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Unformatted text preview: ECONOMICS 201
SPRING 2009 EXAM 2 instructions: Please mark your answers clearly. No class notes or other materials are allowed. 100 points total. Each question is worth
5 points with the exception onuestion 3 (10 points), Question 6 (IS points), Question 13 (10 points), and Question 14 (15 points). i. Characterize the returns to scale (decreasing, constant, or increasing) associated with the following
production functions. a. y = f(X],X2) = 2Xl +33“ b. y =ftX..X2) = minizXn3X2} 2. True or false: When the ﬁrm solves the PMP, and the resulting proﬁt is greater than zero and less than
inﬁnity, it must be that returns to scale are decreasing. Explain. 3. Zeus makes planets using earth (input 1), water (input 2), and ﬁre (input 3). The input prices of earth,
water, and ﬁre are wl , w2 , and W3 respectively. His production function is y = f(X1,X2,X3) = XﬁXng. (10 points) a. What is Zeus” CMP (cost minimization problem)? b. What is the Lagrangian (use p as the multiplier)? What are the ﬁrst—order conditions (FOCs)? c. Use the FOCs to show that Zeus chooses inputs such that MR???” =Input Price Rariou. 4. True or false: Suppose that a ﬁrm solves the CMP and uses lechIIOIOglr"f(Xl ,X2) = aX] + sz (perfect
substitutes). Ifthe price of an input rises, the firm necessarily uses less of it. 5. True or false: Suppose that a ﬁrm solves the CMP and uses technology/(XI ,X2) = minlnXl ,le}
(perfect complements}. If the price of an input rises. the ﬁrm necessarily uses less of it. 6. Dora the Explorer produces two outputs, fun (output I) and leaming (output 2). using one input,X. The
output price offun is PI . and the output price oflearning is P2 . The input price oins w. Dora produces fun using the. production functionyI = (X; l , Where X1 is the amount of the input allocated to output 1.
Analogously. she produces learning using the production functiony2 = [2(X2) , where X2 is the amount
of the input allocated to output 2 (15 pointst a. What is Dora the Explorer‘s PMP (protit maximization problem)? b. What are the ﬁrst-order conditions (with respect to X] and X 2 )'? c. Use the FOCs to demonstrate that when Dora produces both outputs, she does So such that the
marginal amount ol‘input used in the production of fun yields as much revenue as the marginal amount of input used in the production ofleaming. 7. True or false: Ifthere are no ﬁxed costs, it is not possible for the ﬁrm to make negative profits. Explain. ...
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- Fall '08