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2

In the short run, you cannot sell your capital. To do so would be a violation of "short run" and would instead be the "long run". Case 3 (and I would also contest Case 2) are outright forbidden by the definition of short run. I would note that your question has a lot of narrative to it - which is good in some cases, but you want to keep ...


3

The production function has a particular feature: the inputs are perfectly substitutable. One unit of input 1 can be substituted by 2/3 of input 2 to produce the same quantity of output. Intuitively, a producer would optimally use only one output to produce. Suppose the production plan is $(z_1,z_2)$. By choosing $(z_1-1,z_2+2/3)$, the firm produces the same ...


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