A Followup to Opportunity Cost Puzzle

In Opportunity Cost Puzzle the following appeared --

John owns a retail consumer electronics business.

He sells ten 30 inch plasma TV sets per month. His wholesale price is $2000, his markup is $1000 and his selling price is $3000.

If he takes a set home for use in his den, which of these dollar amounts, if any, is his approximate sacrificed opportunity cost. Explain your reasoning.

As was brought forth in the comments, this is fairly ambiguous and depends on various assumptions. However, I believe that this very ambiguity brought forth a larger variety of considerations and viewpoints than would have resulted from a more concise presentation. The vast majority of the comments had useful viewpoints. Taking them all together has improved my level of comprehensive understanding, without saying that an endpoint has necessarily been reached.

I believe that it may be useful to try to set up the problem in a very concise way below and bring out some of the less obvious implications.

The owner sets his retail unit price at $3000 in an attempt to maximize his rate of profit per month, given his estimation of the unit demand that he faces. His supply of units is assumed to be unlimited at a wholesale price of $2000 each. It is assumed that he has an ongoing business with no end in sight and no change in the assumptions above.

To make things precise, assume that the demand faced is defined by the following equation :

q = 40 - p/100, where q is units sold per month and p is the unit price.

Re-arranging, p = 4000 - 100q

TOTAL REVENUE/MONTH = pq = p X (40 - p/100)

or TREV = 40p - (p^2)/100

TOTAL PROFIT = TREV - 2000q = TREV - 2000 X ( 40 - p/100 )

or TPROF = 40p - (p^2)/100 - 80000 + 20p

or TPROF = -80000 +60p - (p^2)/100

Differentiating and setting to zero, dTPROF/dp = 60 - p/50 = 0

The result of this is p = $3000 to maximize rate of profit

at p = $3000, q = 10 units, TPROF = $10000 (max)

at p = $3100, q = 9 units, TPROF = $9900

at p = $2900, q = 11 units, TPROF = $9900

So the rate of profit is indeed maximum at $3000 each for 10 units/month

The decision to be made is an overall business plan in advance to order 10 units every month, plus 1 additional unit every other month. The 10 units are to be sold new at $3000 each and the other unit is to be sold as a demo at cost after being used for two months at home. It is expected that the single demo unit can be sold to someone who would not buy a new one at a price of $3000. It is not relevant whether the wholesale units are paid for in advance or have been advanced on an extension of supplier credit. Since only 10 units can be sold each month without reducing profits, and since ordering is unlimited, the taking of the unit home fails to preclude any sales or profits, and thus does not have an opportunity cost. If the plan were instead to keep one unit home forever, its $2000 cost would never be recovered, and in this case, no opportunity to use the $2000 given up to acquire goods and services that result in future subjective satisfaction would be possible.

All of the above is true for an indefinitely ongoing business. As soon as no more units can be ordered, the unit at home now has an opportunity cost due to the fact that it cannot be sold as new. Alternately, you can say that it always had an opportunity cost, but that it was indefinitely delayed. Whatever inventory existed when ordering became impossible, the unit at home would have increased it by 1. All of the discussion so far has assigned a $3000 opportunity cost to this case, but $3000 is only the profit maximizing price for a sales rate of 10 per month. If the owner were willing to sell out the remaining invesntory at a rate of 1 per month, the opportunity cost would be $3900, or p = $4000 - 100q for other rates of q sales per month.

EXTRA CREDIT

Use the equations above as a starting point to find a general rule between a profit maximizing price and a revenue maximizing price, as a function of a constant marginal cost, if the unit demand vs price curve is linear.

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Since only 10 units can be

Since only 10 units can be sold each month without reducing profits, and since ordering is unlimited, the taking of the unit home fails to preclude any sales or profits, and thus does not have an opportunity cost.
The fact that it's non-optimal doesn't mean there's no opportunity cost. Taking it home still involves forgone sales. The fact that total profits decline just means the opportunity cost is negative. Zero opportunity cost only arises when a resource has literally and strictly no alternative uses.

Phrased another way, consider the choice John faces in ordering 11 units. He can simply sell them all for a total profit of $9900, or follow the open-box scheme and earn $10000 plus some utility from having the TV. In electing the latter he gives up the former, therefore it has the character of an opportunity cost, even if choosing it would be silly (indeed most opportunity costs would be silly if we chose them).

Put yet another way, the fact that he can order unlimited units from his wholesaler is irrelevant in figuring his opportunity costs. Because demand is finite, there is a constraint upon his actions and he must make choices, i.e. sacrifices. Only if demand were also infinite and transacting TV's consumed no time or other resources would there be no opportunity costs. Of course such a case, besides being ridiculous on its face, is pathological to the standard tools of economic analysis. If MB > MC over all X, then you never stop, and all the normal models break down.

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On a different note, have you considered your implicit assumptions in allowing John to sell the open-box unit for a lower price? Specifically, how does he allow someone who wouldn't buy a new unit to buy the open-box, while disallowing someone who would have bought a new unit? For if a new-unit buyer bought the open-box unit instead, John would have one unsold new unit at the end of the month, or be forced into attempting yet another round of (likely unsuccesful) price discrimination.

Noah, Phrased another way,

Noah,

Phrased another way, consider the choice John faces in ordering 11 units. He can simply sell them all for a total profit of $9900, or follow the open-box scheme and earn $10000 plus some utility from having the TV. In electing the latter he gives up the former, therefore it has the character of an opportunity cost, even if choosing it would be silly (indeed most opportunity costs would be silly if we chose them).

Selling all 11 units would violate the definition of an opportunity cost which MUST be the best or highest ranked precluded alternative, or at least the best of the known alternatives.

Put yet another way, the fact that he can order unlimited units from his wholesaler is irrelevant in figuring his opportunity costs. Because demand is finite, there is a constraint upon his actions and he must make choices, i.e. sacrifices.

The fact of finite demand already precludes selling 11 sets instead of 10 because selling 10 ranks higher.

Taking a vacation cruise to Miami doesn't have an opportunity cost of a precluded vacation cruise to Anchorage if it is already precluded by the fact that no such cruise is scheduled.

On a different note, have you considered your implicit assumptions in allowing John to sell the open-box unit for a lower price? Specifically, how does he allow someone who wouldn’t buy a new unit to buy the open-box, while disallowing someone who would have bought a new unit? For if a new-unit buyer bought the open-box unit instead, John would have one unsold new unit at the end of the month, or be forced into attempting yet another round of (likely unsuccesful) price discrimination.

Price discrimination is successfully practiced all the time. You can offer coupons to a limited population, or have a drawing or a contest for the right to buy the discounted set.

Thanks, Don

Selling all 11 units would

Selling all 11 units would violate the definition of an opportunity cost which MUST be the best or highest ranked precluded alternative, or at least the best of the known alternatives.

I don't see how that definition is violated. If you lay out a $5 bill and a $10 bill on the table and tell me I can pick one, the opportunity cost of picking the $5 is picking the $10, and the O.C. of picking the $10 is picking the $5. In the former case it doesn't matter that the choice is irrational, the O.C. is still there.

Electing to pursue an 11-unit open-box scheme precludes pursuing an 11-unit straight-sale scheme, which as far as we know is the highest valued alternative.

The fact of finite demand already precludes selling 11 sets instead of 10 because selling 10 ranks higher.

Wrong. See above.

Taking a vacation cruise to Miami doesn’t have an opportunity cost of a precluded vacation cruise to Anchorage if it is already precluded by the fact that no such cruise is scheduled.

Non sequitur. Selling 11 sets (and thus making sub-maximum profits) is an available option. It would be more analogous to talk about where you go with a cruise ship you own yourself.

Price discrimination is successfully practiced all the time. You can offer coupons to a limited population, or have a drawing or a contest for the right to buy the discounted set.

Of course, but your scheme might fail. Nothing prevents new-unit buyers from entering the contest or clipping a coupon.

I realize this is supposed to be a simple model, but that just means you need to explicitly assume John's ability to price-discriminate. Otherwise, there's some probability that he'll end up with sub-optimal sales, which reduces the expected value of pursuing the scheme (conceptually of course, obviously he doesn't know what that probability is).

Noah, Electing to pursue an

Noah,

Electing to pursue an 11-unit open-box scheme precludes pursuing an 11-unit straight-sale scheme, which as far as we know is the highest valued alternative.

No, we know that a 10-unit straight-sale scheme is ranked higher.

I don't think that we would disagree that if we take the three comprehensive choices, they would be ranked in the following order :

1. Buy 11, sell 10, take one home for 2 months before recovering cost.

2. Buy 10, sell 10.

3. Buy 11, sell 11.

This is the sense in which 3) can't be the opportunity cost for 1).

However, we are really interested in the 'take home' subcomponent choice of 1). Choosing it doesn't preclude the buy 10, sell 10 best alternative, but it subsumes it in the comnprehensive choice.

If I choose to go to the county fair instead of the seashore, then the opportunity cost of riding the ferris wheel may be the preclusion of riding the roller coaster.
But if the fair doesn't have a roller coaster, then riding the ferris wheel has no opportunity cost, at least within the limited definition of the simplified problem.
I think that this parallels the 'take set home' choice.

Of course, but your scheme might fail. Nothing prevents new-unit buyers from entering the contest or clipping a coupon.

But the owner has total control of how he distributes the coupons or sets up the contest. He can minimize the risk almost as much as he wants. If his store is in Buffalo, NY, he can offer the discount set over the phone to people randomly selected from the London (UK) phone book.

Regards, Don