1. Please prove that two indifferent curve cannot cross each other.

2. Please solve the following maximization problem

u = √x₁ + x₂
max√x₁ + x₂, st, p₁x₁ + p₂x₂ ≤ m

3.

a. When prices are (p₁, p₂) = (1, 2) a consumer demands (x₁, x₂) = (1, 2), and when prices are (q₁, q₂) = (2, 1) the consumer demands (y₁ , y₂) = (2, 1). Is this behavior consistent with the model of maximizing behavior?

b. When prices are (p₁, P₂) = (2, 1) a consumer demands (x₁ , x₂) = (1, 2), and when prices are (q₁, q₂) = (1, 2) the consumer demands (y₁, y₂) = (2, 1). Is this behavior consistent with the model of maximizing behavior?

4. If

x(P_x, p_y, m) = 2m/5p_x, m= 1000, p_x = 5, p_y = 20, p_x = 4

Please calculate the income effect and substitution effect

5. if for a two period utility function

u = (c₁, c₂), c₁.c₂ = 2000, m₂ =1000. r = 10%. p = 1

no inflation

Please solve c₁*,c₂* is there any saving? what would happen if r=20%

6.  a. If a consumer's net demands are (5, -3) and her endowment is (4, 4), what are her gross demands?

b. The prices are (p₁, p₂) = (2, 3), and the consumer is currently consuming (x₁, x₂) (4, 4). There is a perfect market for the two goods in which they can be bought and sold costlessly. Will the consumer necessarily prefer consuming the bundle (y₁ , y₂) = (3, 5)? Will she necessarily prefer having the bundle (y₁, y₂)?

c. The prices are (p₁ , p₂) = (2,3), and the consumer is currently consuming (x₁, x₂) = (4, 4). Now the prices change to (q₁, q₂) = (2, 4). Could the consumer be better off under these new prices?

7.

a. A consumer, who is initially a lender, remains a lender even after a decline in interest rates. Is this consumer better off or worse off after the change in interest rates? If the consumer becomes a borrower after the change is he better off or worse off?

b. What is the present value of $100 one year from now if the interest rate is 10%? What is the present value if the interest rate is 5%?

8.

a. The payments of certain types of bonds (e.g., municipal bonds) are not taxable. If similar taxable bonds are paying 10% and everyone faces a marginal tax rate of 40%, what rate of return must the nontaxable bonds pay?

b. Suppose that a scarce resource, facing a constant demand, will he ex¬hausted in 10 years. If an alternative resource will be available at a price of $40 and if the interest. rate is 10%, what must the price of the scarce resource be today?

9.  a. If D(p) = 12 - 2p, what price will maximize revenue?

b. Suppose that the demand curve for a good is given by D(p) = 100/p. What price will maximize revenue?

10.

a. Suppose that all consumers view red pencils and blue pencils as perfect substitutes. Suppose that the supply curve for red pencils is upward sloping. Let the price of red pencils and blue pencils be p, and p_b. What would happen if the government put a tax only on red pencils?

b. The United States imports about half of its petroleum needs. Suppose that the rest of the oil producers are willing to supply as much oil as the United States wants at a constant price of $25 a barrel. What would happen to the price of domestic oil if a tax of $5 a barrel were placed on foreign oil?

11. Suppose that we have two copies of Intermediate Microeconomics to sell to three (enthusiastic) students. How can we use a sealed-bid auction that will guarantee that the bidders with the two highest values get the books?

12.

a. The Cobb-Douglas production function is given by f (xl,x2) = AXYX;. It turns out that the type of returns to scale of this function will depend on the magnitude of a + b. Which values of a + b will be associated with the different kinds of returns to scale?

b. The technical rate of substitution between factors x2 and xl is -4. If you desire to produce the same amount of output but cut your use of xl by 3 units, how many more units of x2 will you need?

13.

a. Suppose a firm is maximizing profits in the short run with variable factor xl and fixed factor x2. If the price of x2 goes down, what happens to the firm's use of xl? What happens to the firm's level of profits?

b. A profit-maximizing competitive firm that is making positive profits in long-run equilibrium (may/may not) have a technology with constant returns to scale.

14. please explain the similarity and the difference between the Soviet model and China's planned economy model.

15. please use micro-economics explain in China's growth model after the late 1990s, why local government usually lease manufacturing land out very cheaply while auction the residential and commercial land out at higher prices.