Since the proof of the method is identical in both

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. Since the proof of the method is identical in both cases, we shall not repeat it here, but simply refer you to the text. In Figure M.6-1, we have used the same labelling system as in Figure A.4-2 (page 597 of the text), with the horizontal intercept of the tangent to the supply curve at C labelled E and the vertical intercept labelled A . Just as with the price elasticity of demand, the price elasticity of supply s = EC/AC = OF/AF = EG/OG. (M.6.3) M6-2 MATH MODULE 6: ELASTICITIES P P P F F F S S S C C C S' S' S' O O O = A = E A A E E G G G Q Q Q (a) Inelastic supply (b) Elastic supply (c) Unitary elasticity of supply FIGURE M.6-1
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Notice, however, that here the directed line segments all go in the same direction, so that (as we would expect) s is positive. In Figure M.6-1 (a), notice that all three ratios are < 1: supply is inelastic. In Figure M.6-1 (b), all three ratios are > 1: supply is price-elastic. And in (c), both A and E coin- cide with the origin: all of the ratios are equal to 1, and the supply curve has unitary price elasticity at C . Application 2: Supply Elasticity: The “Eyeballing” Method If an exact numerical value for price elasticity of supply is not required, then Figure M.6- 2 provides a simple method for gauging whether supply is elastic, inelastic, or of uni- tary elasticity. If the tangent to the supply curve at a point C intersects the vertical axis in the positive quadrant, then supply is elastic at C. If it intersects the horizontal axis in the positive quadrant, then supply is inelastic at C. And if the tangent at C passes through the origin , then ( regardless of its slope ) the supply curve at C has unitary price elasticity. Application 3: Price, Elasticity, and Marginal Revenue In the text (p. 363), we derive the following relation between price ( P ), price-elasticity ( ) and marginal revenue (MR): MR = P (1 + 1/ ), or equivalently, P = MR [ /( + 1)]. (M.6.4) We can use Figure M.6-3 to illustrate one implication of Equation M.6.4. With one exception, if we know any two of the 4 variables P , MR, and the vertical intercept A , we can calculate the other two. To see why, note that = OF/AF and P = OF , so that MR = OF (1 + AF/OF ) = OF + AF = OF – FA. What this says is that A P = P – MR, or MR = 2 P – A , or P = ( A + MR)/2, or (in other words) that P is midway between MR and the MATH MODULE 6: ELASTICITIES M6-3 P O Q C 3 C 2 C 1 ε S > 1 ε S = 1 ε S < 1 FIGURE M.6-2
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vertical intercept ( A ). For example, if the vertical intercept of the tangent to the demand curve is at $24/kg when P = $16/kg, then = OF/AF = 16/(16–24) = –2, and MR = 2 P – A = 2(16) – 24 = $8/kg. The one exception occurs if we are given that MR = 0 and = –1, since these 2 val- ues of these variables are consistent with an infinite number of combinations of P and A : all we know is that A = 2 P.
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