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Mathematics For Economics 3rd Edition Michael Hoy, John Livernois - Solutions

1. For each of the following functions, indicate at which point(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.) =12x+6. (a) f(x) = x
7. How do you describe conditions for continuity of a function defined on a closed interval?
6. What does it mean to say the line x = a is a vertical asymptote of a function?
5. Give two examples of functions that are not continuous.
4. Give two definitions of continuity of a function f(x) at the point x = a.
3. Under what condition does the limit of f(x) as x a exist?
2. What is meant by the expression limx_,g+ f(x)?
1. What is meant by the expression limx_,a— f(x)?
9. Consider the following example of the Bertrand model of price competition. The two firms, 1 and 2, set prices p 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
8. A car production plant has a capacity to produce B cars per hour using N workers. The plant can operate around the clock, including weekends, using three shifts of workers. Let h represent the number of worker-hours used per week and MP the marginal product of workers measured by the number of
7. A car production plant has a capacity to produce 100 cars per hour using 2,000 workers. The plant can operate around the clock, including weekends, using three shifts of workers. Let 11 represent the number of worker-hours used per week and MP the marginal product of workers measured by the
6. In this question we consider two plans, each of which combines the effects of an income-support plan with an income-tax program. Let earned income (before tax) be x and income after taxes and any government transfers be y. Plan A In this plan an individual who earns zero income (x = 0) receives
5. Suppose that the government has been taxing each person’s income at a marginal rate of 0.25 for every dollar in excess of $20,000. That is, the first $20,000 earned is not taxed. The government decides to generate extra tax revenue but wishes to avoid increasing the tax burden on low- or
4. Suppose that a salesperson earns a basic monthly salary of $500 Pl”S a commission of 10% on sales if her monthly sales do not exceed $20000 for the month but receives a commission of 20% (on all sales) if her monthly sales do exceed $20,000. Find the function that relates sales to earnings for
3. Suppose that a salesperson earns a basic monthly salary of $800 plus a commission and possible bonuses based on her level of sales. Suppose that the commission rate is 15% and the possible bonuses are a lump-sum amount of $1,000 if her monthly sales exceed $10,000 and a funher lump sum of
2. Given production function Q(L) = L2, defined on [0. +00). derive the cost function, C(Q), and the profit function. 7r(Q). for a perfectly competitive firm. Given that the production function is continuous (according to definition 4.4), use theorem 4.1 to show that the cost function and the
1. Given the production function Q(L) = bL. b > 0. defined on [0. +00)- derive the cost function, C(Q), and the profit function. 71(Q) for a perfectly competitive firm. Let fixed costs be (‘0, and let w be the unit price of L. Prove that the production function is continuous (according to
8. This question introduces a general definition of a step function. Suppose that we take the following set of points from the interval [a, b]: a = x < x < x < ... < xm-1 < xm = b Suppose that the function f (x) takes on one value, 6,, in the subinterval [x,-_1, x,), i = l, 2, . . . , n — 1 and
7. Suppose that we break up the interval [0, 10] into subintervals, each of length 1, in the following way: ([0, 1). [1, 2), (2, 3)..... [8. 9). [9, 10]} Define the function f(x) as f(x) = k for x = [k-1, k); k = 1, 2, 3,....9 f(x) = 10 for x = [9, 10]Plot this function. Notice that the function
6. Prove that according to definition 4.6, the function, f(x) = 2.x - defined on the interval [0, 1] is continuous. Pay special attention to the points x = 0 and x = 1.
5. (d) f(x)=(x-1)/(x+2x-3) [Hint: Factor the denominator.]2. Prove that according to definition 4.4, the following functions are continuous at every point xe R: (a) f(x)=4x+3 (b) f(x)=mx+b
4. For each of the following functions, indicate at which point(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.) (a) f(x) = -3x+12, x < 4 1-2x+10, x4 (b) f(x)=1/2x (c)
3. For each of the following functions, indicate at which p0int(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 IR in each case.) 2x+3, x
2. For the same functions as in question 1 above, generate the sequence of function values associated with the sequence of x—values, x,, = 2 + 1/11, n = l, 2, . . . . Show that the sequence of function values converges in each case and find the limit. In each case, what does this suggest about
1. For each of the following functions, generate the sequence of function values associated with the sequence of x—values, x" = 2 — l/n, n = 1, 2, 3, . _ .. Show that the sequence of function values converges in each case and find the limit. In each case, what does this suggest about the
7. A student in economics has completed her undergraduate degree and has been accepted into a one-year postgraduate business program. Upon completion of this degree she will earn an additional $2,000 a year for each of the next 40 years. She will give up $18,000 income that she would otherwise earn
6. Suppose that an investment project has an immediate cost of $100 million followed by costs of $50 million at the end of one year and a further $25 million at the end of two years. Net revenues (i.e., revenues in excess of operating costs) accrue in the amount of $16 million at the end of each
5. Evaluate the net present values of the following streams of income: (a) V = $100 per year in perpetuity at an interest rate of 10% (b) V = $100 per year at an interest rate of 10% for 25 years (c) = $100 per year beginning after 25 years at an interest rate of 10% for 25 years
4. *Prove result (iv) of theorem 3.2.
3. (a) Compute the present value of the following amounts of money, given an interest rate of 8%. (i) $100 received one year from now (ii) $150 received five years from now (b) Suppose an individual can earn a rate of return of 8% or can borrow money at this same rate. Explain intuitively why the
2. (c) f(n) =n/(3n+2) (d) f(n) = -n (e) f(n) (n+2n+1)/(n + 1) (f) f(n)=5+1/n (g) f(n) 5-1/n Determine the limit, if one exists, for each sequence given in question 1. If a sequence is divergent, determine whether it is definitely divergent. [Hint: Use definitions 3.2 and 3.4.]
1. Determine the first five terms of each of the following sequences. In each case, draw a graph such as those in figures 3.1 to 3.6. (a) f(n) = 1/n (b) f(n)=2+[(-1)" (1/n)]
1. Explain and provide the notation describing a sequence. 2. 3. What does it mean for a sequence to be bounded? What does it mean for a sequence to have a limit? 4. What does it mean for a sequence to be monotonically decreasing or increas- ing? 5. Why is boundedness not sufficient for a sequence
6. A power company can develop a hydroelectric project at one of two capacity levels, one megawatt or two megawatts, at the cost of $1 billion or $1.75 billion respectively. Construction in either case takes one year's time with the cost being incurred immediately. If the smaller capacity is
5. Suppose that a project has an immediate cost of $10 million and running costs of $1 million per year beginning at the end of a one-year construction period. At the end of this year, annual gross revenue from the project of $1.5 million per year is generated in perpetuity. (You may assume that
4. Evaluate the net present value of the following streams of income: (a) V = $1,000 per year at an interest rate of 5% in perpetuity (b) V = $1,000 per year at an interest rate of 12% in perpetuity (c) V = $1 million per year at an interest rate of 10% in perpetuity (d) V, $1 million per year in
3. By writing out and expanding expressions for s, and ps,, prove that for " lim Sn = lim ap-1 1118 11-00 1=1 - ap 1-p
2. Show that the harmonic series i=1 diverges. [Hint: Group terms from = 2 + 1 to 2k+1, k = 1, 2, 3,.... and note that the sum in each group is greater than 1/2.] Show that this is an example of a sequence for which lim, lan+1/a = 1 and the series generated by it diverges.
1. Consider the trivial sequencea, c, c>0 a constant. Show that this is an example of a sequence for which an+1 lim = 1 12X an and the series generated by it diverges (see theorem 3.4).
3. Prove result (iii) of theorem 3.2.
2. Prove result (i) of theorem 3.2.
1. Use theorem 3.3 to Show that the sequence P V, = V/(l + rt' is convergent when r 2 0 and divergent when —l < 1'
6. Making use of the same situation and assumptions as in question 5 above, find the length of time required for the population in this country to double.
5. Suppose a country with a current population of 100 million is expected to experience a population growth rate of 2% per year for the next 50 years. Assuming continuous compounding, what is the expected size of the population: (a) 5 years from now (b) 10 years from now (0) 20 years from now
4. Prove, according to definition 3.2, that the present value of an amount of money, V, received I periods from now and evaluated at an interest rate r > 0 approaches zero as t —> oo.
3. Suppose that the interest rate (r) is such that the present value of receiving $V2 in t2 years from now is the same as the present value of receiving $V1 1" t1 years from now, t2 > I]. Assume that interest is compounded annually. (3) Show that V2 > V]. (b) Show that the present value of
2. If the interest rate is 10% how much money would one need to receive now to be equivalent to $1 million received two years from now if: (3) Interest is compounded annually? (b) Interest is compounded semiannually? (c) Interest is compounded monthly? (d) Interest is compounded continuously?
1. Find the present value of $100 to be received three years from now. assuming annual compounding of interest, given an interest rate of 12%.
5. Show how to write all terms beginning with the 26th term of the sequence f(n)=(1+r)", n = 1, 2, 3,... using the same domain, n = 1, 2, 3,....
4. Show how the sequence of terms f(n) = n, n = 5, 6, 7,... can be written using the domain n = 1, 2, 3,....
3. Show how the sequence of terms f(n) = 2n, n = 0, 1, 2, ... can be written using the domain n = 1,2,3,....
2. Determine the first 10 terms of each of the following sequences. In each case. draw a graph such as those in figures 3.1 to 3.6. (a) f(n) 5-1/n (b) f(n) n/(n+1) (c) f(n) =c+(-1)" (1/n)] for c constant
1. Determine the first 10 terms of each of the following sequences. In each case. draw a graph such as those in figures 3.1 to 3.6. (a) f(n)=5+1/n (b) f(n) =5n/(2") (c) f(n) = (n+2n)/n
9. Show that the function f(x1, x2) = (x1 + x2)/2, x1, x2 > 0 is concave according to definition 2.25.
8. By using the points x' = 2, x" = 6, and 1/2, illustrate definition 2.25 for the concave function y = 10x2. Use a graph to demonstrate this. (x1+x2)/2,
7. Show that the function y = 10-x is strictly concave according to definition 2.28.
6. Sketch typical level sets of the function y = 10x4x/2 and state whether it is (strictly) quasiconcave or (strictly) quasiconvex. Is the function concave, convex, or neither?
5. Simplify the following expressions: (a) (ab)/a2b (b) a(b/a) (c) 10x0252y1/8 (solve for y in terms of x) (d) log, (b)3
4. (a) passing through (-1, 20) and having slope 2 (b) passing through (-2, 1) and parallel to 3x-4y=2 Find the convex combinations of the pair of points (0.-2. 1.-1) and (-1,3,1,-2).
3. Give the equation and sketch the graph of the line
2. A firm's production set is Y =((x, y) e Ry x) Sketch this set in R. Is it closed? bounded? convex? Explain why we would interpret a boundary point of the set as "efficient" and an interior point as "inefficient."
1. Write out the convex combinations of the following pairs of points: (a) —2 and 2 in R (b) (—2, 2) and (—3, 3) in R2 (c) (0, 0) and (x1, x2) in R2 (d) (—2, 2, 5) and (—3, 3, 8) in R3 In cases (b) and (c), draw a graph to show that the convex combination lies on the line segment between
9. What is the difference between a necessary condition and a sufficient condi- tion?
8. Distinguish between quasiconvexity and convexity.
7. Distinguish between quasiconcavity and concavity.
6. Distinguish between concavity and convexity of a function.
5. Distinguish between closedness and boundedness of a point set.
4. What is a point set? What is a convex set?
3. What is a supremum? What is an infimum?
2. What is meant by "the real line"?
1. How does a Venn diagram help to illustrate the possible relationships between sets and subsets?
9. Construct an example of a strictly quasiconcave function that is not a concave function. 10. Using the points x’ = 1, x” = 9, and A = 5/8. illustrate definition 2.25 for the concave function y = x'”, x > 0. Use a graph in your answer. 11. Show that the function y = xm, x > 0. is strictly
8. (a) Given the strictly quasiconcave function y = f(x1, x2), sketch a typ- ical level set in each of the following cases: (i) The function is increasing in x and decreasing in x2. (ii) The function is decreasing in x and increasing in x2. (iii) The function is decreasing in both variables.(b)
7. Sketch typical level sets of the following functions and state whether they are (strictly) quasiconcave or (strictly) quasiconvex. Then say whether the functions are concave, convex, or neither. (a) y=2x-x1x2+2x (b) y (0.5x+0.5x2)1/2 (c) y = 2x1/2x1/2
6. Simplify the following expressions: (a) a/a (b) ab/ab (c) bxx/cxx 1/b (d) (xx)ble (e) 6x025y04 (solve for y in terms of x) (f) x=(2-1/2)-1/2 (solve for x) ubi xby uby (g) yaxxx (What is log y?) (h) log(b) (i) blog (1/x) (j) log, [b(log, a)]
5. A firm's average-cost function is given by the quadratic y = x - 20x + 120 where y is average cost in dollars per unit of output. The output price is $10 per unit, and is the same at all levels of output. Find the output levels at which the firm just breaks even (i.e., price = average cost).
4. Total revenue is price x quantity sold. Show that the total revenue curve corresponding to the demand curve found in exercise 2 is a quadratic. Is it convex or concave? At what value of x does its maximum or minimum occur?
3. Find the convex combinations of the following pairs of points and, where possible, show them graphically: (a) -2 and 4 (b) (-1, 1) and (3, 4) (c) (-2, 0, 1) and (1, -2, 2)
2. In a class of 120 students, everyone would take two hamburgers if the price were zero, and no one would buy hamburgers if the price were $4 or more. Assume that the class demand curve for hamburgers is linear and give its equation. Explain what this implies about the demand for hamburgers when
1. Give equations and sketch graphs of the lines (a) passing through (0, 1) and having slope -2 (b) passing through (-2, 2) and parallel to y2-5.x (c) passing through (-1, 1) and parallel to x +3=1
8. Prove that the set X: [l,2]U[3,4] CR is not convex.
7. For 6 = 0.1 and e = 10, describe the e-neighborhoods: (a) N6 (- 1) (b) Ns(—1, 1) (c) N6(—l,l,—1)
6. Prove that (a) an e-neighborhood is an open set (b) an open set does not contain its boundary points (c) the intersection of two closed sets is closed (d) the Cartesian product of two closed sets is closed (e) the intersection of two convex sets is convex (f) (from definitions 2.15, 2.16, and
5. Find the Euclidean distance between the following pairs of points: (a) 4 and —5 in R (b) (—6, 2) and (8, — 1) in R2 (c) (5, —3, 0, 8) and (12, —6, 3, l) in R4
4. Aconsumer’s preferences over bundles of two goods (x. _\‘) are represented by the smooth, convex-to-the-origin indifference curves of standard economics textbooks. Take the consumption bundle (x’, y’) and define the better .ret B(x’, y’) = {(x, y) e R1 : (x, y) is preferred or
3. A consumer’s budget set is B = {(X, y) e 1R2+ : plx + my 5 m} where p1, p2 > 0 are prices and m > 0 is income. Illustrate this set. Is it closed? bounded? convex? Consider the set X = B O C where C is defined in exercise 2. Sketch this set. How would you interpret the case X = (A? Is X closed?
2. The consumption set of a consumer is C = {(x, y) e REL :x Z .1" > 0. iv 3 yr > 0} Illustrate this set. Is it closed? bounded? convex? How would you interpret x’ and y’?
1. Form the Cartesian products of the following sets: (a) {1, 2, 3, 4, 5, 6} and {7, 8. 9} (b) 2+ and 2+ (c) The set of even elements of 2+ and the set of odd elements of ZIllustrate these product sets in R3
6. The interest on a loan of $1 is the amount of money that must be paid after a specified period of time over and above the repayment of the $1 loan. Show that the rate of interest is a number expressed only in time units, varying inversely with time. How would you express a rate of interest per
5. A subset of IR has a maximum if it contains its supremum. This supremum is then the maximum of the set. Give examples of subsets of R (including bounded subsets) that do and do not have a maximum.
4. Give the dimensions of the following economic quantities: (a) gross national product (b) average cost of producing a good (c) marginal propensity to consume (d) demand for a good (e) rate of inflation (f) marginal product of capital in producing a good
3. Give the dimension of the variable A in each of the following expressions: ( ) wage rate __ a marginal product _ (b) change in national income = A x change in investment rofit (c) amount of labor used = A (d) tax per unit of a good = % >< elasticity of demand for the good (e) .change in profit =
2. We could state a “completeness property" for Q as: Every nonempty subset of Q that has an upper bound has a supremum in Q. Show by choosing a suitable counterexample that this statement is false.
1. Demonstrate the boundedness or unboundedness of the following sets: (a) z+={1,2,3,...} (b) z={...,—3.—2.—t.0.1.2.3....} (c) R+={XERZXZO} (d) R+=R—R+={XERI.\'
10. Prove that for subsets X, Y, and Z of a given universal set U (a) X—Y=X—(XflY)=(XUY)—Y (b) (X—Y)—Z=X—(YUZ) (c) X—(Y~Z)=(X—Y)U(XflZ) (d) (XUY)—Z=(X—Z)U(Y—Z) (e) X—(YUZ)=(X—Y)fl(X—Z) Illustrate each case on a Venn diagram.
9. A firm’s production set is given by P={(x,y):0:y:fi,0:xsi} where y is output and x is labor input. Sketch and interpret P in economic terms. How would you interpret it?
8. Given two subsets X and Y of a universal set U, prove that (a) XY XUY (b) XUY=Xng(c) X Y Xn (d) XCY implies C X (e) XCY implies XU (Y - X) = Y (f) X-YC XUY (g) XY implies Yn X = Y Illustrate each case on a Venn diagram.
7. A consumer’s consumption set is given by C = {(XI.X2) 3X: _>_ 0-K: 2 0} and her budget set is given by B = {(x.,x2) 1 PIXI + P2X2 E M] where x1 and x2 are quantities of goods, p], [)2 > 0 are prices, and M > 0 is income. Illustrate in a diagram the sets (a) B, C (b) B U C (c) B D C and
6. Prove that the set X = {x : x3 > O and .r < 0} equals the empty set.
5. Are the two sets {1, 2, 3} and {3. 1.2} equal?

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