HOMEWORK 17 questions

QUESTION 1

  1. Consider an one period macroeconomic model which consists of a representative consumer with preferences u(c₁,c₂)=((c₁)^(1-σ))/(1-σ)+((c₂)^(1-σ))/(1-σ))where c₁ and c₂ are two consumption goods traded in this market with a price of p₁ and p₂. The consumer has an income of I irrespective of whether she is working or not.

The market prices of c1 and c2 are $5 and $2, respectively. The consumer has an income of $100 irrespective of whether she is working or not.

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Write down the consumer’s budget constraint and draw it into a graph where c1 is on the horizontal axis and c2 is on the vertical axis. What is the c2 intercept?

 

QUESTION 2

  1. Consider an one period macroeconomic model which consists of a representative consumer with preferences u(c₁,c₂)=((c₁)^(1-σ))/(1-σ)+((c₂)^(1-σ))/(1-σ))where c₁ and c₂ are two consumption goods traded in this market with a price of p₁ and p₂. The consumer has an income of I irrespective of whether she is working or not.

The market prices of c1 and c2 are $5 and $2, respectively. The consumer has an income of $100 irrespective of whether she is working or not.

Write down the consumer’s budget constraint and draw it into a graph where c1 is on the horizontal axis and c2 is on the vertical axis.

What is the c1 intercept?

 

QUESTION 3

  1. Consider an one period macroeconomic model which consists of a representative consumer with preferences u(c₁,c₂)=((c₁)^(1-σ))/(1-σ)+((c₂)^(1-σ))/(1-σ))where c₁ and c₂ are two consumption goods traded in this market with a price of p₁ and p₂. The consumer has an income of I irrespective of whether she is working or not.

The market prices of c1 and c2 are $5 and $2, respectively. The consumer has an income of $100 irrespective of whether she is working or not.

Write down the consumer’s budget constraint and draw it into a graph where c1 is on the horizontal axis and c2 is on the vertical axis.

What is the slope of the budget constraint?

 

QUESTION 4

  1. Consider an one period macroeconomic model which consists of a representative consumer with preferences u(c₁,c₂)=((c₁)^(1-σ))/(1-σ)+((c₂)^(1-σ))/(1-σ))where c₁ and c₂ are two consumption goods traded in this market with a price of p₁ and p₂. The consumer has an income of I irrespective of whether she is working or not.

The market prices of c1 and c2 are $5 and $2, respectively. Preference parameter σ=2.5. The consumer has an income of $100 irrespective of whether she is working or not.

Write down the consumer’s budget constraint and draw it into a graph where c1 is on the horizontal axis and c2 is on the vertical axis.

The slope of the indiffernce curve (i.e., the negative of the marginal rate of substitution or MRS) at the optimum point is:

 

QUESTION 5

 

  1. Consider an one period macroeconomic model which consists of a representative consumer with preferences u(c₁,c₂)=((c₁)^(1-σ))/(1-σ)+((c₂)^(1-σ))/(1-σ))where c₁ and c₂ are two consumption goods traded in this market with a price of p₁ and p₂. The consumer has an income of I irrespective of whether she is working or not.

The market prices of c1 and c2 are $5 and $2, respectively. Preference parameter σ=2.5. The consumer has an income of $100 irrespective of whether she is working or not.

Write down the consumer’s budget constraint and draw it into a graph where c1 is on the horizontal axis and c2 is on the vertical axis.

Calculate the optimal quantity for c1:

 

QUESTION 6

  1. Consider an one period macroeconomic model which consists of a representative consumer with preferences u(c₁,c₂)=((c₁)^(1-σ))/(1-σ)+((c₂)^(1-σ))/(1-σ))where c₁ and c₂ are two consumption goods traded in this market with a price of p₁ and p₂. The consumer has an income of I irrespective of whether she is working or not.

The market prices of c1 and c2 are $5 and $2, respectively. Preference parameter σ=2.5. The consumer has an income of $100 irrespective of whether she is working or not.

Write down the consumer’s budget constraint and draw it into a graph where c1 is on the horizontal axis and c2 is on the vertical axis.

Calculate the optimal quantity for c2:

QUESTION 7

 

  1. Consider an one period macroeconomic model which consists of a representative consumer with preferences u(c₁,c₂)=((c₁)^(1-σ))/(1-σ)+((c₂)^(1-σ))/(1-σ))where c₁ and c₂ are two consumption goods traded in this market with a price of p₁ and p₂. The consumer has an income of I irrespective of whether she is working or not.

Preference parameter σ=2.5. The market prices of c1 and c2 are $5 and $2, respectively. The consumer has an income of $100 irrespective of whether she is working or not.

The government subsidizes consumption good c₁ with a 10 percent subsidy. In addition a lump-sum tax of $15 is charged.

Calculate the new optimal consumption level of c1:

 

QUESTION 8

  1. Consider an one period macroeconomic model which consists of a representative consumer with preferences u(c₁,c₂)=((c₁)^(1-σ))/(1-σ)+((c₂)^(1-σ))/(1-σ))where c₁ and c₂ are two consumption goods traded in this market with a price of p₁ and p₂. The consumer has an income of I irrespective of whether she is working or not.

Preference parameter σ=2.5. The market prices of c1 and c2 are $5 and $2, respectively. The consumer has an income of $100 irrespective of whether she is working or not.

The government subsidizes consumption good c₁ with a 10 percent subsidy. In addition a lump-sum tax of $15 is charged.

Calculate the new optimal consumption level of c2:

 

QUESTION 9

Assume a production function of the form:

Y=z * F(K,N) = z * K^0.3 * (Nd)^0.7, where total factor productivity z=10 and the capital stock in the economy is 100. Given an overall wage rate of $10 per unit worked, solve for the optimal labor demand of this representative firm, i.e. the labor demand where firm profits are maximized.

Nd* =

   
28.4551
30.4551
32.4551
34.4551

QUESTION 10

  1. Consider an one period macroeconomic model which consists of a representative consumer with preferences u(c₁,c₂)=((c₁)^(1-σ))/(1-σ) + a * ((c₂)^(1-σ))/(1-σ)) where c₁ and c₂ are two consumption goods traded in this market with a price of p₁ and p₂.The preference parameter σ=2.5 and parameter a=2.

The marginal utility of an extra unit of consumption of good 1 at point (c₁=3,c₂=3) is:

(Use at least 3 digits after the decimal point for this answer!)

 

 

QUESTION 11

  1. Consider an one period macroeconomic model which consists of a representative consumer with preferences u(c₁,c₂)=((c₁)^(1-σ))/(1-σ) + a * ((c₂)^(1-σ))/(1-σ)) where c₁ and c₂ are two consumption goods traded in this market with a price of p₁ and p₂.The preference parameter σ=2.5 and parameter a=2.

The marginal utility of an extra unit of consumption of good 2 at point (c₁=3,c₂=3) is:

(Use at least 3 digits after the decimal point for this answer!)

 

QUESTION 12

  1. Consider a firm having the following production technology:

 

Y=zF(K,N)=zK^{α}N^{1-α},

 

where z is total factor productivity, K is private capital, N is labor and α∈(0,1) is a parameter governing the income share of capital. Assuming that K is constant. There are competitive goods and labor markets to which the firm can sell its output at price 1 and from which it can hire labor at real wage rates w.

Assume z=1,K=30, α=0.3, and w=$5.

Calculate the marginal product of labor at a labor input of N=6.  Be very precise and report 4 digits after the decimal point.

 

QUESTION 13

  1. Consider a firm having the following production technology:

 

Y=zF(K,N)=zK^{α}N^{1-α},

 

where z is total factor productivity, K is private capital, N is labor and α∈(0,1) is a parameter governing the income share of capital. Assuming that K is constant. There are competitive goods and labor markets to which the firm can sell its output at price 1 and from which it can hire labor at real wage rates w.

Assume z=1,K=30, α=0.3, and w=$5.

Calculate the extra output you can produce when increasing labor from 6 to 7 units. Be very precise and report 4 digits after the decimal point.

Why is this NOT the same as in the previous question?

 

QUESTION 14

  1. If revenue is given by the following function:

REV(t)=t×w×(h-l(t)),

where t is the labor tax rate, w is the wage rate, h is the maximum amount of time available to the household, and l(t) is leisure as an increasing function of the tax rate i.e. if the labor tax rate t increases, leisure increases, so that individuals work less. Assume that

l(t)=min[h,t²].

This simply means that leisure cannot exceed h which is the maximum amount of time available.

Calculate the tax revenue when h=1, w=$5, and the tax rate is 20 percent.

(Use at least 3 digits after the decimal point for this answer!)

 

QUESTION 15

  1. Tax revenue is given by the following function:

REV(t)=t×w×(h-l(t)),

where t is the labor tax rate, w is the wage rate, h is the maximum amount of time available to the household, and l(t) is leisure as an increasing function of the tax rate i.e. if the labor tax rate t increases, leisure increases, so that individuals work less. Assume that

l(t)=min[h,t²].

This simply means that leisure cannot exceed h which is the maximum amount of time available.

Calculate the tax revenue when h=1, w=$5, and the tax rate is 70 percent.

 

QUESTION 16

  1. If tax revenue is given by the following function:

REV(t)=t×w×(h-l(t)),

where t is the labor tax rate, w is the wage rate, h is the maximum amount of time available to the household, and l(t) is leisure as an increasing function of the tax rate i.e. if the labor tax rate t increases, leisure increases, so that individuals work less. Assume that

l(t)=min[h,t²].

This simply means that leisure cannot exceed h which is the maximum amount of time available.

Find the tax rate that maximizes the tax revenue when h=1, w=$5. Be precise and report 4 digits after the decimal point.

 

QUESTION 17

  1. If revenue is given by the following function:

REV(t)=t×w×(h-l(t)),

where t is the labor tax rate, w is the wage rate, h is the maximum amount of time available to the household, and l(t) is leisure as an increasing function of the tax rate i.e. if the labor tax rate t increases, leisure increases, so that individuals work less. Assume that

l(t)=min[h,t²].

This simply means that leisure cannot exceed h which is the maximum amount of time available.

Calculate the tax revenue when h=1, w=$5, and the tax rate is 50 percent.

 

 

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