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[solution]: i need C,and D from question 3 and A,B,C,D,E from question4 and i

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i need C,and D from question 3 and A,B,C,D,E from question4

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I??????????? M????????????? (ECON 3102)

B????? & E?????

S????? 2016

Problem Set 2

Department policy requires that all homework

assignments be typed, except those portions which

are mainly computational/math.

1. (25 points) Imagine a representative household with the following utility function:

u(C, `) =

ln (C) + (1

) ln (`) ,

where 0 < < 1, and C represents consumption of a consumption good (or basket),

while ` represents hours of leisure.

(a) Find the marginal rate of substitution at some arbitrary point, (C, `).

(b) We considered three assumptions that consumer?s preferences must satisfy.

Check whether this utility function satis?es the ?rst two assumptions (namely,

more is be?er and a taste for diversity) or not.

(c) Does this utility function satisfy the Inada conditions?

?is representative household has a budget constraint that looks like

C =wN s + ?

T,

where w represents the real wages, N s the amount of hours worked, T the lump-sum

taxes government imposes, and ? the pro?ts of a representative ?rm which the representative household owns. Finally, the representative household has a total amount

of h hours that she can allocate to working or enjoying leisure time; that is

` + N s =h.

For the rest of this problem, suppose T < ?.

(d) Show that the budget constraint can be rewri?en in real terms (in terms of goods

instead of currency) as

C + w` =wh + ?

T.

(e) Draw the budget constraint in real terms and mark every important element in

it.

(f) Write the maximization problem of the household. (Do not forget to identify the

domain of the utility function, or non-negativity constraints.)

(g) Suppose that = 14 , h = 30, ? = 30, and T = 10. Solve the maximization problem of the representative household. Make sure to argue whether the boundary

points can or cannot be the optima. What will be the optimal level of consumption, leisure, and labor, as functions of real wage, w?

(h) Are leisure and consumption good normal or inferior goods?

2. (25 points) Consider preferences that can be represented by the following utility function;

U (C, `) = a` + bC,

where a and b are positive constants. Leisure and consumption are said to be perfect

substitutes for a consumer who has such preferences.

(a) Illustrate indi?erence curves that correspond to such utility function. Why do

you think leisure and consumption good are said to be perfect substitutes in

this utility function? Do you think it is likely that any consumer would treat

consumption good and leisure as perfect substitutes?

(b) Given perfect substitutes, is more preferred to less? Do preferences satisfy the

diminishing marginal rate of substitution property?

(c) If the budget constraint is given by

C = w (h

`) ,

determine, graphically and algebraically, what consumption bundle the consumer chooses. Show that the consumption bundle the consumer chooses depends on the relationship between a/b and w, and explain why.

2

3. (20 points) Imagine a representative ?rm with the following production function:

Y = zF (K, N ) =zK ? N d

1 ?

,

where 0 < ? < 1, and z represents the total factor productivity, K the amount of

capital that the ?rm owns, and, ?nally, N d the amount of labor it hires. Here, the

representative ?rm is only interested in its pro?t, which is de?ned as

? =Y

wN d .

(a) Write the maximization problem of the ?rm. (Do not forget to specify the domain

of the pro?t function, or the so-called non-negativity constraints.)

(b) Suppose that z = 30, K = 1, and ? = 13 . Solve the maximization problem of

the representative ?rm. Argue whether the boundary point can or cannot be

the optimal choice for the ?rm. What will be the optimal level of hours hired,

production, and pro?ts of the ?rm, as functions of wage w?

(c) Find the labor demand curve of the representative ?rm, as a function of w. Is it

an increasing or a decreasing function?

(d) Using your knowledge on comparative statics, determine the sign of the following derivatives (at a given wage, w);

dN d

dN d

and

.

dK

dz

4. (30 points) Consider the one-period simple economy that we studied in class. Assume

the preferences of the representative consumer are represented by the following utility function,

U (C, `) = ln (C) + `,

and the technology that the representative ?rm uses is represented by the following

production function,

F (K, N ) = zK ? N 1 ? .

> 0 and 0 < ? < 1 are parameters. In addition, assume that the government levies

a tax rate of T = G on the consumer.

(a) De?ne the competitive equilibrium in this economy. Specify which variables are

endogenous and exogenous in your de?nition.

(b) Write down the consumer?s and ?rm?s problems, and solve them. (Keep in mind

that consumer and ?rm are price-takers in our simple mode.)

3

(c) Use your answer in Part (b), and ?nd the competitive equilibrium in this economy analytically. Make sure that you ?nd consumption, leisure, employment,

and real wage.

(d) Graphically, illustrate the steps you take in Parts (b) and (c). Find the competitive equilibrium using this graphical approach. (Your graphs need to be only

approximations of the actual functional forms.)

(e) Show that competitive equilibrium of Part (c) is Pareto optimal.

4

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