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

2. Which of the following systems are inconsistent? (a) X + 2y 2y + z 2 x + y 3 2x + y + 2z -4 (b) x + y + 2x + 2y - 12 N N = 0 = 0 = 0 (c) 12 2*1 -x1 (d) ax y + 2x2 + X3 + by x2 x2 10 - X3 + 2x4 5 + 2x3 0 -4 C + By = y
1. Which of the following systems are linearly dependent? For those systems that are linearly dependent, find the nature of the dependence. For the rest, find their solutions. (a) 2x + 4y — z 5 y + z — 2 x + y + z 7(b) 4x - 2y + 2z = 6 y + Z I 2x 2 (c) 1x1 + 2x2 3.x3 2 - 2x2 3 2x1 - x2 + x3 11
7. An economy has IS and LM curves given by R = 25-2Y M R = -10++Y where R is the interest rate (percent), Y is GNP ($ billions), and M is the money supply. (a) The government has a target for GNP of $7.5 billion. What level of money supply will achieve this and what is the resulting interest rate?
6. There are two markets for goods that are regarded as substitutes and their supply and demand curves are given by q = 2p1, q = 20-P1+ P2 (good 1) (good 2) 9-10+2p2. 9 = 40-2p2+ Pi Find the equilibrium prices and quantities of the two goods.
5. In the two-market model in equations (7.9) and (7.10), what is the interpreta- tion of the coefficient B21 being positive? Give an economic example of this kind of relationship.
4. A closed economy is described by the following simple IS-LM system: C=a+b(1-1)Y-IR 1=7 G = G L = kY-hR M = M (consumption) (investment) (government spending) (money demand) (money supply) where R and Y are the interest rate and real GDP, respectively, and a > 0, 0
3. For which values of the constant,c, does the following system have (a) no solution, (b) one solution? 2x—y=10 —cx+2y=5
2. Where possible, solve the following pairs of equations by substitution and by elimination: (a) x + y = 10 2x = 4 (b) x + y = 0 x — y = 0 (c) 2x + 4y = 2 x + 2y = l (d) 2x + y = 8 x —— z = 2 3 (e) x/2— y = —2 x + 2y 2 15
1. Graph the following pairs of equations and find their solutions: (a) 2x = 4 3y — x = 10 (b) x + y = 1 2x + 2y 2 2 (c) —x + 4y = 10 —x/4 + y = 5 (d) y = 10 — x y = 2x
10. A firm sells in a competitive market at a price of $10 per unit and has the production function x = 241/2 where x is output and L is labor. It has a maximum available labor supply of 16 units. What is its shadow price of labor? Now suppose the firm could hire additional labor at a wage rate of
9. A student is preparing for exams in two subjects. She estimates that the grades she will obtain in each subject, as a function of the amount of time spent working on them are 81 = 20+20 82=-80+312 where g; is the grade in subject i and t, is the number of hours per week spent in studying for
8. Now suppose that in addition to the royalty, the stadium owners, to prevent "price-gouging," set an upper limit of $2 on the price that can be charged for a hot dog. Show the effect this has on the maximum bid for the franchise.
7. In the case described in question 6, suppose that the stadium owners decide to levy a royalty of $0.25 per hot dog sold. Show the effect this has on your answer.
6. A firm wants to bid for the monopoly franchise to sell hot dogs at a baseball game. It estimates the inverse demand function for hot dogs as p=5-0.5x where p is the price in dollars and x is sales of hot dogs in thousands. It also estimates that it can supply the hot dogs at a constant unit cost
5. A profit-maximizing firm has the total-cost function C = 0.5x-3x+6x+50 and sells into a competitive market on which the price is $1.00. What output should it produce? What would your answer be if the price were $2?
4. The market value of a stock of wine grows over time, 1 R+, according to the function v(t), with ' > 0, " < 0, all r. The present value of the stock of wine is given by V = v(t)e" where r is the interest rate. Find and interpret an expression for the point in time at which the present value of
3. Find the stationary values of the following functions, and determine whether they give maxima, minima, or points of inflection. (a) y = 0.5x-3x + 6x + 10 (b) y x-3x +5 (c) y=x4 4x + 16x (d) y=x+1/x (e) y=x-3x-1 (f) y=3x-10x +6x+5 (g) y (1-x)/(1+x) (h) y (3-2) 1/2 (i) y = (2-x)/(x+x-2) (j)
2. Explain why the condition f'(x) = 0 is necessary but not sufficient for x to yield a maximum off, while f'(x) = 0 and f"(x*) < 0 is sufficient but not necessary for x* to yield a maximum.
1. Illustrate in a Venn diagram the relation between stationary values, extreme values, maxima, and minima.
9. Explain what is meant by a shadow price of a constraint.
8. State and explain the first-order conditions for maximization and minimization over an interval.
7. Explain what is meant by binding and nonbinding constraints.
6. When x is restricted to an interval, why is it no longer a necessary condition for a maximum or a minimum that the first derivative is zero at the optimal point?
5. What is the nth derivative test and why is it necessary?
4. What is a sufficient condition for a maximum or minimum. making use of the second derivative?
3. What is the first-order condition for a minimum?
2. What is the first-order condition for a maximum?
1. Distinguish between local and global optima.
3. Jcx, 0 x x |00, x> c = {x That is, it has a fixed production capacity x, below which marginal cost is constant, atc. (a) Sketch the firm's marginal- and average-cost function. (b) Solve for its profit-maximizing output if it sells in a perfectly competitive market. (e) Describe the solution
2. A firm has the cost function
1. Solve the following problems: (a) max 3+2x subject to 0 x 10 (b) max 1 + 10x subject to 5 < x
6. Use the nth derivative test to show that -x has a maximum at x = 0.
5. A firm has the production function x = f(L), where x is output and L is labor input. The firm buys the input in a competitive market. (a) Assuming the firm sells its output in a competitive market, show that setting output where price equals marginal cost is equivalent to setting labor input
4. A monopsonist's revenue as a function of its only input is R = az - bz :0 It is faced with a supply function for the input z = a + Bp. where p is the input price, anda, b,a, > 0. Find the profit-maximizing price and quantity of the input the monopsonist will choose, and compare the analysis to
3. A monopolist has the inverse demand function p=a-bx and the total-cost function C = 10 log.x Give conditions under which there will be a well-defined. profit-maximizing output and explain your answer in a diagram.
2. A firm in a competitive market discovers a new production process which gives it the total-cost function C = 10log x where x is output. Explain as fully as you can, in both mathematical and economic terms, why there may be a breakdown of perfect competition in this market.
1. For each function in question 1 of section 6.1 exercises, now use second- order conditions to determine whether each stationary value you found is a maximum, minimum, or point of inflection.
5. A monopolist faces a linear demand function. Show that if it maximizes sales revenue, it sets an output exactly half that it would produce if it "sold" its output at a zero price.
4. The demand function facing a monopolist is x apb -b What range of values must b lie in for a solution to the profit-maximization problem to exist?
3. Find the supply curve of a competitive firm with the total-cost function C = 0.04x + 3x + 80
2. Show that a profit-maximizing monopolist's output is unaffected by a propor- tional profit tax, but is reduced by a tax of $t per unit of output. Explain these results.
1. Find the stationary values of the following functions and state whether they yield a local maximum, local minimum, or point of inflection (sketch the function in the neighborhood of the stationary value): (a) y=x-3x + 1 (b) y=x-4x + 16x - 2 (c) y 3x3 3x-2 = (d) y=3x-10x +6x+1 (e) y=2x/(x+1)
10. For the following functions, find the Taylor series formula for n = 2 (i.e., the remainder term involves the second derivative as in equation (5.9)). Determine whether using the differential dy = f'(x) dx to estimate the impact on y of a change in x of an amount dx leads to an underestimate or
9. Let C(y) y 12y2 +50y+ 20, y 0 be a firm's cost function. Find the interval over which it is concave and the interval over which it is convex. Use this information and a table such as that of example 5.19 to draw this function.
8. For the same production function as in question 7,q = aLb, show that the cost function is convex (concave) if the production function is concave (convex). Relate your answer to the answer in question 7. =
7. A firm uses one input (L) to generate output (q) according to the production function q = aL”, a > 0, and b > 0 (also L 2 O). The input price is wand fixed costs are Q). Show that dq/dL is rising if dC/dq is falling, dq/dL is falling if dC/dq is rising, and dq/dL neither rises nor falls if
6. Suppose that a firm’s total product function is y = 40L2 — L3. Show that the average product of labor, AP(L), rises when marginal product of labor, MP(L), exceeds AP(L), falls when MP(L) is less than AP(L), and is horizontal at the point where MP(L) = AP(L).
5. Find the expression for the point elasticity of demand 6 (with respect to own price) for the demand function y = 200 — 5 p. Determine the ranges of prices for which 6 is less than 1 and greater than 1. Illustrate on a graph of this demand function.
4. Suppose that two firms, A and B, behave as competitive firms in deciding how much output to supply to the market. Firm A ’5 cost function is CA = atq+flq1 and firm B’s cost function is C3 = yq + qu. Assumea. [3, y,a) > Oand V 2 01- (a) Find the supply functions, defined on q 3 O, for each
3. Find the slope ofeach of the following production functions. _\' = f(L). Graph the functions and their derivative functions. Give the economic significance of the slope of the derivative functions (i.e. whether the derivative is increasing or decreasing in L). In each case L > 0. (a) y = 64L1/4
2. Suppose that a salesperson has the following contract relating monthly sales, S, to her monthly pay, P. She is given a basic monthly amount of $500, regardless of her sales level. On the first $10,000 of monthly sales she earns a Q 10% commission. On the next $10,000 of monthly sales she eams
1. From the definition of the derivative (definition 5.3), find the derivative of the function f(x) =x2+3x—4
10. Use the Taylor series expansion to show that using the tangent line (or differential) at a point x = x0 to estimate the value of the function at some other point x 7e x0 leads to an overestimate if the function is strictly concave and an underestimate if the function is strictly convex.
9. Explain with the use of a graph why the second derivative of a differentiable concave function is less than or equal to zero.
8. Explain with the use of a graph why the second derivative of a differentiable convex function is greater than or equal to zero.
7. Write out the 13 rules of differentiation given in this chapter.
6. Why is it the case that if a function f(x) is differentiable at a point x =a, then it must also be continuous at that point?
5. Use the concepts of left-hand and right-hand derivatives to indicate when a function f(x) is differentiable at a point x = x0.
4. Define and explain left—hand and right-hand derivatives of a function f(x) at a point x = x0.
3. What is the relationship between a tangent line, the derivative of a function, and total differential?
2. How can a tangent line be defined in terms of a sequence of secant lines?
1. What is the difference between a secant line and a tangent line?
4. For the function f(x) = x, find the Taylor series formula for n = 2 (i.e.. the remainder term involves the second derivative as in equation 5.9). Show that using the differential dy = f'(x) dx to estimate the impact on y of a change in x of amount dx leads to an underestimate of the actual
3. For the function f(x) = x/2, x 0, find the Taylor series formula for n = 2 (i.e., the remainder term involves the second derivative as in equa tion 5.9). Show that using the differential dy = f'(x) dx to estimate the impact on y of a change in x of amount dx leads to an overestimate of the
2. Use the Taylor series expansion formula to find an estimate for the function f(x) = In(1+x), x > -1, for any value x belonging to the interval [0.1]. Choose x = 0, and ensure that your computation is correct to within 0.00! (see example 5.24).
1. Use the Taylor series expansion formula to find an estimate for the function f(x)e for any value x belonging to the interval [0. 11. Choose x=0 and ensure that your computation is correct to within 0.001 (see exam- ple 5.24).
4. For the function f(x) = x, find the Taylor series formula for n = 2 (i.e.. the remainder term involves the second derivative as in equation 5.9). Show that using the differential dy = f'(x) dx to estimate the impact on y of a change in x of amount dx leads to an underestimate of the actual
3. For the function f(x) = x/2, x 0, find the Taylor series formula for n = 2 (i.e., the remainder term involves the second derivative as in equa tion 5.9). Show that using the differential dy = f'(x) dx to estimate the impact on y of a change in x of amount dx leads to an overestimate of the
2. Use the Taylor series expansion formula to find an estimate for the function f(x) = In(1+x), x>-1, for any value x belonging to the interval [0.1]. Choose x = 0, and ensure that your computation is correct to within 0.001 (see example 5.24).
1. Use the Taylor series expansion formula to find an estimate for the function f(x)=e for any value x belonging to the interval [0. 1]. Choose x=0 and ensure that your computation is correct to within 0.001 (see exam- ple 5.24).
5. Suppose that a firm in a competitive market sells a product at price p = 45 and has the same cost function as in question 4 above, namely C(y)=y3-9y+ 60y+10, y 0. For this firm's profit function find the interval over which the function is concave and the interval over which it is convex. Use
4. Let C(y)=y3-9y2 + 60y+ 10, y 0 be a firm's cost function. Find the interval over which this function is concave and the interval over which it is convex. Use this information and a table such as that of example 5.20 to draw this function.
3. Let y = x/, x > 0, be a production function, where y is output and x is a single input. Derive the cost function, C(y) = co+rg(y), where x = g(r) is the inverse of the production function, co is the fixed cost, and r is the unit cost of the input. Show that the production function is strictly
2. Show that the function f(x) = x/4, x>0 satisfies the definition of a strictly concave function (definition 5.10).
1. Show that the function f(x) = x satisfies the definition of a strictly convex function (definition 5.8).
8. A bakery advertises its bagels by noting either the price per dozen or per bagel and doesn't offer any quantity discounts. Thus, for example, if the price is $4.80 per dozen (i.e., for 12 bagels), then it is $0.40 per bagel. Since these prices are the same, the baker is not surprised to find
7. A bakery advertises its bagels by noting either the price per dozen (i.e.. 12 bagels not a "baker's dozen" of 13) or per bagel and doesn't offer any quantity discounts. Thus, for example, if the price is $4.80 per dozen (ie. for 12 bagels), then it is $0.40 per bagel. Since these prices are the
6. Suppose that a monopolist faces inverse demand function p = a - bq. Find its marginal revenue function. Plot both the demand function and the marginal revenue function on a single graph.
4. For the total cost function TC(y) 3y+7y+24, y >0 show that (and illustrate on a graph):(a) MC is less than AC where AC is falling. (b) MC = AC at the point where the AC curve is horizontal. (c) MC exceeds AC where AC is rising. A firm uses one input, L, to generate output. q. according to the
3. Suppose that two firms, A and B, behave as competitive firms in deciding how much output to supply to the market. Firm A's cost function is C^= 10q+2qq0, and firm B's cost function is CB = 15q + q2.q0. (a) Find the supply functions, defined on q 0, for each firm and draw them on the same graph.
2. Find the slope of each of the following production functions, y = f(L). Graph the functions and their derivative functions. Give the economic significance of the sign of the slope of the derivative functions (i.e., whether the derivative is increasing or decreasing in L). (a) yaL, a > 0 (b)
1. Find the slope of each of the following production functions, y = f(L). Graph the functions and their derivative functions. Give the economic significance of the sign of the slope of the derivative functions (i.e., whether the derivative is increasing or decreasing in L). (a) y = 10L (b) y=8L1/3
5. Suppose that a salesperson has the following contract relating monthly sales, S, to her monthly pay, P. She is given a basic monthly amount of $600, regardless of her sales level. On the first $10,000 of monthly sales she earns a 10% commission. On any additional sales she earns a commission of
4. Consider the following income tax scheme: The first $5,000 of income is not subject to any tax. The next $15,000 is subject to a tax rate of 20%. The next $30,000 is subject to a tax rate of 35%. Any additional income is subject to a tax rate of 50%. (a) Find and graph the tax function, 7 (y),
3. Consider the following income tax scheme: The first $6,000 of income is not subject to any tax. The next $10,000 is subject to a tax rate of 20%. The next $30,000 is subject to a tax rate of 30%. Any additional income is subject to a tax rate of 40%. (a) Find and graph the tax function, T (y),
2. The following are examples of functions which are not differentiable at some point. Explain in each case why the function is not differentiable according to definition 5.5. That is, find the left- and right-hand derivatives at the point of nondifferentiability. -2x+20, x 4 (a) f(x)= -x+16, x>4
1. The following are examples of functions that are not differentiable at some point. Explain in each case why the function is not differentiable according to definition 5.5. That is, find the left- and right-hand derivatives at the point of nondifferentiability.(a) f(x)= [3x+2, x 5 (x+12, x> 5
4. Return to the example in question 2 of exercise 5.1. Recall (or compute) the sequence of changes Ay, associated with changes Ax, with reference to the point P (20, 400). Compare the results of this formula for Ay,, the actual change in y, with the estimated change in y using the differential dy
3. Return to the example in question 1 of exercise 5.1. Note that the derivative of this function, f(x) = x, is f'(x) = 2x. Use the differential to estimate the changes in y between P = (20, 400) and each of the 5 points Q.." = 1, 2, 3, 4, 5. Find the percentage error defined as e = (Ay-dy)/(Ay) x
2. From the definition of the derivative (definition 5.3), find the derivative for each of the following functions: 3. (a) f(x)=6x (b) f(x)=12x-2 (c) f(x)=kx for k a constant
1. From the definition of the derivative (definition 5.3), find the derivative for each of the following functions: (a) f(x)=3x-5 (b) f(x)=8x (c) y = 3x
4. The slope of the tangent for the function y = x is 1/(2x). Find the equation of the tangent line at the point x = 1. Illustrate on a graph.
3. The slope of the tangent for the function y = x is 2x. Find the equation of the tangent line at the point x = 3. Illustrate on a graph.
2. As for question 1, compute a sequence of ratios Ay/A.x for the function y=x with respect to the fixed point P = (20. 400). This time use A.x, = 1/m. n = 1,2,3.... to generate a sequence of points Q = ((20+ 1/m). (20+ 1/n)) and so a sequence of ratios Ay (20+1/n) - (20) A.x,, (20+1/n)-(20) Show
1. Suppose that we choose the point P = (20, 400) on the function y = x2. Find the ratio Ay / Ax for each of the line segments (secants) found by connecting each of the points Q1 = (25, 625), Q2 2 (24,576), Q3 = (23, 529), Q4 = (22, 484), and Q5 = (21, 441), and arrange in a table as illustrated
6. Consider the following example of the Bertrand model of price competition. Two firms, 1 and 2, set prices pl 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
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 relationships between the cost of the energy required by the locomotive and the number of carriages it pulls are as
4. Let y = x be a production function relating input x to output y. Let & 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 the revenue function, the cost function, and the profit function for the
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
2. Prove that according to definition 4.4, the function, f(x) = 3x, is continuous at every point x E R.

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