Key Concepts asymptote continuous from the left continuous from the right continuous functions discontinuous functions intermediate-value theorem left-hand limit marginal product right-hand limit vertical asymptote

Review Questions

2. What is meant by the expression limA fix)?

3. Under what condition does the limit of f(x) as x — a exist?

4. Give two definitions of continuity of a function fix) at die point x — u

5. Give two examples of functions that are not continuous.

6. What does it mean to say the line x = a is a vertical asymptote of a function?

7. How do you describe conditions for continuity of a function defined on a closed interval? 8. State intuitively the intermediate-value theorem.

Review Exercises

1. For each of the following functions, indicate at which poini(s) the function is discontinuous and explain which of the conditions of definition 4.3 is not satisfied. In each case, graph the function. (The domain is R in each case.)

(b) f(x) = U + 1 )/(.v2 - I ) [Hint: Factor the denominator.)

2. Prove thai according to definition 4.4, the function, f(x) = 3.r, is continuous at every point x e R.

3. Suppose that the government has been taxing each person's income at a marginal rate of 0.4 for every dollar in excess Of $25,000 with the first $25,000 earned not taxed, in addition the government imposes a lump-sum surtax of $2,000 on every person who earns $100,000 or more. Write out and graph income after tax, _y, as a function of income before tax, x. Indicate why, according to definition 43. the function has a point of discontinuity at a = $100,000. Discuss any incentive effects on hours worked thai may arise due to this discontinuity in the tax schedule.

4. Let y = x2 be a production function relating input .v to output v. Let c represent the unit cost of input x, and assume that total cost equals fixed costs, Co, plus the cost of input x. Let p be the unit price of y. Find ihe revenue function, the cost function, and the profit function for the firm. Given that the function l'(x) = x2 is continuous, are these functions continuous? (Use theorem 4. | to answer this question.) Discuss.

5. A railway company runs a train from A to B. The wages of the engineer and guard for one trip total $500, Each carriage on the train holds an absolute maximum of 50 passengers. The relationship between the cost of the energy required by the locomotive, and the number of carriages it pulls, are as follows:

1 carriage 1,000

2 carriages 1.800

3 carriages 2.400

4 carnages 2.600

5 carriages 3.400

6 carriages 4.600

7 carriages 8.000

ft makes no difference to these costs whether the carriages are empty or full.

(a) What is the cost incurred to transport just one passenger from A to B ?

(b) What is the increase in cost created by taking a second passenger?

(c) What is the increase in cost created by taking the 51st passenger?

(d) Draw the functions relating: (i) total costs

(Si) average cost per person to the total number of passengers, over the range 0 to 350 passengers, and comment on the continuity properties of these functions.

6. Consider the following market demand and supply functions".

Graph Dip) and S(p) on one diagram and zip) = Dip) - Sip) on another. Find the equilibrium price and quantity for this market and illustrate on both graphs. Show that these demand functions satisfy the requirements for existence of a positive equilibrium price, as specified in theorem 4.3.

7. Consider the following market demand and supply functions:

Nonce that as price falls to $45, a jump in demand occurs. This may be due to a new group of consumers deciding to enter the market once the price falls to this level.

Graph Dip) and Sip) on one diagram and :.{p) — Dip) - Sip) on another. Which of the conditions of theorem 4.3. that guarantees existence of a (positive) equilibrium price, is absent? Discuss.

8. "Consider the following example of the Bertrand mode) of price competition. Two firms, 1 and 2. set prices pi and p2, respectively. The firm offering the lower price captures the entire market. If they charge the same price, then they share the market equally. Assume that market demand is determined by the function y = 50 - p, and each firm faces cost function C(y,) = 10.v/, which implies constant unit cost of S10.

fa) If firm 2 charges a price p2 = 20, find firm I \ revenue function. A', (/>i), and profit function, JTi (p\ )■ Draw a graph of each of these functions and explain why each is discontinuous at p\ = 20.

(b) For a general price charged by firm 1 of />,. where p2 > 10, do the same exercise as for part (a).

(c) Upon considering the results above, why is it the case that the outcome of each firm charging a price of $10 is the equilibrium for this model? Explain.

♦Consider the following example of Hotelling's location model. Each of two firms sells a homogeneous product and charges a price of $25 while facing constant unit cost of$ 15. There are N = 1.000 consumers who are uniformly (evenly) distributed along a street one mile in length, represented by the unit interval [0, l]. The two firms, A and B, will choose locations. LAand ¿B respectively, on the line |0, I ] in such a way as to maximize market share, and hence profit.

(a) Assuming that firm B establishes its location first at Lb = 0.8, find and graph firm A's market-share function and profit function 7ta(La). Discuss why each function is discontinuous at point LA - 0,8.

(b) For any genera) choice of location Ln < 0.5 by firm B, do the same exercise as part (a).

(c) For any general choice of location Lb > 0.5 bv firm B, do the same exercise as part (a).

(d) For choice of location LB = 0.5 by firm B. do the same exercise as part (a).

(e) Since firm B can deduce where firm A would locate, conditional on firm B's own choice of location Lfi, where would firm B locate? What is the equilibrium outcome of diis model? Discuss.

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