28 Apr 2016

A Possible Flaw in the Way Economists Typically Solve Problems

Economics 26 Comments

I was helping some students study for an exam–the thing I was best at in my life was taking tests, so I always find ways to go back to that activity–and I encountered what may be a subtle inconsistency in the way economists typically solve these types of problems. But, before I decide whether I’ve actually stumbled on something worth pursuing, I first want to make sure other professional economists went down the same path I did.

So, in the comments, if you want to participate please tell me whether you have a degree (and what level) in economics, and then what you think the answers are to the questions.


There are 1,000 firms. Each firm must decide whether to produce corn or wheat. For firm i, the production function for corn is given by:

C_i = 2L_i^(2/3)

where _ denotes subscript and ^ denotes exponentiation. L_i is the number of units of labor hired by firm i.

For wheat the production function is:

W_i = ( 160L_i^(1/2) ) / n

where n is the number of firms who have chosen to go into wheat production.

The market prices for corn and wheat output are $3 and $5 per unit, the wage rate for labor is $4, and the interest rate is 10% per year. Each firm is a price taker in whatever line it goes into.

==> Calculate the profit-maximizing levels of output for a corn- and a wheat-producing firm, respectively.

==> In equilibrium, how many of the 1,000 firms will go into corn and how many into wheat production?

26 Responses to “A Possible Flaw in the Way Economists Typically Solve Problems”

  1. Major.Freedom says:


    The fundamental problem with these functions is the ambiquity if not meaninglessness of the units of measure. The units are incommensurable and cannot coherently be equated, multiplied or divided to “solve for the unknowns”. The “units” of labor, and “units” of corn and “units” of wheat are not commensurable. They can only be ranked.

    I’ll leave it to the technical experts to solve the problem the way we were taught.

    • Bob Murphy says:

      MF I agree that economists are notorious for ignoring units, but that’s just sloppiness, not a conceptual problem, right?

      • Major.Freedom says:

        Perhaps, but it is why Caplan isn’t an Austrian. Rothbard devoted a large part of his critique of neoclassical economics to the problems of units of measure.

        Maybe what you recently realized is more fundamental.

        • Bob Murphy says:

          MF just to be clear, I’m not saying your point was trivial. I think this is a huge deal, especially in the capital & interest literature, where getting the dimensions right would show how much confusion there had initially been.

          But, what I mean is that in general I don’t think it’s crazy to have a model where a function maps from (units of labor hours, fertilizer, and water) to (units of output).

          • guest says:

            “But, what I mean is that in general I don’t think it’s crazy to have a model where a function maps from (units of labor hours, fertilizer, and water) to (units of output).”

            That model would have to be completely subjective to the individual involved in production.

            It’s not that labor, fertilizer, water are commensurable with either themselves or with output, but that the costs of the input are only economically relevant to the producer insomuch as they bear on the profitability from the sale of the output.

            If, for example, the producer is able to procure fertilizer, but not water or labor, and if he cannot foresee procuring the other inputs in time to make his desired profit, then it doesn’t matter how much the fertilizer’s cost contributes to the cost of production – unless he processes the fertilizer in conjunction with the other inputs, he’s not going to sell his output which never got produced.

            All this to say that this is another example of marginal units vs. objective units, the marginal unit being the maximum acceptable cost associated with producing one unit of output, with the costs of the individual units of the various inputs only being relevant to that maximum acceptable cost.

            Joseph Salerno explained this with his own example in the following video:

            [Time stamped]
            The Birth of the Austrian School | Joseph T. Salerno

            “But Menger wanted to go further, he says, ‘Wait a minute. That doesn’t explain how each individual unit …’ – Remember, the unit of the good is what’s important – ‘… each individual unit of a factor of production is valued, priced’ …”

            Joseph Salerno says that Menger asked a very insightful question that unravelled the puzzle: How much would the total product decrease if a unit of the factor of production was subtracted from the process of production?

            The production function for wheat is:

            1,000 W = 90 L + 2 H + 1 P + 40 A + 500 F

            (Aside: You can’t really add those together; The plus sign is a convenient way of saying you need that many units of all those inputs to get 1,000 bushels of wheat.)

            W = wheat (bushels)
            L = labor (days)
            H = horses
            P = plough
            A = land (acres)
            F = fertilizer (lbs.)

            Menger asks: What is the value of a 100 lb. sack of firtilizer?

            Menger’s solution: Assuming a reduction of output of 40 bushels for a reduction of 100 lbs. of fertilizer, the value of a 100 lb. sack of fertilizer is the marginal utility of 40 bushels of wheat to the farmer.

            In money terms, it would be the money value of 40 bushels of wheat.

            Joseph Salerno goes on to say that the farmer would be willing to pay up to the money value of 40 bushels of weat for a 100 lb. sack of fertilizer.

            The marginal unit of factors of production, then, is *not* each individual factor unit, but all of the factor units required to make a unit of output.

    • Tel says:

      In an exam situation, you are pretty much helpless other than to accept the equations they give you. Once you accept that, the units don’t matter too much.

      If you want to be a stickler there’s a constant “2” in the first equation which would have a unit something like “pounds of corn * hours of labour ^(-2/3)” and that might be a mighty strange unit but anyway you could do it. Hiding units inside mystic constants is well accepted practice.

      Seems to me there are other peculiarities about this problem: market prices never change, so suppose you decide to go into the corn business, you can guarantee to buy labour at a certain fixed price and also you can guarantee to sell corn at a certain fixed price… in effect none of the other firms exist for you anymore. Every corn producing firm is completely isolated from every other firm. Egats, now I know what bugs me, it’s an LK “fixprice” economy.

      The wheat producers are almost isolated, they only notice another firm when a corn producer decides to change over to wheat production (or vice versa). But note that the “/n” term really just changes the productivity constant “160” so you can easily see that some sort of economy of scale makes wheat production lean towards a monopoly producer (i.e. W = (160/n) L^(1/2) and the “i” index is irrelevant).

      Thus, you solve the corn producer first (because that solution never changes) then you solve the case of 1 wheat producer, 2 wheat, etc until there is no incentive for any corn producer to switch over to wheat production… that should be stable I would guess.

      Without even starting to plug any numbers (or reading other people’s solutions), I’d expect the number of wheat producers to come out on the lowish side, maybe 10 or 20 in that ballpark. Obviously nowhere near 1000 wheat producers.

    • Gene Callahan says:

      Similarly, we could not possibly have an equation in physics involving grams, meters, and calories: those things aren’t commensurable! Physicists should just be ranking their preferences for certain grams and certain meters.

      • Major.Freedom says:

        Then you should be able to enlighten us on what the calculated economic value is for 4 hamburgers divided by 3 t-shirts plus 5 computers.

        I will be able to tell you the calculated OBJECTIVE value for the variable in question with that equation involving grams, meters and calories.

        The commensurability and incommensurability of units ultimately stems from whether those units are associated with the material, objective world as opposed to the mental, subjective world. Grams, meters and calories are in principle commensurable in the objective world because they are all connected by the same subject who understands them. The subject can commensurate the variables and at the same time rank those variables with his own self as the standard of measure.

        The reason why the unit called “Newton”, which using SI base units is “kg*m/s^2”, is a unit that can be measured, and is commensurable with other objective variables is because MEASURING is an act of the subject. Measuring connects them. Measuring is also a choice that entails opportunity costs. The fact you measured something at a particular time and place is a revealed preference of subjectively ranked ends.

        Murphy’s problem above is en economic problem, not a physics problem.

        To take what is argued on the subjective side of things, and pretend you are making a clever or snarky comment by responding with “similarly”, shows you really don’t understand the concepts here.

        Maybe you should spend more time learning about the Turing Test so that your comments on that can be, unlike here, supported by actual understanding. You don’t have to go into attack mode every time Rothbard is mentioned in a way other than contempt or disagreement. You really don’t like Rothbard, do you? Your passions are overpowering your ability to reason, hence your comment above. Sad.

        • Bob Murphy says:

          MF go look again at the problem. It was saying that certain physical units of inputs transform into certain physical units of outputs. The market values come in via prices, which are given. I don’t think there is anything horrible going on with these assumptions, in the way you’re suggesting.

          • Adrian Gabriel says:

            Dr Murphy, you bring up an interesting point about the profit maximization models taught at universities. As I read MF’s response to Callahan I was curious whether the disregard for subjective nonmonetary costs (disutilities) are unimportant entities of which to disregard? To me MF has done an excellent job in previous comments to others pointing out the reification of mathematical modelling. Indeed you are correct that it is a model, and is it not true that these models are simplifications, and perhaps even complete oversimplifications of our complex economic systems?

  2. Keshav Srinivasan says:

    I have no degree in economics, and I’ve never taken any economics courses. And let me say at the outset that I don’t understand why the interest rate plays any role at all, so I solved this problem without reference to the interest rate.

    In any case, my answers are for part A are 2 and 40,000/n^2 respectively. And my answer for part B is that there will be either 141 or 142 wheat-producing firms, and the rest will be corn producing.

    Is that right?

    • Keshav Srinivasan says:

      Oh, I think I know what role the interest rate may play: instead of putting his dollars into either wheat or corn, a business man can choose to instead put his money into a bank account and earn a 10% return on his investment. Am I on the right track?

    • Bob Murphy says:

      Keshav, right, that’s what I think the correct answer. Thanks.

      But let’s see if other economists agree.

    • Tel says:

      Yes, I pretty quickly got 2 units of corn as the ideal output for the corn producer which is irrespective to anything other firms do and makes $2 profit.

      Took me a bit longer to get ideal W is 16000 / (n^2) making a profit of $40000 / (n^2)

      Then just figure out where the wheat maker profit hits $2, which should be 141 firms I would think (if it goes to 142 then one of them is better off going back to corn again). Depends on whether the corn producer can “look ahead” to know that they would reduce the productivity of wheat… we normally presume perfect knowledge, so therefore the corn producer knows it isn’t worth becoming #142 wheat producer.

      The interest rate of 10% seems unrelated because we don’t know the capital cost of setting up any business here. A corn producer spends $4 on labour (presumably per hour) and makes $2 profit (presumably also per hour) which seems more attractive than 10% interest. Let’s suppose he works 1900 hours per year that means spending $7600 for a return of $3800 is well above the going interest rates.

      In fact, seems to me that the interest rate of 10% would necessarily imply the other equations must be wrong, or at least that market prices would be forced to fall. If simple corn production got a guaranteed return like that then everyone would be doing it and no one would buy the corn.

  3. Keshav Srinivasan says:

    Bob, I also have a guess as to what the flaw you’ve identified is, just because I know about your past work: it is that economists assume that the interest rate is equal to the marginal product of capital, but in this case it is actually equal to the marginal product of labor. Is that right?

    • Bob Murphy says:

      Keshav, I didn’t want to type in the whole question. Later on they ask how much a wheat producing firm could sell for, versus a corn producing one. So clearly the interest rate is involved there.

  4. Chris says:

    Working on PhD in economics

    I agree with Keshav’s first answer, but not with his second (I’m not sure which part you were responding to when you said it was correct). I don’t think the interest rate should play a role in this problem. The firm doesn’t have any money to invest at that rate. It pays the wages of the workers with the output that it generates in production. Maybe that’s what you have a problem with? If so I definitely agree that economists tend to be very sloppy with timing in their models.

    • Major.Freedom says:

      Minor quibble, but I think wages are paid before the output is ready.

      It takes about 60 to 100 days to produce a crop of corn. Workers are being paid during those months. The employers are the ones who have to wait.

      Then there is the money required to acquire and maintain the capital the workers utilize. That had to have been paid for too.

      • Chris says:

        In real life yes. But in most basic macro models firms don’t own the capital and don’t have any wealth. This period’s output is the only thing they have to pay with. Profit maximization is purely a static problem. As I said, I don’t necessarily think that is the right way to go, but that’s how it’s usually done.

        • Major.Freedom says:

          Fair enough.

          In the traditional Cobb Douglas production function though, K (for capital) is a variable. Here the point is to conceptualize decision making, in a static framework, between more capital at the expense of labor, or vice versa. There are cost inputs and revenue inputs, which you might think implies time, but it is still all concurrent.

          This unfortunate prevalence of static frameworks in economics was the combination of hypostatization, which is very common, and mathematics infiltration.

          I am so thankful I was able to learn about German Idealism, and its syntheses such as praxeology.

      • Tel says:

        Yeah, the problem completely omits any consideration of capital.

        I think there’s a worse assumption in there, in terms of this type of scenario:

        * With only 1 wheat producer, the optimal output is 16000 units of wheat and profit is $40000.

        * Thus, massive incentive for one corn producer to decide to switch and make wheat… no problem but now the productivity instantly goes down for the big producer.

        * With 2 wheat producers, the optimal output (each) is 4000 units of wheat and profit (each) is $10000.

        * Total wheat in the market has dropped by half, but the price didn’t change (!) and a huge number of workers got laid off and no effect on the labour market (!) and the new producer just somehow managed to destroy the productivity of the incumbent big producer (how?!?!) who is then forced to drastically cut production (but why cut production if prices are fixed anyhow?)

        That is generally the problem with exercises intended for student consumption, they require suspension of disbelief. Like watching a Holywood movie, once you have suspended disbelief it’s difficult to figure out what is a legitimate plot twist and what is just some loose end that never got thought about.

    • Bob Murphy says:

      Chris, yep, you are hitting on exactly what my problem with this question was. The reason it jumped out at me here is that it gave us the interest rate and it was agricultural production, which clearly involves a gestation period. But I concluded that the “correct” answer was to assume production was instantaneous and that financial capital is tied up for exactly 0 seconds in the production process.

      To repeat, this isn’t a problem unique to this question; I think in general economists “solve” the firm’s problem in a way that is technically correct but wrong in spirit.

  5. Adrian Gabriel says:

    Google PhD

    It’s very cool that Mr Srinivasan was able to calculate the correct answers using effective mathematical simplification. Typically with the profit maximizing process you set MR=MC. To find the equilibrium you set both equations equal. I honestly did not do the math nor plug out the process but I am curious as to if businesses actually do this process themselves in their accounting methods. I am aware that certain computer programs help businesses, specifically in the food industry, calculate labor costs and thus they want to keep those low during the day or over time. Not sure they ever do any form of calculations that seek to maximize profits with production functions. I am aware that economics basically helps people understand the processes of how economies work with mathematics, which is pretty cool.

    One further disclaimer here is that there is a LARGE difference between what Dr Murphy is forced to teach in his class, something I learned in Econ courses in the university as well, and what is taught in Austrian Economics (specifically what we read in Rothbard’s MES Chapter 7). We are presented a more realistic approach to factors and pricing of factors through the MVP and DMVP. I feel that the individual analysis of factors of production being employed in an individual firm, the subjective valuations of these factors, and the prices involved make more sense in the real world of business decisions. I’m not sure business owners care about finding a production function in the manner described in econ courses, I feel this is simple theory which simply allows people to push forth government intervention in the market.

    I know I have veered off the path of answering the question at hand, but I am finding more and more people believing math can solve all answers to all questions in the world, when it is basically a theoretical way to describe life. Indeed math modeling or computer programming of mathematical models allows certain people to get closer to their modeled estimate in the real world, but real life is not so simple and can actually be starkly different than what was mathematically predicted. Even the idea of calculating the price spread is so difficult based upon the entrepreneur’s access to buying and selling prices. As life goes on I am realizing academia might be stressing a certain dynamic of academics that veers people away from creativity and more onto the path of machination and socialist calculation.

  6. steven landsburg says:

    Without looking at any of the comments and without looking at your followup post, I get:

    Corn firms maximize profit at $L=1$, where profit is $2$.

    Wheat firms maximize profit at $L=10000/n^2$, where profit is $40000/n^2$.

    To equate profits across industries, $n=100\sqrt{2}$.

    This makes many assumptions, e.g. that there are an unlimited number of firms, so entry to the wheat industry does not entail exit from the corn industry, etc. It seems to me perfectly reasonable to relax those assumptions.

    Now I will look at the comments and your followup to see what I missed.

  7. steven landsburg says:

    Okay — now I’ve read the comments.

    I haven’t seen the actual statement of the problem, but I am guessing that somewhere in the probblem statement, it says “Answer in the long run”.

    The standard intermediate-level assumption is that in the long run, we can treat the cost of moving from one industry to another as insignificant. That makes the interest rate irrelevant here.

    There are a number of possible justifications for that assumption. The one I prefer is that old firms are constantly dying off and being replaced by new firms. A new firm enters either the corn or the wheat industry. There is no cost of switching; they just enter one industry once and stay there. Newcomers always choose the industry with greater profit, so in long run equilibrium, profits are equalized.

    I agree that one could tell more sophisticated stories, and I believe there are about eighty gazillion examples of such stories in research papers, but probably not in intermediate level textbooks.

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