A Gasoline Selling Problem

You are the owner of Joe's Desert Service, the only gas station for 150 miles in any direction in the Southwest US.

Under normal conditions, you charge $2.50 per gallon of regular.

Your full service attendent is an illegal alien who works 40 hours a week for the minimum wage. Your payroll and income tax arrangements are known only to the two of you.

You mutually agree to change from a wage payment to a commission payment of 4 cents per gallon pumped.

1. If the total take home pay falls by 10%, what is the new price per gallon charged for regular from the list below?

2. Same question, but the total take home pay increases by 10%?

a. $2.75
b. $2.58
c. $2.54
d. $2.52
e. $2.50
f. $2.48
g. $2.46
h. $2.42
i. $2.25

Why?

Added HINT - Assume profit maximizing pricing behavior.

end

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In both cases it stays at

In both cases it stays at $2.50, because the price you charge for gas is market driven, not cost driven. A change in your labor cost structure doesn't affect the demand for gas.

But his example posits that

But his example posits that there is no market - this guy has a local monopoly. And while monopolies certainly can set a profit maximizing price, in practice there are reasons not to - fear of government action, fear of armed people desperate for gas and with a serious sense of grievance, "shame" at "gouging," and perhaps more likely , fear of attracting competition. Obviously this guy is NOT maximizing profits, because the only gas in the middle of the deser is worth easily twice what he's charging.

e. $2.50

e. $2.50

Dylan, no, because the price

Dylan, no, because the price will already reflect that stuff.

Sorry, that wasn't clear --

Sorry, that wasn't clear -- what I mean is that yes, he is maximizing profits within his percieved constraints. If people will riot and steal my product if I set the proce above $X, then $X still maximizes my profits within those constraints.

Hint added. $2.50 a gallon

Hint added. $2.50 a gallon may be a profit maximizing price in a low income market, for example.

Regards, Don

given that the illegal

given that the illegal immigrant is probably one of the main customers at Joe's Desert Service, and his income will affect how much gas he can afford/waste, if his salary goes up, gas prices go up, if his salary goes down, gas prices go down.

Your question is vague to

Your question is vague to the point of meaninglessness.
If you assume that the number of gallons sold is independent
of whether the employee works on a fixed cost or a comission,
then you should charge more if you're paying the employee
on comission (although you can't say how much more without
knowing how elastic the market for gas is).

But that's a boneheaded assumption, because the whole point
of paying on comission is that it gives the guy an incentive
to try harder to sell more. Although in this particular case
it's hard to see what the guy's going to do.

One thing I can tell you: if, after making this deal, you raise your
prices and your profits go up while your employee's wages go down,
he will feel ripped off.

George, But that’s a

George,

But that’s a boneheaded assumption, because the whole point
of paying on comission is that it gives the guy an incentive
to try harder to sell more. Although in this particular case
it’s hard to see what the guy’s going to do.

You are correct. The word commission was misleadingly used when only a technical description of the pay arrangement vs gallons sold was intended.

Regards, Don

Ok, let me see if I can

Ok, let me see if I can clarify what you're asking:
The guy used to be making minimum wage, which is 5.15 an hour, so at 40 hours a week that's 206.

Now after the wage change he's making 10% less, which is 185.40, right? So at 4 cents a gallon he's selling 4630 gallons.

But we don't know how many galllons he was selling at the old price. If we assume that at the old price the change in payment structure wouldn't have affected take home pay (not stated but I think it may have been implied), then at the old price 5150 gallons were sold. It stands to reason that the price went up if gallons sold went down, but I don't see how you can say by how much, without additional assumptions. Presumably the owner is making more after than before (otherwise why bother changing price), but I don't see how you can say how much. I think if you assume a linear relationship between gallons sold and price, and assume the owner was maximizing profits at both prices, you can come up with an answer, but I don't see how you can justify such an assumption.

And the second case seems even weirder. Maybe I'm missing something,
but it seems "obvious" to me that since you're trying to maximize profits per gallon * gallons sold, then the change in payment structure should always make the owner want to raise prices, since by changing a fixed cost into a variable one he's decreasing the profit per gallon unless he raises the price. Unless you mean to assume he's maximizing profit after the price change but wasn't before, which seems like a really weird assumption, and also doesn't seem to give enough info to get an answer.

George, And the second case

George,

And the second case seems even weirder. Maybe I’m missing something,
but it seems “obvious” to me that since you’re trying to maximize profits per gallon * gallons sold, then the change in payment structure should always make the owner want to raise prices, since by changing a fixed cost into a variable one he’s decreasing the profit per gallon unless he raises the price. Unless you mean to assume he’s maximizing profit after the price change but wasn’t before, which seems like a really weird assumption, and also doesn’t seem to give enough info to get an answer.

You've got a number of correct assertions, but you are correct that there isn't anywhere near enough information to use numbers to find the answer. I think that the key to the answer is buried in your post.

The assumption is that you are pricing to maximize total profits for a given projected market demand structure both before and after the change in the compensation method, whether or not the take home pay goes up or down. There is no assumption about the change in compensation method being better for either you or the employee. The only number that is needed to give an answer is the 4 cent per gallon 'commission.'

Thanks, Don

OK, one last try: If we

OK, one last try:

If we don't know how the change in salary structure
would have changed wages if the price had stayed constant,
then the bit about the worker's take home pay is
completely irrelevant, since the owner is indifferent.

If we assume a profit maximizing price before and after, since the owner's variable costs have gone up, he will want to increase the price, but not by the full 4 cents since increasing the price will decrease the number of gallons sold. So the answer is between 2.50 and 2.54, so pick 2.52, not because there's any reason to split the difference evenly, but because it's the only answer within the range among your list of choices. Is that what you were looking for?

George, If we don’t know

George,

If we don’t know how the change in salary structure
would have changed wages if the price had stayed constant,
then the bit about the worker’s take home pay is
completely irrelevant, since the owner is indifferent.

Exactly (almost). It is not primarily that the owner is indifferent, but that the profit optimizing price at the time and place of sale is only a function of the perceived market demand structure and the marginal (or variable) costs.

If we assume a profit maximizing price before and after, since the owner’s variable costs have gone up, he will want to increase the price, but not by the full 4 cents since increasing the price will decrease the number of gallons sold. So the answer is between 2.50 and 2.54, so pick 2.52, not because there’s any reason to split the difference evenly, but because it’s the only answer within the range among your list of choices. Is that what you were looking for?

Yes, except that there is a reason to split the difference exactly, at least for an artificial problem.

The profit maximizing price for a monopoly supplier with zero marginal costs is also the revenue maximizing price.

If the marginal costs become positive, the profit maximizing price must rise.

In this situation, setting a profit maximizing price is the same thing as selling those units, and only those units whose marginal revenue exceeds their marginal cost. The unit price and the quantity sold are connected by the market demand structure.

If we start at the revenue peak, selling one less unit by raising the unit price will reduce the cost by the marginal unit cost, but the revenue will hardly fall at all for small enough marginal units. Thus, the price must be increased enough to drive the revenue down its slope so that it can catch up with the marginal cost reductions.
The last unit sold will be the last one whose marginal revenue still exceeds its marginal cost.

As a result from the literature (at least one), the profit maximizing price exceeds the revenue maximizing price by 1/2 the marginal cost if the demand relationship between price and quantity is linear in the region of prices above the revenue peak. It should not be too difficult to prove (or disprove) this.

Since pricing for maximum profit requires what is at best an educated guess about the market demand structure, it would be hard to justify anything more complex than a linear relationship between price and demand in the region of interest.

It is in this sense that an increase of 4 cents in marginal cost should produce an increase of 2 cents in the profit maximizing price. Profit maximizing is a goal, not an achievement.

Thanks for your fruitful effort, Don