You seem to be treating this like a university lecture hall, and one in which the professor doesn't have 1 on 1 office hours or need to do grading.
But most learning, especially K-12 learning, requires labor intensive supervision by teachers. At a minimum they have to keep the kids under control. At the higher end students may need one on one instruction from time to time.
And so something like school vouchers, which took kids out of the public classroom, may or may not increase effective per student funding and on different timelines. The school can lay off teachers if it sees sustained lower enrollment (in theory, ignoring politics) or freeze new hiring. And it can slow down new school construction or order fewer of those overflow trailers. It could even shutter or combine schools in the long run.
Obviously, this isn't something that the K-12 system would want to do, but it wouldn't be the end of the world. Talk about "taking money out of the schools" is basically bogus, as eventually the labor and capital investment will correlate to the new enrollment. Teachers that didn't get work in public school could find demand for their services in new private schools (if they are good at their job).
P.S. In the northeast spending per pupil is over $20k, and this often excludes gargantuan pension liabilities. Most increases in government spending have been at the state/local level, often the K-12 school systems. Whenever people tell me we don't have money for giant child tax credits, I remind that all I really need is to re-direct the K-12 budget directly to parents.
Yes, most vouchers are far below total cost (they usually include only state funding since it's done at the state level). This makes the anti-voucher case even weaker, but they still spout the same nonsense about it pulling money out of public schools. The implicit assumption is that the marginal cost per student is less then the voucher, whatever it is, and rhetorically they act like the marginal cost is $0.
One thing you aren't considering on vouchers, and it may not work like this everywhere, but here in Colorado, charter schools (not exactly the same as vouchers, but close enough for this example) receive the per-student money, but not a share of the infrastructure money. They are split into separate piles. So the charters have to come up with infrastructure funding another way (either by spending less of the per-student money or fundraising). My daughter's school was founded about a decade ago but rented classroom space until just before covid, when their building opened. They are expanding it now, some classes are in trailers, but will all be in building this fall.
My point is, if the amount of a voucher is less than the total per-student cost, this actually increases the per-student spending on the remaining students in the public schools. At a certain point it will cause problems, as schools will need to contract or whatever, but on the margin, a student leaving and taking even slightly less than the full amount with them helps the remaining students. Its sort of the reverse of Bryan's argument. Every student leaving helps the remaining students until the student that tips it to an astronomical cost.
Minor point: since the current number of students in the classroom is an integer, the set {0, 1, ... , 30} contains 31 points and so a uniform distribution defined on it implies a probability of 30/31 that marginal cost =0 and probability 1/31 that it is the full $500,000. So the expected value does not quite equal the average cost. Admittedly this does not affect the qualitative assertions made in the post.
The mistake, I think, that economists make when talking about "marginal cost" is that we forget to specify exactly which margin we're talking about. Take as an example software, which is often put forward as a "zero marginal cost" industry. Well, I suppose the marginal cost of licensing one more copy of your software might be zero AFTER you've developed the software, beta-tested it, marketed it, and so on. (Although, even this ignores the costs of ongoing customer support! It's not hard to imagine that your marginal customers might require greater customer support, in which case marginal costs aren't just positive but increasing.)
But what is the marginal cost producing v2.0 of the software? Well, that's a different margin isn't it, and the cost of this margin is certainly not zero.
Anytime someone invokes "zero marginal cost" they are usually talking about a margin in the extreme short run. The marginal cost of an additional student in an undersubscribed college economics course might be close to zero (although this ignores grading, office hours, etc.). But the marginal cost of adding one more student to the economics major is a different story. And the MC of adding one more student on campus is different still.
In other words, we need to be careful about precisely which margin we are talking about when we casually invoke the phrase "marginal cost."
This describes all of what capital budgeting analysts called "step up costs." Most costs of a business are step up costs because they are caused when exceeding process capacity. Very few costs are indirect (software programming or R&D) or direct (cost of goods at a retailer or unit shipping).
The complication with applying marginal cost analysis in this case is that classrooms are durable capital goods that can serve students in many different classes over many decades. At the point in time when a school administrator is making investment decisions regarding classroom construction, a relevant marginal cost calculation would be the additional cost of building a classroom with X+1 seats instead of X seats, which would be justified if the expected discounted marginal revenues generated by the additional seat over the entire lifetime of the classroom exceed that marginal cost.
It is the durability of the capital good in this instance that makes it likely that errors in expectations or fluctuations in the short-run demand for seats could give rise to a situation where a particular class might have empty seats that could be filled at no additional cost to the school (although other services typically bundled with classroom instruction of a student, like the labor required for grading, might incur additional expenses). This short-run margin, however, is not the long-run margin that is relevant to classroom construction decisions.
In the scenario described, the $500k spent for building a second classroom would not simply make it possible to seat the 31st student in the short run; it would make it possible to seat additional students in numerous large classes over a period of decades, while still retaining the capacity to seat up to 30 students in the old classroom for smaller classes. Moreover, there are many different possible values of X that might be considered for the second classroom's size. Assuming that a second classroom with an X > 30 were viable, the marginal revenue from the 31st student would only likely account for a very small portion of the $500k. It is revenues from future generations of students accommodated by the new classroom that would be needed for justifying most of the construction costs.
The difference between average and marginal cost is core to so much microecon and business design. It's surprising how often it's forgotten in other realms of discussion.
Yes, there is definitely a step function. I've been on a number of school boards and we wrestled with this all the time.
In terms of K-12, what you describe so well places a premium on figuring out the "right" number of students per teacher. The size of a classroom is also a constraint.
The marginal cost to the additional student might be zero, but the marginal cost to current students could be high. More students can lead to more distractions and fewer opportunities to interact with the professor. It’s more of a common property category that a pubic good category.
I come from an engineering background. In my work, the cost of an increment in demand depends on the size of the increment. (One can then obtain average incremental cost per unit of the increment, but that's just for convenience.)
So consider a mobile telecommunications network with a fixed number of users. If a user increases his demand by 1 Gbit, the cost is zero. We will not invest any more money to accomodate this. The only effect is an imperceptible increase in congestion (if any) and so an imperceptible decrease in quality of service for all users. But the magnitude is so small that nobody worries about it.
Suppose now that another supplier, Company X, wants to allow its customers to roam over my network. That might involve a projected 100 million Gbits every month. The increment is now much, much larger. If I don't increase my network's capacity, there will be a noticeable degradation in service quality. So I will invest $Y in increased spectrum, or cell splitting, or whatever is the minimum cost way to increase capacity. Once again, for convenience, I express this as an average incremental cost of $Y/$X.
So asking "What is the cost?" without any further information is meaningless. I have to look at the decision I am contemplating (including level of service, quantity of increment, and timing).
Of course, none of this is new. James Buchanan laid it out in the early part of his little book, Costs and Choice.
The correct way to model this is a stochastic process where often adding someone is 0 and sometimes it's a lot so you pick a point in time at random according to some distribution. This distribution can be calibrated based on availability of resources so in fact even the expected marginal cost is time-dependent and situation-dependent.
This gives a framework to differentiate between education and pro-immigration say.
It seems we are leaving out some things.
1. Grading homework and tests is a cost for the teacher
2. Textbooks which are provided for free in K-12
3. Disruption of other kids learning. Some of those extra kids will be disruptive, causing learning loss.
You seem to be treating this like a university lecture hall, and one in which the professor doesn't have 1 on 1 office hours or need to do grading.
But most learning, especially K-12 learning, requires labor intensive supervision by teachers. At a minimum they have to keep the kids under control. At the higher end students may need one on one instruction from time to time.
And so something like school vouchers, which took kids out of the public classroom, may or may not increase effective per student funding and on different timelines. The school can lay off teachers if it sees sustained lower enrollment (in theory, ignoring politics) or freeze new hiring. And it can slow down new school construction or order fewer of those overflow trailers. It could even shutter or combine schools in the long run.
Obviously, this isn't something that the K-12 system would want to do, but it wouldn't be the end of the world. Talk about "taking money out of the schools" is basically bogus, as eventually the labor and capital investment will correlate to the new enrollment. Teachers that didn't get work in public school could find demand for their services in new private schools (if they are good at their job).
P.S. In the northeast spending per pupil is over $20k, and this often excludes gargantuan pension liabilities. Most increases in government spending have been at the state/local level, often the K-12 school systems. Whenever people tell me we don't have money for giant child tax credits, I remind that all I really need is to re-direct the K-12 budget directly to parents.
robc,
Yes, most vouchers are far below total cost (they usually include only state funding since it's done at the state level). This makes the anti-voucher case even weaker, but they still spout the same nonsense about it pulling money out of public schools. The implicit assumption is that the marginal cost per student is less then the voucher, whatever it is, and rhetorically they act like the marginal cost is $0.
One thing you aren't considering on vouchers, and it may not work like this everywhere, but here in Colorado, charter schools (not exactly the same as vouchers, but close enough for this example) receive the per-student money, but not a share of the infrastructure money. They are split into separate piles. So the charters have to come up with infrastructure funding another way (either by spending less of the per-student money or fundraising). My daughter's school was founded about a decade ago but rented classroom space until just before covid, when their building opened. They are expanding it now, some classes are in trailers, but will all be in building this fall.
My point is, if the amount of a voucher is less than the total per-student cost, this actually increases the per-student spending on the remaining students in the public schools. At a certain point it will cause problems, as schools will need to contract or whatever, but on the margin, a student leaving and taking even slightly less than the full amount with them helps the remaining students. Its sort of the reverse of Bryan's argument. Every student leaving helps the remaining students until the student that tips it to an astronomical cost.
Minor point: since the current number of students in the classroom is an integer, the set {0, 1, ... , 30} contains 31 points and so a uniform distribution defined on it implies a probability of 30/31 that marginal cost =0 and probability 1/31 that it is the full $500,000. So the expected value does not quite equal the average cost. Admittedly this does not affect the qualitative assertions made in the post.
The mistake, I think, that economists make when talking about "marginal cost" is that we forget to specify exactly which margin we're talking about. Take as an example software, which is often put forward as a "zero marginal cost" industry. Well, I suppose the marginal cost of licensing one more copy of your software might be zero AFTER you've developed the software, beta-tested it, marketed it, and so on. (Although, even this ignores the costs of ongoing customer support! It's not hard to imagine that your marginal customers might require greater customer support, in which case marginal costs aren't just positive but increasing.)
But what is the marginal cost producing v2.0 of the software? Well, that's a different margin isn't it, and the cost of this margin is certainly not zero.
Anytime someone invokes "zero marginal cost" they are usually talking about a margin in the extreme short run. The marginal cost of an additional student in an undersubscribed college economics course might be close to zero (although this ignores grading, office hours, etc.). But the marginal cost of adding one more student to the economics major is a different story. And the MC of adding one more student on campus is different still.
In other words, we need to be careful about precisely which margin we are talking about when we casually invoke the phrase "marginal cost."
See also: "we don't need to build any extra houses, some houses are currently empty."
This describes all of what capital budgeting analysts called "step up costs." Most costs of a business are step up costs because they are caused when exceeding process capacity. Very few costs are indirect (software programming or R&D) or direct (cost of goods at a retailer or unit shipping).
The complication with applying marginal cost analysis in this case is that classrooms are durable capital goods that can serve students in many different classes over many decades. At the point in time when a school administrator is making investment decisions regarding classroom construction, a relevant marginal cost calculation would be the additional cost of building a classroom with X+1 seats instead of X seats, which would be justified if the expected discounted marginal revenues generated by the additional seat over the entire lifetime of the classroom exceed that marginal cost.
It is the durability of the capital good in this instance that makes it likely that errors in expectations or fluctuations in the short-run demand for seats could give rise to a situation where a particular class might have empty seats that could be filled at no additional cost to the school (although other services typically bundled with classroom instruction of a student, like the labor required for grading, might incur additional expenses). This short-run margin, however, is not the long-run margin that is relevant to classroom construction decisions.
In the scenario described, the $500k spent for building a second classroom would not simply make it possible to seat the 31st student in the short run; it would make it possible to seat additional students in numerous large classes over a period of decades, while still retaining the capacity to seat up to 30 students in the old classroom for smaller classes. Moreover, there are many different possible values of X that might be considered for the second classroom's size. Assuming that a second classroom with an X > 30 were viable, the marginal revenue from the 31st student would only likely account for a very small portion of the $500k. It is revenues from future generations of students accommodated by the new classroom that would be needed for justifying most of the construction costs.
The difference between average and marginal cost is core to so much microecon and business design. It's surprising how often it's forgotten in other realms of discussion.
Yes, there is definitely a step function. I've been on a number of school boards and we wrestled with this all the time.
In terms of K-12, what you describe so well places a premium on figuring out the "right" number of students per teacher. The size of a classroom is also a constraint.
A beautiful and simple insight. I wish I'd understood this half my life ago.
The marginal cost to the additional student might be zero, but the marginal cost to current students could be high. More students can lead to more distractions and fewer opportunities to interact with the professor. It’s more of a common property category that a pubic good category.
I come from an engineering background. In my work, the cost of an increment in demand depends on the size of the increment. (One can then obtain average incremental cost per unit of the increment, but that's just for convenience.)
So consider a mobile telecommunications network with a fixed number of users. If a user increases his demand by 1 Gbit, the cost is zero. We will not invest any more money to accomodate this. The only effect is an imperceptible increase in congestion (if any) and so an imperceptible decrease in quality of service for all users. But the magnitude is so small that nobody worries about it.
Suppose now that another supplier, Company X, wants to allow its customers to roam over my network. That might involve a projected 100 million Gbits every month. The increment is now much, much larger. If I don't increase my network's capacity, there will be a noticeable degradation in service quality. So I will invest $Y in increased spectrum, or cell splitting, or whatever is the minimum cost way to increase capacity. Once again, for convenience, I express this as an average incremental cost of $Y/$X.
So asking "What is the cost?" without any further information is meaningless. I have to look at the decision I am contemplating (including level of service, quantity of increment, and timing).
Of course, none of this is new. James Buchanan laid it out in the early part of his little book, Costs and Choice.
Not to mention the cost of education presented to a student themselves, or the cost of educational labor afforded to teachers…
The correct way to model this is a stochastic process where often adding someone is 0 and sometimes it's a lot so you pick a point in time at random according to some distribution. This distribution can be calibrated based on availability of resources so in fact even the expected marginal cost is time-dependent and situation-dependent.
This gives a framework to differentiate between education and pro-immigration say.