A firm operating in a perfectly competitive market has the option to use two different technologies to produce a…

A firm operating in a perfectly competitive market has the option to use two different technologies to produce a…

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A firm operating in a perfectly competitive market has the option to use two different
technologies to produce a good. The first technology exhibits constant returns to scale, where the
output (Y) is equal to the product of a productivity measure (A) and the allocated flow of
machine-hours (K). The second technology demonstrates decreasing returns to scale, and the
output (Y₂) is the square root of the allocated flow of machine-hours (K₂). The firm has a total
flow of machine-hours (K) available. The objective is to maximize the sum of Y and Y₂, while
adhering to the resource constraint that the sum of K and K₂ must equal K. (Consider that both
A,K > 0)

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To maximize the sum of Y and Y2 subject to the resource constraint we need to determine the optimal
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