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WELLESLEY COLLEGE DEPARTMENT OF ECONOMICS ECONOMICS 201-03 JOHNSON Answers to Problem Set #5 1. q = KL-O.8K 2 -0.2L 2 a. When K = 10, q = lOL - 80 - .2L 2 . To graph marginal productivity = dq = 10 - .4L= 0, maximum at L = 25 dL 2 d q = -.4 , :. Total product curve is concave. dL 2 APL= q/ L= 10-801L-.2L ).. ZEEEL42:---:: To graph this curve: dAPL = 80 -.2 = 0, maximum at L = 20. L - When L = 20, q = 40, AP L = ° where L = 10, 40. Quantity (q 45 - - - - - - - - - - - - - :::-- --r-__ Labor input (L) .L 40 Labor I input (L) 10 10 20 25 30
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MPL==10-.4L, 10-.4L==0, L=25 See above graph. K==20 q=20L-320-.2L 2 c. 320 AP L = 20 - - - .2L ; reaches max. at L = 40, q == 160 . L MP L = 20 - .4L, = a at L = 50. d. Doubling of K and L here multiplies output by 4 (compare a and c). Hence the function exhibits increasing returns to scale. To calculate the elasticity of substitution, it's best to express the MRTS as some function of the capital-labor ratio (K1L). SOMETIMES, this isn't possible (as in the case where the MRTS is NOT well-defined, and other times, the MRTS may be constant and therefore NOT depend on K1L. Let's then proceed and see where it takes us: a. Here, we know that the MRTS is NOT well-defined because the production function is not differentiable. HOWEVER, we know that for perfect complement (Leontief) technologies, the finn will ALWAYS use K and L in the same FIXED proportion (here 2L for every 5K, so a K1L ratio of2/5)! Thus, because the percentage change in K1L is ALWAYS 0, we know that the elasticity of substitution will be a as well. This result makes sense in that NO MATTER WHAT happens to the MRTS (as it goes around the isoquant and moves from infinity to zero) the effect on the K1L ratio is always O. Thus, with fixed proportion technologies, K and L are NOT AT ALL substitutable, so the EASE this substitution (and therefore the value of cr) is O. b. Here, we know that the MRTS is CONSTANT (and equal to 16/37 for those you following along at home)! Thus, the denominator in my elasticity expression is now 0, because it (the MRTS) NEVER changes. One can certainly have different K1L ratios here as we move around an individual isoquant, so this situation implies an elasticity of substitution that's infinite! Make sense? Sure, when one realizes that with perfect substitutability, the ease with which K can be substituted for L is invariant to where one is on the isoquant. . ...thus, that ease is absolute (and infinite in tenns of cr ). Now, perhaps, the FUN begins. Find an expression for the MRTS here, and convince yourself that: MRTS = K1L as one might expect with a Cobb-Douglas looking production function. So, given the fact that the K1L ratio is log-linear in the MRTS, we have that this elasticity substitution is 1 . Why? Well, as we move around the isoquant, the MRTS changes proportionally to the change in the K1L ratio used. This implies EASIER substitutability than in (a) but NOT as easy substitutability as in (b). Reflect on the fact that cr is essentially tracking how movements in the MRTS drive movements in the cost-minimizing K1L ratio to produce a given quantity.
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