only when output is larger. Profit is the difference between price and average cost at the equilibrium output level, which is shown as the shaded region for production level x2*.
(b) Suppose that p = $15 and C(x) = 0.125x2+7x+4. Find the optimal output level, x*. Calculate the firm’s equilibrium profit level.
If p = $15 and C(x) = 0.125x2+7x+4, the optimal solution, x*, maximizes:

= 15x – 0.125x2 – 7x – 4

=> FOC: 15 – 0.25x - 7 = 0 8 = 0.25x

x* = 8/0.25 = 32

SOC: -0.25 < 0 (√)

The firm’s equilibrium profit level is: * = 15(32) – 0.125(32)2 – 7(32) – 4 = $124

5. (40 points). A firm produces in an industry facing input prices for capital and labor given by r = $1 and w = $5, respectively, and output is sold for $p per unit. The firm’s production function is:

  1. If the firm produces with = 4 in the short run, derive the short-run supply function.

First find the cost function, then maximize profit to find supply.
Cost = wL + rK = 5L + K = 5L + 4
The short-run cost function, c(y), requires finding L(y) –the amount of labor required to produce an output level of y units-- when = 4. From the production function:

y = 2 + L0.5 => y - 2 = L0.5

Squaring both sides and solving for L: L(y) = (y-2)2

The short-run cost function is: c(y) = 5(y-2)2 + 4
Supply solves the profit maximization problem for the competitive firm:

= py –5(y – 2)2 – 4

FOC:  =p – 10(y – 2) = 0 => y(p) = 2 + p/10
(b) Characterize the long-run supply function.
Firm’s Problem (FP): min. C = wL + rK s.t.

L = 5L + K + [y - K0.5 - L0.5]

FOC: (1) dL /dL = 5 - L-0.5 = 0

(2) dL /dK = 1 - K-0.5 = 0

(3) dL /d = y - K0.5- L0.5 = 0

From (1), (2) (K/L) 0.5 = 5  K = 25L

which means you should be employing 25 times as much Capital as Labor inputs at the cost-minimizing input mix. Substituting K =25L into the production function, we find

y = (25L)0.5 + L0.5 = 5L0.5 + L 0.5  y = 6L0.5 => L*(y) = (y/6) 2

K* = 25L* K*(y) = 25(y/6) 2
The long-run cost function is: c(y) = 5L*(y) + K*(y) = 5(y/6) 2 + 25(y/6) 2 = 30(y/6) 2 = (5/6)y 2

Supply solves the profit maximization problem for the competitive firm:

= py –(5/6)y2

FOC:  =p – 5/3y = 0 => y(p) = 3p/5

6. (20 points). Suppose Nokona and Rawlings compete in a Cournot duopoly market for baseball gloves with the cost functions c(qN) = 40qN for Nokona and c(qR) = 40qR for Rawlings. Player’s view Nokona and Rawling gloves to be perfect substitutes and inverse demand for baseball gloves is P = 520 − 0.5Q, where Q = qN + qR is the total quantity of gloves produced by both firms.

(a) Calculate the reaction functions, qNR and qRR. Show the equilibrium quantity of gloves produced by each firm as the intersection of two reaction functions on a graph.

Nokona’s Profit: N = [520– 0.5qN – 0.5qR]qN – 40qN

FOC: N' = 520 – qN – 0.5qR – 40 = 0

Reaction function: qNR = 480 – 0.5qR Similarly: qRR = 480 – 0.5qN

(b) Find the equilibrium quantity of gloves produced in a Cournot duopoly. What is the market price of a baseball glove in a Cournot duopoly?

Cournot equilibrium is the intersection of the reaction functions.

qN = 480 – 0.5[480 - 0.5qN] = 240 + 0.25qN => qN* = 320 = qB*
Market price: P* = 520 – 0.5Q* = 520 – 0.5(qN* + qR*) = 520 - 320 = $200

7. (25 points) Pete Rose is seen to place an all-or-nothing $100,000 bet on the Raiders to win the AFC West in 2009; that is, if the Raiders win the AFC West, Pete Rose wins $100,000 and if the Raiders do not win the AFC West, Pete will lose $100,000). If Pete has a logarithmic utility-of-wealth function, , and his current wealth is $500,000, what must he believe is the minimum probability that the Raider win the AFC West?

Pete must receive greater expected utility from the bet than his utility level from not placing any bet at all. If he does not bet, his utility is:


If Pete bets, he will have either $1,100,000 or $900,000 in wealth, depending on the outcome of the bet. His expected utility is:

Because Pete is seen to make the bet, Pete must believe that 13.71 + 0.2p > 13.82, or p>0.55.

8. (25 points). A local community has 100 graduating high school seniors. Each high school senior has a different work ethic, from slacker to superstar, and there is exactly one student of each type. Let  denote type (productivity), where  is distributed uniformly on the interval [0, 100] from total slacker to total superstar. Each graduating seniors consider going to college. The advantage of going to college is the higher wage earned by college graduates. There are two wages in the economy, w0 for high-school graduates and w1 for college graduates (with w1 > w0). Wages in each market are equal to the average productivity (type) in the market. Going to college is costly, and the cost of going to college depends on type. Specifically, the cost of going to college is given by c() = 180 – 2.

5 points each

(a) If no one goes to college, what is the equilibrium wage in a single market for high school graduates? With separate markets for high-school graduates and college graduates, show how wages depend on the value of the critical type (*) who is indifferent between going to college?

Equilibrium wage in a common labor pool, w* = E(productivity) = (0+100)/2 = $50.
Workers with a terminal high-school degree have uniformly distributed productivity on the interval [0,*] and college graduates have uniformly distributed productivity on the interval [*, 100]. The equilibrium wage in each market is:

w0 = (0+*)/2 = */2.

w1 = (*+100)/2.
8(b) For a given difference in wages, the benefit of going to college for all types is the wage premium, w1 – w0. Solve for the value of the critical type (*) who is indifferent between college and no college as a function of the wage premium.
When the cost of going to college is given by c() = 180 – 2, the critical type equates the benefit of the higher wage to the cost of college, and therefore solves:

w1 – w0 = 180 - 2; or * = 90 – ½(w1 – w0)

(c) Solve for the critical type, *, and the equilibrium wages w0* and w1* in the economy. How many high-school graduates go on to college?

Plugging the equilibrium wages in part (b) into your answer for part (c), the critical type, *, solves

* = 90 – ½((*+100)/2 – */2) = 90 – ½(50)  * = 65.

That is, the top 35 out of 100 high-school graduates go on to college.
The equilibrium wages w0* and w1* in the economy are: w0* = */2 = 65/2 = $32.5

w1* = (*+100)/2 = 165/2 = $82.5

(d) Now suppose a new technology is introduced (e.g., websites with plagiarized essays) that lowers the cost of going to college to c()= 130 – 2. What effects does the new technology cause, if any, on the wage paid to workers in each labor market? Explain.

If a new technology is introduced that lowers the cost of going to college to c()= 130 – 2, proceeding as above = 40 (60% of high school graduates now go on to college) and the new wages are

w0 =  /2 = $20

w1 = (+100)/2 = $70

Wages are lower for everybody! Why? As education costs decline more people go on to complete college. This reduces average productivity among those who do not seek further educational attainment after high-school (hence w0* declines). The higher number of college graduates, in turn, reduces average productivity among college graduates (hence w1* declines). If costs fell to zero, 100% of high-school graduates would go on to college, w0* would decline all the way to zero, and w1* would decline to $50, the expected productivity in the entire population. (At which point the market for college would unravel…)
(f) Clearly articulate why signaling is inefficient.
Signaling is inefficient, because it does not increase average productivity in the economy. To see this, note that in your solution to part d, the average wage earned by a worker is

E(w) = 0.65(32.5) + 0.35(82.5) = 50.

Hence, on average, signaling has no value to workers (the average wage is still $50). Signaling does not increase productivity either, so it has no value to firms. Signaling creates private value to highly productive individuals (the top 35% of workers are better off incurring the cost of education to earn a $82.5 wage than they are earning a $50 wage). However, this comes at the expense of the less productive workers, who now stand to earn only a $32.5 wage. Benefits are no larger either way, although an exactly-offsetting redistribution of benefits takes place from less productive to more productive workers. Because signaling produces no benefit at all in the population but has cost, signaling is inefficient.

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