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# Problem Set 3 Answer Key - 11.4 Recall the following...

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11.4 Recall the following assumptions introduced with regard to the pro- duction technology sets for a typical firm j, Y j : P.I. Y j is convex for each j. P.II. 0 ∈ Y j P.III. Y j is closed. P.VI Y j is a bounded set for each j H . Maintaining these assumptions, we can show two properties of Y j : (a) Y j displays no scale economies. If y ∈ Y j then it follows that ( 1 2 ) y ∈ Y j also. (b) Firm j ’s technology Y j is unable to deal with very large inputs (recall that inputs are represented by negative co-ordinates of elements of Y j ). For example, if y ∈ Y j , there is y prime < y (the inequality applies co-ordinatewise) with | y prime | sufficiently large so that y prime / ∈ Y j . Demonstrate properties (a) and (b). Explain what they mean. Suggested Answer : (a) 0 and y ∈ Y j . By P.I (convexity), it follows that any convex combination of 0 and y also ∈ Y j . Then 1 2 0 + 1 2 y = 1 2 y ∈ Y j . Hence there are no scale economies in Y j . Any activity in Y j can be undertaken at any positive proportional reduction in scale. (b) Y j is bounded according to (P.VI). Choose y << 0 (that is, y is strictly negative) and | y | > sup {| x || x ∈ Y j } . Then y / ∈ Y j . Even though y is an abundant input and requires no significant output from firm j , accepting the input y

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Problem Set 3 Answer Key - 11.4 Recall the following...

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