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Your question is somewhat wrong in the sense that a firm needn't cover its unavoidable costs (sunk costs). If there are fixed costs that are somewhat avoidable or recoverable (for instance selling an old machine for a salvage value), we need to cover them. Now coming back to the main question. We can express the profit of a firm using the following equation ...

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$W(4) = \max\left\{4+bW(4),\frac{1}{2}\left(4 - k + bW(4)\right) + \frac{1}{2}\left(16 - k + bW(16)\right)\right\}$ $W(16) = \max\left\{16+bW(16),\frac{1}{2}\left(4 - k + bW(4)\right) + \frac{1}{2}\left(16 - k + bW(16)\right)\right\} = 16+bW(16)$ First solve for $W(16)$ to get $W(16) = \frac{16}{1-b}$. Then substitute it in $W(4)$ to solve for $W(4)$ as a ...

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This is some dynamic supply-demand model (I am not aware of it having some special name). The first equation gives you the evolution of prices. It says that there will be inflation if there is excess demand $d_t>s_t$ and deflation if there is excess supply (that’s why the first equation has ($d_t-s_t$). The second equation tells you how supply changes ...

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