MATH20330: Optimization for Economics Homework 1: Solutions

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1 MATH0330: Optimization for Economics Homework 1: Solutions 1. Sketch the graphs of the following linear and quadratic functions: f(x) = 4x 3, g(x) = 4 3x h(x) = x 6x + 8, R(q) = q q. y = f(x) is a line with y-intercept 3 and slope 4,cutting the x-axis at x = 3/4. y = g(x) is a line with y-intercept 4 and slope 3,cutting the x-axis at x = 4/3. y = h(x) = (x )(x 4) is a parabola with x-intercepts at x = and x = 4 and lowest point (3, 1). R = q q = (10 + q)(40 q) is a parabola with R-intercept 400, q-intercepts at q = 10 and q = 40 and highest point at (15, 65).. Using only the definition of a limit of a function at a point, prove that lim x 3 (4x 7) = 5.

2 Let f(x) = 4x 7. Observe that f(x) 5 = (4x 7) 5 = 4x 1 = 4 x 3. Suppose given ɛ > 0. Take δ = ɛ/4 (for example). Then if x 3 < δ we have f(x) 5 = 4 x 3 < 4δ = ɛ and we re done. 3. Using only the definition of the limit of a function at a point, prove that 1 lim x x = 1 : Thus you are being asked to prove that lim x a f(x) = L where f(x) = 1/x, a = and L = 1/. (a) Relate f(x) L to x a in this case. (b) Show why it follows that f(x) L 1 x a when x 1. (c) Given ɛ > 0 (and you may assume ɛ < 1 why?) what choice of δ will guarantee that f(x) L < ɛ as long as x a < δ? Prove your assertion. (a) f(x) L = 1 x 1 = x x x =. x (b) If x 1 then x = x and hence 1 x 1. Therefore f(x) L = x x 1 x = 1 x a. 4. Let (c) For δ we may choose any positive number which is ɛ. For then x a < δ = f(x) L 1 x a 1 δ 1 ɛ = ɛ. f(x) := x x (x 0). (a) Calculate the values f(1), f(), f(5), f(0.3), f(5.18) (b) Calculate the values f( 1), f( 7), f( 58.31), f( 0.001). (c) Draw the graph of f(x). (d) What are the points of discontinuity of f(x)? Do the left-hand and right-hand limits exist at these points?

3 (a) If x > 0 then x = x and hence f(x) = x/x = 1. So f(1) = f() = f(5) = f(0.3) = f(5.18) = 1. (b) If x < 0 then x = x (note that x > 0) and hence f(x) = ( x)/x = 1. So f( 1) = f( 7) = f( 58.31) = f( 0.001) = 1. (c) The graph of y = f(x) is the horizontal line y = 1 to the right of the y-axis and the horizontal line y = 1 to the left of the y-axis. (d) x = 0 is the only point of discontinuity. We have lim f(x) = lim ( 1) = 1 and lim x 0 x 0 f(x) = lim 1 = 1. x 0+ x A salesperson earns a basic monthly salary of 500 euro. If her monthly sales do not exceed 0, 000 euro she receives a commission of 10% of sales. If her monthly sales exceed 0, 000 euro, her monthly commission is 0% of sales. (a) Express her monthly income I as a function of her monthly sales S. Draw the graph of I against S. (b) What, if any, are the points of discontinuity of I? Calculate the left- and right-hand limits (if they exist) at any such points. (a) I(S) = { S, S S, S > 0000.

4 (b) There is a discontinuity at S = 0000: while lim I(S) = = 500 S 0000 lim I(S) = = S Consider the following two government income tax/income support plans: Plan A: An individual who earns zero income receives a tax-free government transfer (income support) of 6000 euro annually. No income support is received by anyone who has a positive annual income. Those who earn in excess of 6000 euro per annum pay an income tax at the rate of 0% on each euro earned in excess of 6000 euro. Plan B: Everyone receives a basic supplement of 6000 euro. (This is called the demogrant.) Each person then pays income tax at the rate of 40% for every euro of earned income ie. not including the demogrant. Let x be the amount of earned income. Let y be income after government transfers and taxes. (a) In a single diagram, draw the graph of y as a function of x for each of the two plans. Determine where the graphs intersect. (b) Which plan appears more favourable from the point of view of the earner? The answer depends on annual income; explain. (c) Which plan provides the greater incentive to increase earnings? Explain your answer. (a) Let y A be the net income in plan A and y B be the net income in plan B. Then 6000, x = 0 y A = x, 0 < x 6000 x.(x 6000) =.8x + 100, x > and y B = x. Note that y B > x = y A for 0 < x < The graphs intersect when x > To determine the x-value of the point of

5 intersection we solve y B = y A i.e x =.8x for x: This gives.x = 4800 so that x = = 4, 000. So y A > y B when x > (b) If your earned income is 0, neither plan is preferred (both give 6000). If your earned income lies between 0 and 4000 plan B is preferable (since y B > y A in this interval). If your income exceeds 4000 then plan A is preferable. (c) If your earned income is 0 plan B provides a greater incentive to earn more. If your earned income is above 0, plan A always provides greated incentive to earn more: When your income is below 6000 you take home the whole of each additional earned euro and if your income is above 6000 you take home 80c from each additional earned euro. Under plan B, on the other hand, you take home only 60c for each additional earned euro. (To summarize: The marginal (net) income is larger under plan A. Note that the marginal income is precisely the slope of the (straight line) graph.) 7. Consider the following example of the Bertrand model of price competition. Two firms, A and B, set prices p A and p B, respectively. The firm offering the lower price captures the entire market. If they set the same price, they share the market equally. Suppose that the market demand function is y = 50 p and that the cost function is the same for A and B: C(y) = 10y.

6 (a) Suppose that firm B fixes a price p B = 0. Draw a graph of the cost, revenue and profit functions for firm A, each as a function of the price p A. Find lim pa 0 for each of these functions. (b) At which common price p A = p B will neither firm have any incentive to lower their price? Explain your answer. (a) The cost is 10(50 p A ) = p A, p A < 0 C A = , p A = 0 p A > 0. The revenue is p A (50 p A ) = 50p A p A, p A < 0 R A = 0 1 0, p A = 0 p A > 0. The profit is π A = R A C A = (50p A p A ) (500 10p A) = 60p A p A 500, p A < = 150, p A = 0 0, p A > 0. We have lim C A = lim (500 10p A) = = 300, p A 0 p A 0 and lim R A = lim (50p A p p A 0 p A A) = = lim π A = lim (R A C A ) = = 300. p A 0 p A 0 (b) For either firm the graph of profit π = R C = 60p p 500 crosses the p-axis at p = 10. At price below 10 either firm makes a loss. At prices above 10 (and below 50) either firm will make a profit provided it can undercut the other in price. 8. Determine a value of the parameter b for which the function { 31 bx, x 3 f(x) = 3x x b, x > 3.

7 is continuous. Draw the graph of f(x) for this value of b. For any value of b, f(x) is continuous on x < 3, since it is linear on this interval, and on x > 3 since it is quadratic. Now lim f(x) = lim (31 bx) = 31 3b = f(3) (since linear functions are continuous). x 3 x 3 On the other hand, lim f(x) = lim x 3+ x 3+ (3x x b) = b = 1 b. If f is to be continuous we must have therefore 31 3b = 1 b which gives b = 5.