Draw a correctly labeled graph showing a natural monopoly. On your graph, label the price and quantity the monopoly will choose if unregulated as

$$P_U$$

and

$$Q_U$$

What happens to the size of consumer surplus when the firm is required to charge

$$P_R$$

rather than

$$P_U?$$

Use the formula for the cosine of the difference of two angles to solve. Find the exact value of each expression.

$$\cos \left( 120 ^ { \circ } - 45 ^ { \circ } \right)$$

You want to determine whether a tall tree will fit onto your 75-m-long logging trailer after you cut it down. You throw a rock straight up with a speed great enough that the rock just reaches the top of the tree, releasing the rock 2.0 m above the ground. You observe that the time interval from the instant you release the rock to the moment it hits the ground is 5.0 s.

Symbols are appealing because they often carry powerful associations and emotional overtones. Write a brief essay analyzing the symbolism in "Through the Tunnel." First, consider the three settings: the wild beach, the safe beach, and the tunnel. Tell what you think each setting must symbolize for Jerry. Then, consider the larger symbolic meaning of Jerry's passage through the tunnel. What does his trip represent? Why is it a powerful symbol? Support your analysis, and elaborate with direct references to details in the story. (You can take notes in a chart.)

The area of one face of a cube is 144 square inches. What is the volume of the cube?

Imagine that the city of Townville has 1,000 residents and pays $100,000 per year to rent a public good for the benefit of every member of the community.

Identify and explain the two characteristics shared by every public good.

Read the information in A Closer. Do you think Jerry's experience can be viewed as an initiation rite? What similarities and differences can you find in his experience and the coming-of-age rites in other cultures?

Let X have the conditional geometric pmf $\theta ( 1 - \theta ) ^ { x - 1 } , x = 1,2 , \ldots$ where $\theta$ is a value of a random variable having a beta pdf with parameters $\alpha$ and $\beta.$ Show that the marginal (unconditional) pmf of X is $\frac { \Gamma ( \alpha + \beta ) \Gamma ( \alpha + 1 ) \Gamma ( \beta + x - 1 ) } { \Gamma ( \alpha ) \Gamma ( \beta ) \Gamma ( \alpha + \beta + x ) } , \quad x = 1,2 , \ldots$ If $\alpha = 1$ we obtain $\frac { \beta } { ( \beta + x ) ( \beta + x - 1 ) } , \quad x = 1,2 , \ldots$ which is one form of Zipf ’s law.

Give the units of the rate constants for reactions with the following overall orders:

first order;

The purchase of antivirus software by one person provides benefits to other people because they are less likely to receive a virus from the software purchaser. Draw a correctly labeled graph showing how the market will determine the quantity of anti virus software purchased. On the same graph, show the socially optimal quantity of antivirus software. Shade and label the area that represents dead weight loss. List two different government policies that could bring about the optimal quantity of antivirus software.

How can small business owners find ways to raise investment they need to start a business?

Which of the following statements is true of a monopoly as compared to a perfectly competitive market with the same costs? I. Consumer surplus is smaller. II. Profit is smaller. III. Deadweight loss is smaller. A. I only B. II only C. III only D. I and II only E. I, II, and III

Draw a correctly labeled graph showing the market determined quantity of pollution, and explain why that quantity will be chosen in the absence of intervention and private deals. On the same graph, show the socially optimal level of pollution.

Which of the following government actions is the most common for a natural monopoly in the United States? A. prevent its formation B. break it up using antitrust laws C. use price regulation D. public ownership E. elimination of the market