Partial Elasticities

6.IK. Given Q - 700 - 2/' ♦ 0,02 V. where /' - 25 and Y - 5000. Find (a) the price elasticity of demand and (f>) the income elasticity of demand.

wbcrc -2 and V • 700 - 2(25) ♦ 0 02(5000) « 750.

6.19. Given Q 400 HP 0 05Y. where V - 15 and Y = 12.000 Find (a) the income elasticity of demand and (b) the growth potential of the product, if income is expanding by 5 percent a year. (c) Comment on the growth potential of the product rf) Q - 400 - 8(15) - 0 05(12.000) «80 and aQOY 0.05. Thus.

KcArrangin({ terms and »ubstilutinjt the known parameters.

llic demand (or the good will increase by 3.4 percent.

€) Since 0< », < I. it can be expected that demand lor the good will increase with national income, but the increase will be Ion than proportionate fhuv while demand grows absolutely, the relative market «hare of the good will decline in an ctpanding economy If tr> 1. the demand (or the product would grow taster than the rale of ciparmon in the economy, and increase its relative market share And if <0. demand for the good would decline as income increavev

6.20. Given Q] - 100 - /*, + 0.75/\ - 0.2SP} + 0.0075Y. Al Px = 10. P7 = 20. Py = 40. and Y = 10.000. 0, = 170. Find the different cross price elasticities of demand.

6.21. Given (?, = 50 - AP, - 3P, + 2P, + 0.001 Y. At /\ = 5. Pz = 7. Pj - 3, and V = 11,000, £>, = 26. (u) Use cross price elasticities to determine the relationship between good 1 and the other two goodv (b) Determine the cffcct on of a 10 percent price increase for each of the other goods individually.

With c,j negative, goods 1 and 2 are complements. An increase in P, will lead to a decrease in Qx. With <,j positive, goods 1 and 3 are substitute*. An increase in P> will increase

Rearranging terms and substituting the known parameters.

If P: increases by 10 pcrcent, Qt decreases by 8.1 percent.

Cri "1

If Px incieaseji by 10 pcrcent, Q, increases by 2.3 percent OPTIMIZING ECONOMIC FUNCTIONS

6.22. Given the profit function v * 160.r - 3*2 - 2xy - 2y2 + I20y - 18 for a firm producing two goods x and y, (a) maximize profits, (b) test the second-order condition, and (c) evaluate the function at the critical values x and y.

a) tr. - 160 - 6r - 2y - 0 «r, - -2* - Ay + 120 ■ 0 When solved simultaneously, * - 20 and y = 20.

b) Taking the second parnaK

- -6 v,y - -4 w,, = -2 W'uh both direct second partial* negative, and nx.vwv>(rrn)2. n is maximized at * = y = 20.

6.23. Redo Problem 6.22. given it - 25* - x2 - xy - 2y* + 30y - 28.

With jt„ and tt„ both negative and ir„ tr„ > (w,t)j. it a maximized, f) r = 172

&24. A monopolist selJs two products x and y for which the demand functions are x - 25 - 0.5PS (6.29)

and the combined cost function is c = rJ + 2ry + /+20 (6J/)

Find (a) the profil-maximizing level of output for each product, (b) the profit-maximizing price for each product, and (c) the maximum profit

Substituting in (6.32), n = (50 - 2x\x + (30 - y)y - (x2 + 2*y ♦ y1 + 20) « 50* - 3X2 + 30y -2y* - 2xy - 20 (6.35)

The first-order condition for maximizing (6.JJ) «

Solving simultaneously. * - 7 and y » 4. Testing the seennd-order conditon. - -6. ir,r = -4. and n.y « -2. With both direct partiab negative and „jr. v a maximized.

P. - 50 - 2(7) »36 P, - 30 - 4 - 26 c) Substituting x » 7. y = 4 in (6.35). v « 215.

6^5. Find the profit-maximizing level of (a) output, (b) price, and (c) profit for a monopolist with the demand functions x « 50 - 0.5P, (6.36)

y = 76- P, (6.37) and the total cost function c = 3r + 2xy + 2yJ + 55. a) From (6.36) and (6.37).

it - (100 - 2x)x + (76 - >•)>• - (Ix3 +2xy* 2/ + 55) - lOQx - Sx3 + 76y - 3>J - 2xy - 55 (0.40)

Maximizing (6.40), v, = 100 - lOar - 2y = 0 v, =• 76-6y-2x - 0

Ihus. x - 8 and ,y - 10 Cheeking the second-order condition. n„ - -10. ryv - -6. and w,y « -2. Since *„. jr,y < 0 and ir„ > (w,y)\ tt is maximized at the critical values. b) Substituting x = 8. y = 10 in (A38) and (6..»),

6.26. Find the profit-maximizing level of (a) output, (b) price, and (c) profit for the monopolistic producer with the demand functions

a- - 36 - jPj (6.42) and the joint cost function c - (ft + 20,02 + (fi + 120.

rr - (74 - 1.50,)0, ♦ (72 " 2&H?a " (0? + 20,0, - 05 + 120) - 740, - 2.50i ♦ 720j - 30? - 20,0; - 120 (6.45)

The first-order condition for maximizing (6 45) is ir. = ?4 - 50, - 202 = 0 », = 72-60,-20, = 0

Thus. 0, = 1134 and 0. =• 8,15. Testing the second-order condition, ttu - -5. v^ - -6. and fr,2 ■ -2. Thus. *,,, ffi.<0. w,, os>(ir,^. and n is maximized.

b) Substituting the critical values in (6.43) and (6.44).

P, - 74 - 1.5(11.54) - 56.69 P2 - 72 - 2(8.15) - 55.70 e) rr - 600 46

6.27. Find the profit-maximizing level of (a) output, (b) price, and (c) profit when

02 = 8200 - 20P2 (6.47) and c = O.10i +0.10,02+ O.20t +325 a) From (6.46) and (6.47)

Thus. tt - (520 - 0.1 Q,)Q, + (410 - 0.05QX)Q3 - (0.1 <?? + 0.1 (?, (?2 ♦ 0.205 + 325)

- 520Qi - 0.2\$ +410C?i- 0.25 - 0.1Q, Q, - 325 (6.50)

Maximizing (6.50).

7t| — 520 — 0.4{?, - 0.1G; = 0 TTj = 410 - 0L5& - 0.1 <?, = 0

ITius. Qx ■ 1152.63 and & - 589.47. Checking Ihc second-order condition, w„ - -0.4. tra - -0.5. and it,, - -0.1 - wal. Sincc w3<0 and W|,irij>(»|i)J. rr a maximized at Q, - 1152.63 and Q: = 589.47.

P, - 520 - 0.1(1152.63) - 404.74 - 410 - 0.05(589.47) - 380.53