Paying $n$ people such that the ratio of their pays matches their quality ratio and subject to minimum wage constraints

Question

You are trying to pay $n$ people. Each person $i \in \{1, \ldots, n\}$ has an expected minimum wage $w_i > 0$ and a quality $q_i > 0$. Denote the payment for each person as $p_i$.

We require:

(1) Each person must be paid at least their expected minimum wage.

(2) For each pair of people, the ratio of their payments must be equal to the ratio of their qualities, i.e., $\frac{p_i}{p_j} = \frac{q_i}{q_j} \ \ \forall i \neq j \in\{1,2\ldots, n\} $.

We want to minimize $\sum_i p_i$ subject to the above 2 questions.

My question is do these conditions ensure that at least 1 person will be paid their minimum wage? And if so, how do we determine who that person/people is/are?

Answer 1

Let $\alpha$ be the maximum value for $\frac{w_i}{q_i}.$ Then pay person $i$:

$$p_i=\alpha q_{i}.$$

Note that since $\frac{w_i}{q_i}\leq \alpha,$ we have $p_i\geq w_i.$

Basically, if $\frac{w_1}{q_1}=\alpha,$ you have to pay person $1$ at least $p_1\geq w_1,$ and all other values are decided by that one value:

$$p_i=\frac{q_i}{q_1}p_1\geq \frac{q_i}{q_1}w_1$$

So any value of $p_1>w_1$ yields a larger sum $\sum p_i.$

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You can think of this more clearly if you think about $q_i$ instead as units of work. If person $i$ does $q_i$ units of work, the condition $\frac{p_i}{p_j}=\frac{q_i}{q_j}$ is the statement that person $i$ is paid the same amount per unit of work as person $j.$ That is, the price per unit of work must be uniform.

The maximum value $\frac{w_i}{q_i}$ is thus the minimum possible uniform price for a unit of work that can possibly be paid.

Answer 2

Suppose otherwise that the minimal solution $(p_1,\cdots,p_n)$ pays every person above their expected minimum wage. Then you can find an $0<\epsilon<1$ (pick $\epsilon$ very close to 1) such that $(\epsilon p_1,\cdots,\epsilon p_n)$ also pays everyone their minimum wage. Since the constraint $p_i/p_j=q_i/q_j$ is scale invariant for $p$, this new solution satisfies all constraints and gives a lower sum $\sum_i^n p_i$. Hence the minimum solution pays at least one person their expected minimum wage.