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BENEFIT ANALYSIS HOMEWORK 1 FALL 2011 QUESTION 1 PARETO EFFICIENCY Consider a world in which there are only two agents: agent 1 and agent 2. Assume that each agent has a strictly positive marginal utility over wealth, and each agent i is endowed with a wealth of w i . Finally, assume you are evaluating a project that will generate a social wealth of W . €
a)
What condition must be satisfied by W to be considered a Potential €
Pareto Improvement? Is this a sufficient condition for being an actual €
Pareto Improvement? Explain why. b)
Characterize the condition € W should satisfy to get the project that
approved in a standard Cost
Benefit Analysis. How is this answer related to your previous answer in part a)? c)
Do you need to know €hat are the Potential Efficient Allocations to w
answer part a) and b) in this question? Explain why. QUESTION 2 AGGREGATE DEMAND Consider a world with n consumers living in it. Each consumer has the following utility function: u( x, y ) = xα y1−α €
Assume each consumer i is endowed with a wealth of w i and they are price takers in the markets of goods. €
a)
Is € this utility function homothetic, quasilinear, both, or neither? You do €
not need to show your result here. b)
Find consumer’s i demand functions for goods x and y . c)
Find the market demand functions for goods x and y . Are they a n function of ∑i =1 w i ? Does the distribution of wealth affect the markets’ €
€
€
demands? €
€ Imagine that there is a government project that will increase the amount of x available €
in this world by λ units and it costs c to implement it. € € € d)
e)
f)
g) Under a ceteris paribus assumption, what is the gross social benefit derived from this project? Give the range of λ that will make the project be approved in a Standard Cost
Benefit Analysis. Under a ceteris paribus assumption, is this project Pareto Improving? Explain. €
Under a ceteris paribus assumption, is this project a Potential Pareto Improvement? Show it. QUESTION 3 AGGREGATE DEMAND Repeat question 2 but using the following utility function: u( x, y ) = min{αx, (1 − α ) y} QUESTION 4 AGGREGATE DEMAND €
Repeat question 2 but using the following utility function: u( x, y ) = αx + (1 − α ) y QUESTION 5 AGGREGATE DEMAND €
Repeat question 2 but using the following utility function: u( x, y ) = α log( x − bx ) + (1 − α ) log y , where bx > 0 . Note: bx is known as the minimum requirement of consumption to stay alive. €
€
QUESTION 6 AGGREGATE DEMAND €
Repeat question 2 but using the following utility function: u( x, y ) = −α (bx − x ) 2 , where bx > 0 . QUESTION 7 SOCIAL PREFERENCES €
€
State Arrow’s Impossibility Theorem. QUESTION 8 AGGREGATE SUPPLY Consider a world with n firms living in it. Each firm employs the following technology to transform inputs into outputs: F (K , N ) = AK α N 1−α €
Assume each producer i is a price taker in the market of goods and in the markets of factors of production, and α ∈ (0,1) . €
a)
D
€ oes this technology show Constant Returns to Scale, Increasing Returns to Scale, Decreasing Returns to Scale, or none of the above? €
b)
Find producer’s i demand function for N in the short run. c)
Find producer’s i demand functions for N and K in the long run. d)
Find producer’s i supply function in the short run. e)
Find producer’s i supply function in the long run. €
€
f)
Find the market supply function in the short run. €
€
€
g)
Assuming Free Entry Condition and that all firms own the same amount €
of capital, what is the value of n in the short run? €
Imagine that there is a government project that will increase the amount of goods produced in this market by λ units €nd it has a social gross benefit of c to implement it. a
h)
Under a ceteris paribus assumption, what is the gross social cost derived from this project in the short run? €
€
i)
Give the range of λ that will make the project be approved in a Standard Cost
Benefit Analysis. j)
Under a ceteris paribus assumption, is this project Pareto Improving? Explain. €
k)
Under a ceteris paribus assumption, is this project a Potential Pareto Improvement? Show it. QUESTION 9 AGGREGATE SUPPLY Assuming that every firm owns the same amount of capital in the short run, repeat question 8 but using the following production function: F (K , N ) = min{ AK K , AN N }
€ QUESTION 10 AGGREGATE SUPPLY Consider a world with n firms living in it. Each firm employs the following technology to transform inputs into outputs: F ( N ) = max{ A( N − F ), 0} €
Note: F is the minimum amount of labor needed to start producing. €
a)
Does this technology show Constant Returns to Scale, Increasing Returns to Scale, Decreasing Returns to Scale, or none of the above? b)
Find producer’s i demand function for N . c)
Find producer’s i marginal cost function, average variable cost function, and average total cost function. d)
Find producer’s i profits function. €
€
e)
Find the market supply function. €
f)
Can these firms be price takers in the market of goods? €
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 Fall '11
 Triece
 Economics, Utility

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