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Long, long, ago in a galaxy far, far, away, there existed an empire which spanned many solar systems. The majority of these planets are inhabitable to some extent, and we're populated by humans during the great exodus from Earth. However, this empire is somewhat constrained by the absence of wormhole or hyperdrive technology.

Industries which specialized in the transport of much needed equipment and personnel is the only way to get supplies from planet to planet. Ships are mounted with warp drives, specially built drives that can speed up travel in space. Crew members are put in suspended animation for the journey and are defrosted when reaching the location. Then, supplies are unloaded and payment is collected. However, the journey can take decades or even centuries before reaching the final destination.

How can an industry built on transporting equipment operate under these circumstances?

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    $\begingroup$ Try reading Larry Niven's Known Space novels and short stories, especially the ones set before the Man-Kzin Wars. They should prove very useful. $\endgroup$ – The Daleks Jun 1 at 1:18
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Your empire's main constraint seems to be the large lag time. This is obviously problematic because it makes it very difficult for anyone to order anything for delivery since conditions may have changed significantly by the time the item is received (if the purchaser is still alive, which they likely won't be).

Your economic system, therefore, revolves around a small-ish set of standardized materials that are traded and that are known to be highly useful and versatile throughout the Empire, and unlikely to become obsolete. People don't stop needing wood and D4-pills just because a few decades have passed. Last year's iPhone is already obsolete. Each individual planet (or star system) is required to more or less have the capacity to manufacture pretty much anything it needs as long as it gets its regular shipments of whatever raw materials don't occur in abundance there. Forget about the Aldebaran II telecom industry introducing a new QuasarPhone 9 and hoping to sell it to up-and-coming residents of Sirius X. Aldebaranians know that they have a surplus of latinum and lack significant quantities of vespine gas, so whenever a century-ship arrives, they buy up lots of gas and sell as much latinum as will fit in.

If information can travel significantly faster (and, if your ships are 1x lightspeed, it probably can't), then you can have businesses sell blueprints off-world to local manufacturers (which could be received quickly) rather than shipping physical product (which could take centuries).

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    $\begingroup$ Plus one for recognizing that inter-galactic trade would never be in goods or materials. It just doesn't make economic sense to ship material between star systems, when each star system would be self-sufficient in any materials. Planets are big. Why import materials that can far more easily be mined, and fabricated in-system? Intergalactic trade would be in information and intellectual property rights, and in royalties. Someone invents a better mouse trap, and licences the manufacturing rights. $\endgroup$ – Justin Thyme the Second Jun 1 at 3:45
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The industry probably was able to transport materials because they already started the journey ten years before to make up for the time delay. The industry could also try selling the things it needs to sell to places closer than were they want to sell to make additional profit in the short term too. Also maybe time dilation probably made 10 years seem like a few years only.

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Provided technical knowledge is evenly spread:

It can't.

Given a need for equipment or personnel, it is always faster to build your own than to send off for it, given the time involved.

Or can it?

What can work is a steady stream of deliveries. If you receive an annual delivery of sufficient equipment to meet the years need, there is no reason to build your own. For this to work, a number of planets different production capabilities are needed. For instance, one does mechanical contrivances, another makes better and better livestock, a third better and better grains, and a fourth might do medicines.

Additional cargo

Every ship would also probably carry something akin to a technical version of wikipedia, for knowledge sharing purposes.

Ships would carry people who wished emigrate to somewhere.

Ships would carry reproductions of art and literature. These might also be sent by radio or laser.

Some ships might do long haul, to transport knowledge, seeds, and some livestock long distances. This would be faster than planet to planet spread as it eliminates the stop-and-trade delays.

Other thoughts

You will need well established and stable interstellar law. Planets deciding to institute a tariff is a real problem.

One thing that might not work is any sort of interstellar currency. So how do the ships get profit? Simple: somebody retires at each stop, and gets the profit. New crewmen are hired regularly. Though round-trip flights could accumulate wealth at the endpoint planets.

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