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Given Information Budget: 20 , 000   birr 20,000birr Wage rate: 𝑤 = 1 , 000   birr per worker w=1,000birr per worker Price of implements: 𝑟 = 4 , 000   birr per unit r=4,000birr per unit Production function: 𝑄 = 10 𝐿 𝐾 0.5 Q=10LK 0.5 a) Marginal Product Functions The marginal product of labor ( 𝑀 𝑃 𝐿 MP L ) and the marginal product of capital ( 𝑀 𝑃 𝐾 MP K ) are derived by taking partial derivatives of 𝑄 Q with respect to 𝐿 L and 𝐾 K, respectively: Marginal Product of Labor ( 𝑀 𝑃 𝐿 MP L ): 𝑀 𝑃 𝐿 = ∂ 𝑄 ∂ 𝐿 = 10 𝐾 0.5 MP L = ∂L ∂Q =10K 0.5 Marginal Product of Capital ( 𝑀 𝑃 𝐾 MP K ): 𝑀 𝑃 𝐾 = ∂ 𝑄 ∂ 𝐾 = 10 𝐿 ⋅ 1 2 𝐾 − 0.5 = 5 𝐿 𝐾 − 0.5 MP K = ∂K ∂Q =10L⋅ 2 1 K −0.5 =5LK −0.5 b) MRTS The marginal rate of technical substitution ( 𝑀 𝑅 𝑇 𝑆 MRTS) measures how much of one input can be reduced when the other input increases, keeping output constant. It is calculated as: 𝑀 𝑅 𝑇 𝑆 𝐿 𝐾 = 𝑀 𝑃 𝐿 𝑀 𝑃 𝐾 MRTS LK = MP K MP L 𝑀 𝑅 𝑇 𝑆 𝐾 𝐿 = 𝑀 𝑃 𝐾 𝑀 𝑃 𝐿 MRTS KL = MP L MP K 𝑀 𝑅 𝑇 𝑆 𝐿 𝐾 MRTS LK : 𝑀 𝑅 𝑇 𝑆 𝐿 𝐾 = 10 𝐾 0.5 5 𝐿 𝐾 − 0.5 = 2 𝐾 𝐿 MRTS LK = 5LK −0.5 10K 0.5 =2 L K 𝑀 𝑅 𝑇 𝑆 𝐾 𝐿 MRTS KL : 𝑀 𝑅 𝑇 𝑆 𝐾 𝐿 = 5 𝐿 𝐾 − 0.5 10 𝐾 0.5 = 𝐿 2 𝐾 MRTS KL = 10K 0.5 5LK −0.5 = 2K L c) Optimal Combination of 𝐿 L and 𝐾 K The firm maximizes production subject to the budget constraint: 𝑤 𝐿 + 𝑟 𝐾 = 20 , 000 wL+rK=20,000 1 , 000 𝐿 + 4 , 000 𝐾 = 20 , 000 1,000L+4,000K=20,000 Divide through by 1 , 000 1,000: 𝐿 + 4 𝐾 = 20 L+4K=20 𝐿 = 20 − 4 𝐾 L=20−4K At the optimal combination, the MRTS must equal the ratio of input prices: 𝑀 𝑅 𝑇 𝑆 𝐿 𝐾 = 𝑤 𝑟 MRTS LK = r w 2 𝐾 𝐿 = 1 , 000 4 , 000 = 0.25 2 L K = 4,000 1,000 =0.25 𝐾 𝐿 = 0.125 L K =0.125 𝐾 = 0.125 𝐿 K=0.125L Substitute 𝐾 = 0.125 𝐿 K=0.125L into the budget constraint: 𝐿 + 4 ( 0.125 𝐿 ) = 20 L+4(0.125L)=20 𝐿 + 0.5 𝐿 = 20 L+0.5L=20 1.5 𝐿 = 20 1.5L=20 𝐿 = 20 1.5 = 13.33 ≈ 13 L= 1.5 20 =13.33≈13 Using 𝐾 = 0.125 𝐿 K=0.125L: 𝐾 = 0.125 × 13.33 = 1.67 ≈ 2 K=0.125×13.33=1.67≈2 Thus, the enterprise should hire 13 workers and buy 2 units of implements. d) Maximum Number of Armchairs Produced Substitute 𝐿 = 13 L=13 and 𝐾 = 2 K=2 into the production function: 𝑄 = 10 𝐿 𝐾 0.5 Q=10LK 0.5 𝑄 = 10 × 13 × 2 0.5 Q=10×13×2 0.5 𝑄 = 10 × 13 × 2 Q=10×13× 2 𝑄 = 10 × 13 × 1.414 ≈ 184 Q=10×13×1.414≈184 Thus, the enterprise will produce approximately 184 armchairs. Check the answer please