Econ100B PS1

# Econ100B PS1 - 3 Suppose that the production function q = L...

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ECO 100B Problem Set 1 Answers will be posted on April 11, 2008 1. For each of the following, say whether the production technology satisﬁes (i) Diminishing MRTS, (ii) Diminishing Marginal Product in each input, and (iii) Constant Returns to Scale (assume throughout that L,K > 1): (a) f ( L,K ) = L + K (b) f ( L,K ) = LK (c) f ( L,K ) = ( LK ) 2 (d) f ( L,K ) = LK 1 2 . (e) f ( L,K ) = 10 KL K + L 2. Suppose that two ﬁrms, Firms 1 and 2, use the same technology but Firm 2 is only 75% as productive as Firm 1: for Firm 1, q 1 = f ( L,K ) and for ﬁrm 2, q 2 = 3 4 f ( L,K ). At a particular level of inputs, L and K , how does the Marginal Product of Labor diﬀer across the ﬁrms? How does the Marginal Rate of Technical Substitution diﬀer?
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Unformatted text preview: 3. Suppose that the production function q = L 1 4 K 3 4 . When k = 1 and q = 2, what are the MP L and the MRTS ? 4. Suppose that a production function, f , is homogenous of degree γ , i.e., f ( λL,λK ) = λ γ f ( L,K ). (a) Under what conditions does such a production function have constant returns to scale? (b) Prove that MP L is homogenous of degree γ-1. (c) Prove that L df dL + K df dK = γf ( L,K ). 5. For what values of a and b will the production function f ( L,K ) = ( L a + K a ) 1 b for a , b > exhibit increasing, constant, or decreasing returns to scale? 1...
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