That is an excellent question. If I taught economics, I think I would use that example as a class exercise. It is not trivial.
Quick answer is that the guy driving the backhoe will be paid more than the guy with the shovel — if the job is big enough.
This is because a shovel is a much cheaper item than a backhoe, even if you rent the backhoe (I’ve rented a trencher before, it is a similar decision to make). There is some small level of training to use the backhoe, but a competent person could learn this in an hour (again, a trencher is a bit easier, but somewhat comparable and I was able to learn it quickly). But experience makes the job go more quickly.
So, you bid the job for $500. Now, how do we maximize profit? Suppose it is $200 + $50 an hour for the backhoe. You have three laborers available. Person A will shovel, work for $40 an hour, and will take 10 hours. The other two drive a backhoe. Person B has 20 years experience and is paid $200 an hour, and will take 1 hour. Person B has 5 years experience, makes $80 an hour, and will take 2 hours.
So, A will cost $400. B will cost $200 + $50*1 hour + $200*1hour, or $450. C will cost $200 + $50*2 hours + $80*2 hour, or $460.
So in this case, if you could afford a 10-hour job, the shovel is cheaper. If however you have to pay double time for overtime over 8 hours, then A is $40*10 + $40*2, or $480, and the 20-year-experience guy is the cheapest, even though he gets paid the most.
As a child, my brother and I took a job one time digging up a pipe in a guy’s front yard, because the professional wanted $1500. The two of us took 3 full days, and got paid next to nothing, so the friend of the family came out way ahead, and my brother and I made good money.
When I ran my own pipes in my house years later, i rented the trencher, but then my neighbor wanted to play with it so I let him drive it. He was pretty experienced, so it took him almost no time, and charged me nothing, and I got a 150-foot pipe laid for the $300 rental price. And he dug me a couple of trenches to drain the water from my downspouts.
Next time I list all the jobs I did in my life, I’ll have to remember the infamous ditch-digging. On the end of the 3rd day, we were just cleaning up the last part of the hole, I took a big swing with the pick-axe, and put a hole in the supply line, which the “miss utility” people had mismarked as being 10 feet away. On the other hand, the pipe was easy to fix because the water coming cleaned out all the dirt around the pipe. And even with the plumber cost, the homeowner came out way ahead hiring us.
Oh?
Why?
They BOTH remove the same amount of dirt.
Unless there is a time clause in the contract; it would make no difference if one man used a spoon.
Tell me about it!
Last summer I trenched from the house to the barn, about 90'.
I had overhead wires to the barn and no water, and I wanted water and no overhead.
My yard has lots of trees, and the roots REALLY messed me up; both timewise and direct route diversions.
No rain for weeks, so the ground was not mucky clay.
It took me so long, that the placement of the plumbing and the wire (in plastic conduit for extra protection) was postponed 'til the next day.
Yup, it started raining before I got the stuff in the ground and a lot of the dirt got back in the trench, making it too shallow for the plumbing to be freeze proof.
It took to more weeks of jacking around with a homemade plow device to scoop the glommy mess out of the bottom and FINALLY get the job done.
(Next time the wife says, "You SURE you can do that by yourself?", she'll get a different answer!)