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The statement I'm confused about says the following:

The best surplus we can get from trading $n$ objects equals $$\sum_{k=1}^n (v_k - c_k) = V(n) - C(n)$$

Any efficient trading volume thus maximizes $TS(n) = V(n) - C(n)$. The smallest such maximizer equals $n^*$ and the largest such maximizer equals $n^{**}$. Thus $TS(n)$ is a constant between $n^*$ and $n^{**}$.

If anyone has an economics background and can possibly explain what the $n^*$ and $n^{**}$ means, that would be great, however anyone's input would be appreciated.

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up vote 2 down vote accepted

It is common in economics to denote the optimal value of a variable by using a superscript star. Thus, $n^*$ should be a choice for $n$ which makes $TS(n)$ as large as possible. Sometimes there isn't a single best choice, though, and there could be several different choices for the variable $n$ which provides the same maximal value $TS(n)$. Convention isn't unanimous on what notation to use, but if you have two different values of $n$ which both give the same maximal value $TS(n)$, it makes sense to call one $n^*$ and the other $n^{**}$.

In your question, $n^*$ is supposed to be the smallest maximizer and $n^{**}$ is the largest maximizer, and it is stated that $TS(n)$ is constant between $n^*$ and $n^{**}$. For this to make sense, your objects should be ordered according to their contribution to the total surplus, or the value of $v_k-c_k$, in decreasing order. Then $n^*$ would be the number of objects such that $v_k-c_k>0$, and $n^{**}-n^*$ is the number of objects such that $v_k-c_k=0$. If there are no objects with exactly zero surplus then you will have $n^*=n^{**}$. You would of course want to trade all objects with positive surplus, and you are indifferent to trading objects with zero surplus.

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