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Production function: f(x1, x2) = [min{x1, 2x2 }]^2

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1. Consider a firm that uses two inputs, the quantity used of input 1 is denoted as x1 and the quantity used of input 2 is denoted by x2. The firm produces and sells one good, using the production function: f(x1, x2) = [min{x1, 2x2 }]^2. The prices of each input are w1=$2 and w2=$3. The market for the product is competitive and output price is P=$5.
a) Does the production function exhibit constant, increasing or decreasing returns to scale? Why?
b) Based on your answer to part (a), does the firm have a long-run profit maximizing plan?
c) What is the cheapest way to produce 36 units of output? How much is the cost in this case?
d) What is the firms profit if it manages to sell all 36 units product at the market price?

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This solution answers questions about the production function: f(x1, x2)=[min{x1, 2x2 }]^2. All calculations are shown in detail, and the results are illustrated with an Excel spreadsheet.

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a) The production function is a squared function, meaning that if you double all the inputs, the output will increase by a factor of 4. That means that the function exhibits increasing returns to scale.
b) The firm's long-run ...

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