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W14 MIC Problem Set 5 Solutions

# As before assume an interior optimum a applying

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Unformatted text preview: n C[ w, r, Q ] = 2 Qwr − w , then deriving the input demand equations and finally the production function. As before, assume an interior optimum. a) Applying Shephard’s Lemma to the cost function, find the factor demand equations, L[w,r,Q] and K[w,r,Q]. b) Using the two factor demand equations, show that the corresponding production function is Q = KL + K. Hint: Try to eliminate both w and r at the same time. Answer: 5) a) Shephard’s Lemma involves taking the partial derivatives of the long run total cost function with respect to the factor prices: ∂C ∂ # 1 Qr = 2 Qwr − w% = 2 Qr w −1/2 − 1 = − 1 = L[ w, r, Q ] & ∂w ∂w \$ 2 w ∂C ∂ # 1 Qw = 2 Qwr − w% = 2 Qw r −1/2 = = K [ w, r, Q ] & ∂r ∂r \$ 2 r b) Our goal is to eliminate the factor prices, leaving only Q[L,K]: Qr L +1 r L= −1 ⇒ = w w Q K= Qw K w Q r ⇒ = ⇒ = r r K w Q L +1 Q = ⇒ Q = K ( L + 1) = KL + K K Q...
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