TL/DR
There isn't one, or at least not a good one. Slavery has built in inefficiencies and brutalizing effects for both slave and free populations.
What are our goals and values?
There are a large number of competing goals and values that people could have. In this answer, I'm assuming the following goals or values.
- Rich people want to stay rich, and they want their children to be rich.
- The laboring class is part of society.
- Generally speaking, people prefer happiness to misery.
- People prefer a stable society to an unstable society.
However, the arguments I'm making really only require that we accept any one of these premises. Slavery is a bad idea whether you accept all four, or any given one, or really, any premise that I can imagine beyond being having a fetish for holding slaves, and even in that case you're probably better off renting free labor to pretend to be slaves.
The economic case
Slavery is a system in which some people in society are forced to do specific work at the command of other people, and have no freedom to migrate within the society or to seek other work. In a slave system, laborers must be strongly and strictly controlled. Slave economies combine inequality with a command market for labor, while turning the entire economy into a prison economy.
A command economy for labor is not efficient. The actual demand for labor is controlled by market forces, but in a slave society, what labor people will do and who they will do it for is strictly controlled. While it is true that slave labor can be rented to where need is greatest, such a system is rife with cronyism and other problems. Self-organized labor is able to follow demand more closely, price itself more appropriately, and adapt to circumstances more easily.
If there is a goal of getting work done, a command economy for labor is objectively worse at accomplishing that goal than a market economy for that same labor.
A prison economy has additional problems. Not only does a prison economy inherit the problems of a command economy, it also necessitates additional costs. For example, those in charge must provide food, clothing, and shelter. They also have to pay a security force to prevent the prisoners from seeking freedom.
We can see these problems when we examine modern slavery in the United States. Convict labor depends heavily on state subsidies to cover these extra costs and to provide the illusion of cheap labor. This illusion has the effect of depressing wages for certain work in the non-prison labor force, while simultaneously forcing costs on that labor force.†
If there is a goal is to get work done, a free market for labor is objectively more efficient for society than a prison market for labor.
Increasing the corporate bottom line at the expense of everything else is also a losing strategy long term for the profits of the corporation.††
Finally, a slave society also suffers from the economic problem of extreme economic inequality. It is an objective fact that societies with extreme economic inequity are less stable economically and politically than societies with less inequality.†††† It is also simply bad for the market. Slaves are extremely limited in their opportunity to contribute to the economy by trading goods and services in an open market. In contrast to paid labor, slave labor is able to contribute much less to overall economic activity.†††
Societies with extreme inequality are also prone to large scale violence. As an example, the richest slaveholder in Haiti in 1790 was a corpse in 1804. A stable economy is a better economy to be rich in, if the goal is to stay rich.
Strictly practically speaking, the only upside to slave labor is marginally reduced labor costs, and those reduced costs come with enormous additional costs that are forced on the non-slaveholding class, both free and slave, by the slave-holding class. This inequality is bad for the market, bad for profits, dangerous to the slave-holding class, and inefficient. It serves none of our four goals.
The social case
From a sociological standpoint, a slave society brutalizes each and every member. A slave society is an open air prison. Any non-slave in a slave society has to be actively complicit in supporting the system by acting as a guard, for example by looking for and turning in slaves attempting to escape the prison. So in addition to having their wages depressed and opportunities for work reduced, the labor class in a slave society is compelled to act as a bounty hunter in their spare time. In addition, such a society lives under the constant threat that the slaves will attempt to free themselves by violence. This extra stress is not good for the health of the non-slaves.
The slaves, of course, live under even more stress than the non-slaves and are brutalized to a much greater degree. Every slave lives under the constant threat of violence, is constrained in their movements, frustrated in their ambition, insecure in their person and property. If we look at the overall psychological health of the slave population, we're going to see significant negative effects rising from this condition. As they are members of the society, the sociological negatives are obvious.
So if we have an abstract goal such as happiness for any member of the society, a slave society has extreme stresses that will inevitably detract from that happiness. If we have a goal of a society where more people enjoy greater phsyical health, the stressors of a slave society are detrimental to that. These stressors are largely missing from more egalitarian societies.
When we examine how happy people are, people in more egalitarian societies are happier than those in less egalitarian societies. This applies to those who are well off and those who are not particularly well off.
Slavery has large economic and social costs and few benefits. In short, slavery is just a bad way to organize a society.
† https://legaljournal.princeton.edu/the-economic-impact-of-prison-labor-for-incarcerated-individuals-and-taxpayers/
†† https://hbr.org/2009/12/why-profit-shouldnt-be-your-top-goal
††† For a brief review of how economists have thought about slavery in the ante-bellum south, see this article: https://www.econlib.org/library/enc/usslaveryandeconomicthought.html
†††† https://en.wikipedia.org/wiki/Effects_of_economic_inequality