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ECON 200 - Spring 2013

Final exam

NAME SECTION

There are 4 required questions plus an extra-credit question. Each required question is worth

50 points for a total of 200 points. The extra credit question is worth a total of 20 points. Read

each question entirely before starting to answer it.

Problem 1 A consumer is choosing between consuming a composit good c (which costs 1

dollar per unit) and leisure l. She can supply labor L up to an endowment of 12 hours a day at

a wage w= 10 dollars per hour. She has an additional endowment of 180 dollars.

The consumers preferences over consumption and labor are the following:

u(c, L) = 0.8log(c) + 0.2log(12 L)

where log(x) represents the natural logarithm ofx.

a) Set up the consumers UMP with respect to c and l with all the appropriate constraints.

Derive the feasibility and the tangency conditions of an interior solution (label each condition

explicitly and clearly to receive the full credit). (5 points)

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b) Solve for the Marshallian demands for c and l and for labor supply. Graphically illustrate

the budget constraint and the optimally chosen bundle. (5 points)

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Now, allow the consumers wage to increase to w = 20 dollars per hour.

c) Solve for the Marshallian demands for consumption and leisure and for labor supply.

Graphically illustrate the budget constraint and the optimally chosen bundle. (5 points)

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d) Calculate the substitution effect from this wage change. Explain what would allow you to

establish the sign of the substitution effect if you had not solved for it. (10 points)

e) Calculate the income effect from this wage change. (10 points)

f) Is there any other effect that you have to compute to explain how the consumers demand

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We expect that the substitution effect would lead the amount of leisure consumed to decrease as a result of an

ncrease in its price (the wage): at m' the initial optimal bundle is still affordable to the consumer. Hence, byWARP, she would not optimally choose any bundle that as previously also available but not chosen, such as an

undle where the consumption of leisure is greater than the one chosen before the price change.

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for leisure changes when the wage changes? Relate this discussion to the Slutsky equation.(10

points)

g) If the consumer had no additional income, what would her labor supply be? Explain the

economic intuition behind this result. (5 points)

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Problem 2 The income of a risk-averse farmer is uncertain because of the weather shocks

the farm is exposed to. In the good state of nature, which happens with 20% probability, her

yield will be worth $10,000, in the bad state of nature, her yield will be worth $5,000. The farmer

is off

ered to purchase an insurance product which will give her $5,000 in the bad state of nature.She has no other insurance mechanism available. The farmers utility function is u(c) =log(c).

a) Set up the farmers UMP in the absence of the insurance product. Calculate the farmers

expected utility in the absence of insurance. (5 points)

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e) Would the farmer buy the product if it costed $3,500? Why? (10 points)

f) Assume now that the farmers utility function is linear, i.e. u(c) = c. Is she risk averse,

risk neutral or risk-loving? Would she buy the product if it costed $4,500? Why? (10 points)

Problem 3 Calculate the equilibrium relative prices in an economy composed of two indi-

viduals, Wilbur (W) and Zaira (Z), which have the following preferences

uW(xW1

, xW2

) = (xW1

)0.3 (xW2

)0.7, uZ(xZ1

, xZ2) = (xZ

1)0.7 (xZ

2)0.3.

They each have endowments (W1

,W2

) = (6, 4) and (Z1

, Z2) = (4, 6) respectively.

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a) Calculate the equilibrium price good 1 relative to good 2 and the equilibrium allocations.

(10 points)

b) Draw the Edgeworth box associated with this problem (you can provide a qualitative

representation of the indifference curves). (5 points)

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c) Suppose that both Wilburs and Zairas endowments of good 2 double, while the rest of

the problem stays the same. Without solving for equilibrium explicitly, can you say what would

happen to the equilibrium relative price of good 1 compared to the initial case (i.e. would it be

higher or lower)? Why? (10 points)

d) Suppose that you now learn that Wilbur and Zaira have new preferences, while the rest

of the problem stays the same:

uW(xW1

, xW2

) = (xW1

)0.2 (xW2

)0.8, uZ(xZ1

, xZ2) = (xZ

1)0.6 (xZ

2)0.4.

Without solving for equilibrium explicitly, can you say what would happen to the equilibrium

relative price of good 1 compared to the initial case (i.e. would it be higher or lower)? Why?

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f) Assume that good 1 is listening to music, and when one person listens to music, that benefits

the other person who can also hear. For example, that could imply that Wilburs preferences are

now

uW

(xW1

, xW2 ) = (

xW1 )

0.25

(xW2 )

0.65

(xZ1)

0.1

and similarly for Zaira. Is competitive equilibrium is Pareto efficient in this case? Justify

your answer. (10 points)

Problem 4Indicate whether the statements below are true or false. Explain your reasoning

and always relate it precisely to the concepts you have learned in this class. The points will only

be credited solely based on your justification. You can use figures to clarify your argument.

a) Strong separability of consumption over time is an assumption that can straightforwardly

be applied to the behavior of all consumers in all circumstances because it imposes little restric-

tions on the consumers preferences. (10 points)

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b) A wage increase always raises labor supply. (10 points)

c) Imagine an economy with two goods and two agents which has a unique competitiveequilibrium. You observe this economy with two different sets of prices, first (p1, p2) = (3, 4) and

second (p1, p2) = (9, 12). Then, because the equilibrium is unique, you know that at least one of

these prices must not be the equilibrium prices. (10 points)

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d) When preferences are convex, any Pareto efficient allocation is a competitive equilibrium

for some prices. (10 points)

e) The certain equivalent of a given lottery is decreasing in the consumers risk aversion. (10

points)

Problem 5 - EXTRA CREDIT John is a stay-at-home dad who has to manage the

household resources by minimizing how much the household is spending on groceries ( g which

costs 1 dollar per unit) and other items (a composite good cwhich costs 1 dollar per unit). Thedual problem of this EMP is a UMP which depends on the household income m.

The household has Cobb-Douglas preferences gac1a and every month it spends exactly 10%

of its budget on groceries.

Like every month, John has to achieve a target utilityu = 1.704m.1 Using duality, write down

1Note that 0.10.1

0.90.9 = 1.704.

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