eco204_HW_5 - ECO 204 2008‐2009 Ajaz Hussain HW 5 Suppose...

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Unformatted text preview: ECO 204 2008‐2009 Ajaz Hussain HW 5 Suppose we analyze a company with 2 inputs capital (K) and labor (L). The production function is Q = f(K, L). In lecture 6 we discussed‐‐ amongst others‐‐ these production technologies: • • • Imperfect Substitutes Inputs Technology: Q = Lα Kβ Perfect Substitutes Inputs Technology: Q = αL + βK Complements Inputs Technology: Q = min(αL, βK) In lecture 6 we also discussed the CES production function: Q = [α L(σ – 1)/σ + β K(σ – 1)/σ ] σ/(σ – 1) where σ is the elasticity of substitution. We discussed how the iso‐quants of these technologies were approximated by the iso‐quants of: • Imperfect Substitutes Inputs Technology: lim σ → 1 of Q = [α L(σ – 1)/σ + β K(σ – 1)/σ ] σ/(σ – 1) • Perfect Substitutes Inputs Technology: lim σ → ∞ of Q = [α L(σ – 1)/σ + β K(σ – 1)/σ ] σ/(σ – 1) • Complements Inputs Technology: Q = min(αL, βK): lim σ → 0 of Q = [α L(σ – 1)/σ + β K(σ – 1)/σ ] σ/(σ – 1) You are not responsible for proving the statements above. Question 1 Assume 0 ≤ σ ≤ ∞ and derive the MRTS = (dK/dL) = (dQ/dL)/(dQ/dK) of the CES production function. Question 2 Consider a company using K and L as imperfect substitutes: Q = Lα Kβ. (a) Derive the MPK = dQ/dK and interpret it. (b) Derive the MPL = dQ/dL and interpret it. (c) Derive the MRTS = dK/dL = (dQ/dQ)/(dQ/dL) = MPK/MPL. Interpret this. (d) Substitute σ = 1 in the MRTS of the CES production function: do you get the MRTS of Q = Lα Kβ? (e) Suppose α = 1/3 β = 2/3. What are the returns to scale for this technology? (f) In consumer theory, we saw the utility function U = Q11/3 Q22/3 represents the same preferences as (say) U = Q12/3 Q24/3. By the same token, does the production function Q = L1/3 K2/3 represent the same technology as Q = L2/3 K4/3? (g) Suppose α = 1/3 β = 2/3. What are the returns to scale for this technology?s (h) For what values of α and β will this technology exhibit increasing returns to scale? Decreasing returns to scale? Question 3 Consider a company using K and L as perfect substitute:s Q = αL + βK. (a) Derive the MPK = dQ/dK and interpret it. (b) Derive the MPL = dQ/dL and interpret it. (c) Derive the MRTS = dK/dL = (dQ/dQ)/(dQ/dL) = MPK/MPL. Interpret this. (d) Substitute σ = ∞ in the MRTS of the CES production function: do you get the MRTS of Q = αL + βK? (e) Suppose α = 1/3 β = 2/3. What are the returns to scale for this technology? (f) For what values of α and β will this technology exhibit increasing returns to scale? Decreasing returns to scale? ...
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