PPFs and Supply curves (why PPFs are bowed out and supply curves slope up)

This is a simple "teaching" post. As a public service, and because I'm a teacher. I am just trying the get the presentation right. [The bit at the end is harder, and I'm not sure I'm happy with it.]

What determines the curvature of the Production Possibilities Frontier, and how is it related to the elasticity (or slope) of supply curves?

1. Start with a very simple economy. There is only one resource; I will call it "land" (though you could call it "labour" if you like). All land is identical. The country has 100 acres of land. The technology lets you grow 1 apple per acre per year, or 2 bananas per acre per year.

Put quantity of apples A on the vertical axis, and quantity of bananas B on the horizontal axis. The PPF looks like this:

Any point on or inside the PPF is feasible; any point outside the PPF is not feasible.

The PPF in this example is a straight line, with a slope of -1/2. The opportunity cost of an apple is 2 bananas (if you grow one extra apple you lose 2 bananas). The opportunity cost of a banana is 1/2 an apple (if you grow one extra banana you lose half an apple. The slope of the PPF (or the inverse of the slope) represents the opportunity cost of one good in terms of the other.

Now suppose there are 100 farmers, each owning 1 acre of land. It's a market economy, and each farmer is small, so takes the price of apples (and bananas) as given. Each farmer chooses whether to produce apples or bananas, and chooses whichever gives the highest profit. Let Pa and Pb be the prices of apples and bananas, so that Pa/Pb is the (relative) price of apples in terms of bananas.

What would the supply curve of apples (and bananas) look like? (We want to figure out how many apples the farmers would want to produce as a function of Pa/Pb.)

Put Pa/Pb on the vertical axis, and A on the horizontal axis. (Yes, economists have the axes for supply and demand curves the wrong way round, for complicated historical reasons, and it's too late to change us now, so get used to it).

The supply curve for apples would be perfectly elastic (horizontal) at a relative price of 2, for any quantity of apples between 0 and 100 (it goes vertical at 0 and 100). For Pa/Pb < 2, no farmer would want to grow apples (all will grow bananas). For Pa/Pb > 2, all farmers would want to grow apples. For Pa/Pb = 2, farmers don't care which they grow.

So, if we assume that, at a price Pa/Pb =2, people would want to eat some of both fruits, we know that in competitive equilibrium Pa/Pb = 2, and the demand curve determines how many apples are grown.

(It's the same for the supply curve of bananas, which is horizontal at Pb/Pa = 1/2.)

Now generalise this simple example just a little. Assume you get a apples per acre and b bananas per acre. The slope of the PPF is now -a/b. The apple supply curve is now horizontal at b/a.

2. Now let's generalise this simple example a lot. Drop the assumption that all land is identical. Think of the country as a long thin island, on a North-South axis, where a/b is not a constant, but is an increasing function of latitude. The further North you go, the bigger is a/b, which means that any acre of land has a comparative advantage in growing apples over a more southerly acre, and a comparative advantage in growing bananas over a more northerly acre.

[If I were teaching this in Australia I would reverse North and South; if I were teaching this in a hilly country I would replace latitude with elevation. Which shows it doesn't really matter; all that matters is that I can line up acres of land by a/b.]

If we are on the PPF, it must be the case that there is a line somewhere on the island, and all land South of that line is growing bananas, and all land North of that line is growing apples. Otherwise we could switch some southerly land from apples to bananas, and switch some more northerly land from bananas to apples, and get more of both apples and bananas, which would mean we were not originally on the PPF. Call that line the "apple/banana margin".

At the apple/banana margin, the slope of the PPF is -a/b. But as you grow more and more apples, so the apple/banana margin moves South, a/b falls, which means the PPF gets flatter. Which means the PPF is now curved, and bowed out. Which means the marginal opportunity cost of an extra apple rises, the more apples you grow. (And the marginal opportunity cost of a banana rises the more bananas you grow.

Which means that the supply curve of apples will slope up. (So will the supply curve of bananas).

3. Now let's generalise the original example in a different direction. Assume you need both land and labour to grow apples (and bananas). But all land is identical, and all labour is identical (to keep it simple). Will the PPF be a straight line?

If you use all the land and labour to grow apples, you can grow Amax apples. If you use all the land and labour to grow bananas, you can grow Bmax bananas. (That's just a definition of Amax and Bmax.)

Now suppose you use half the land and half the labour to grow each. And suppose that lets you grow 0.5Amax apples and 0.5Bmax bananas (I'm assuming constant returns to scale). But you could almost certainly grow more apples and bananas than that. Because it is very unlikely that apples and bananas are exactly equally labour-intensive. If apples are more labour-intensive than bananas, you could grow more of both by switching some labour to apples and some land to bananas. (Or vice versa if apples are less labour-intensive than bananas.) Which means you were inside the PPF when you split the land and labour 50-50 between the two crops. Which means the PPF cannot be a straight line and must be bowed out. Like this:

Which means the supply curve of apples will slope up. Unless by sheer fluke apples and bananas are exactly equally labour-intensive.

4. If we drop the assumption that all land is identical and all labour is identical, that gives us an even bigger reason for the PPF to be bowed out and supply curves to slope up.

[This bit is harder.

5. If we drop the assumption of constant returns to scale, and assume Increasing Returns to Scale instead, then we might (or might not) get a PPF that is bowed in (it depends whether IRS is big enough to offset the other effects I talked about above). But then with IRS we do not get a competitive equilibrium with lots of little farms. We may get only one apple monopolist. And monopolists do not, strictly speaking, have supply curves. Because the profit-maximising quantity of apples depends not just on price but also on the elasticity of demand. Only if we assume the elasticity of demand is a constant, and does not change when the demand curve shifts, can we define a "pseudo supply curve". And that pseudo supply curve is the monopolist's Marginal Cost curve, plus a vertical markup that depends on elasticity of demand (the markup approaches zero as the demand curve becomes perfectly elastic). But IRS does not mean the MC curve slopes down; it only means that the MC curve is below the ATC curve.

Assume there is a fixed cost of growing apples, so we have IRS for apples, but not for bananas. The only difference that makes is that the PPF goes horizontal for a short stretch when it hits the banana axis. Because growing just one apple per year means you must use resources to cover that fixed cost so you lose a lot of bananas. The rest of the PPF can be bowed out, for all the above reasons. And the apple monopolist's pseudo supply curve will still slope up, for all the above reasons. A shift in demand towards apples means the monopolist will push the apple/banana margin South, which will raise his marginal opportunity cost of apples, so the relative price of apples will rise (unless it is offset by an increase in the elasticity of demand for apples.]

Comments

[Sean]

Is "bowed in"/"bowed out" the standard economist terminology? I'm familiar with concave/convex.

[Herbert]

Nick, the mathematical theory of the slope of the PPF was developed in the 1950s, I believe. The main theorem, which fits your presentation, says: With two factors of production and to constant-to-scale production functions, the PPF is linear if and only if the factor intensities are identical. Otherwise, it is strictly concave.

The old proofs relied on second derivatives and such stuff. An easier and more elementary way is to use convex set theory.

[Nick Rowe]

Sean: real economists say "concave/convex". But I always find it confuses others, and me. The production set is convex; the PPF is concave to the origin. Something like that. Bowed out is easier. Everyone understands it.

Herbert: thanks. Yep, my guess is this stuff is nearly 150 years old, if you search hard enough. The only trick is trying to present it simply and intuitively with minimal math. I'm aiming at first year students, and layperson blog readers (and teachers of intro economics).

[notsneaky]

Herbert beat me to it! I was going to make the comment that sometimes we teach this wrong (and textbooks get it wrong) by saying "PPF is bowed out due to diminishing returns to capital (or land) and labor". But we can have diminishing returns and linear PPF if factor intensities are the same. I actually prefer (cuz Malthus) to call "factor intensities" "rates at which diminishing returns set in" which I think is more intuitive.

I have a textbook sitting next to me - which I won't name, because otherwise it's a pretty good book - in which authors consistently use a 2 good example with 2 Cobb-Douglas production functions with same coefficients but where they keep drawing an upward sloping supply curve.

[Nick Rowe]

notsneaky: Yep. The intro textbooks are not great on this, in my experience. And I've come across a couple of people on the econoblogoshere over the years who don't seem to get it, and I don't think it's all their fault.

This is how I would describe it: With Constant Returns to Scale production functions, and with equal factor intensities in the two goods, the Long Run PPF would be linear (and the LR supply curve horizontal), and only the "Short Run" (holding the allocation of one factor constant) PPF would be bowed out (and only the SR supply curve upward sloping) due to diminishing marginal product of the variable factor.

But in the realistic case of unequal factor intensities, (not to mention heterogenous factors), the LR PPF will be bowed out too.

[Mike Sproul]

Another reason supply curves slope up is that demand curves slope down.

[Nick Rowe]

Mike: you lost me there. Ah! You mean the demand curve for leisure slopes down? Right! So if we put Pa/W on the axis, we would get an upward slope.

But if we separate the technology/resource side from the preference side (like I'm doing here), and call the former "supply" and the latter "demand" (though that's arguable) we wouldn't get that.

[Mike Sproul]

I was thinking that farmers who want to produce more apples must bid land away from orange farmers. Since the demand for oranges slopes down, the apple farmers must pay more for each additional acre that they bid away from the orange farmers.

[Nick Rowe]

Mike: Ah! again. But I think that effect is already captured in the Pa/Pb relative price.

[Mike sproul]

Hmm. I think you're right about that. Think of a ppf, and a budget line tangent to it, and then an indifference curve tangent to the budget line. The ppf determines the supply curves, while the indifference curves determine the demand curves, and the two curves are independent of each other. It's only when you start to draw net demand and net supply curves that you get interdependence of supply and demand curves.

[rsj]

1) If both goods are IRS, then the PPF is concave, not convex. You can produce a lot of good A and no good B, or a lot of good B and no good A at the ends, with the lowest amount of production being of both good A and good B in equal amounts. With a multiple goods, and a mix of IRS, CRS, and DRS, the PPF will in general be neither convex nor concave. A straight line joining two points that are IRS will not be within the simplex. I think Nick cheated a bit here by using a two dimensional model. Two dimensions are special and should not form your intuition. The maximum utility reached will in general not satisfy a tangency condition, although there will of course be a maximum as the simplex is still convex and finite dimensional. For infinite dimensions, there may not be a maximum utility.

2) If there are nonzero switching costs, then the PPF will have a pinch and will be clipped at the ends, and may not have a tangency to the utility function although there will be a maximum utility. But because it doesn't satisfy a tangency condition, there may be no way to reach it by making marginal adjustments. Switching costs introduce state. E.g. if you start out at (.4, .6), and there is a cost to converting apple land to banana land, then there will be a pinch at 0.4, 0.6. For more than one generation, you may be stuck at that state forever, or you may asymptotically approach (but never reach) the ideal state. I think state is important here, and there are all sorts of contortions (such as introducing revolving capital in Fischer's model of present versus future goods) done to erase path-dependence.

That's enough crankiness for tonight!

[rsj]

OK, one more comment!

Switching costs are important things. Countries (as well as firms) may need a "push" from outside, because you may not get to the state that you want to be by making marginal improvements -- e.g. this has policy implications.

[Nick Rowe]

Bloody Windows decided to reconfigure, and I lost a whole long comment!

Mike: Yep, I was separating the resources & technology side from the preferences side.

rsj: "1) If both goods are IRS, then the PPF is concave, not convex."

That is only true in the one factor model, where all land is the same. It won't work with heterogenous factors. Take an extreme example, where the South land won't grow apples and the North land won't grow bananas. The PPF is bowed out L-shaped, regardless of IRS.

" For infinite dimensions, there may not be a maximum utility."

You lost me there. Suppose there are infinite possible goods, but each good has a fixed cost. To maximise the SWF we produce a finite number of goods, and produce 0 of all the rest.

"2) If there are nonzero switching costs, then the PPF will have a pinch and will be clipped at the ends, and may not have a tangency to the utility function although there will be a maximum utility. But because it doesn't satisfy a tangency condition, there may be no way to reach it by making marginal adjustments. Switching costs introduce state. E.g. if you start out at (.4, .6), and there is a cost to converting apple land to banana land,..."

Very roughly, that is the Intro textbook model of the short run, except the textbook lets you switch labour from apples to bananas, and you don't switch land until the "long run", when all the trees die and must be replaced anyway. So the Short Run supply curve is less elastic than the long run. If you can switch neither land nor labour, you have the very short run, where supply curves are perfectly inelastic.

A bit of crankiness is good, in moderation.

Where's Avon by the way? I need someone who can remember the difference between concave and convex.

[rsj]

Nick,

L shaped is also concave. E.g. flip the L and you get a convex shape. It's not important, though, the point is that a line between two points doesn't lie within the simplex.

"You lost me there. Suppose there are infinite possible goods, but each good has a fixed cost.".

The proper formulation requires dated goods. I.e. The PPF needs one axis per good, and when you add investments, you have apples today versus apples 1 year from now. And apples today versus apples 2 years from now, etc. Only in the special case of revolving capital can you just think of this year and next year. When you add stuff like buildings that last N years, you need to date beyond the current year.

"Very roughly, that is the Intro textbook model of the short run, except the textbook lets you switch labour from apples to bananas, ".

I wasn't thinking of *banning* switching, only imposing a cost to switching. Say you have to retrain a worker, so they are only 2/3 as productive for the first year when they switch to the new job. That's enough to add a kink. If the land has to be first plowed up and the apples removed before you can plant bananas there, you add a kink, etc. If there is any non-zero cost to repurposing capital goods, you add a kink. But as there will always be a kink, you can't rely on a tangency condition -- e.g. there may not be a sequence of pareto improving moves to get to the better point on the PPF. This obviously depends on local analysis -- the slightest kink around an existing local maxima will make you really stuck, but a kink at a really bad spot for utility may not make you stuck.

The first criticism effectively says that there is no fixed point process that will get you there, so navigating to the right spot may too difficult to do. The second criticism says that you can't rely on people individually improving their situation to get you to the right spot. So there isn't a basis on which to believe that the "long run" PPF is nicely behaved. This invocation of a "long run" also rubs me the wrong way, as it seems to be by fiat rather than by any plausible convergence process. You can't say things are non-convex in the short run but turn out to be convex in the long run.

[Nick Rowe]

rjs: I wasn't clear on the L-shape. What I mean is that the feasible set is a rectangle, and so is convex. A line between two points in the rectangle must also lie within the rectangle. And in that sort of model, where supply curves are vertical, it is the demand curve tangency that determines price (though the supply curve determines quantity demanded in equilibrium, so tells us where we are on that demand curve).

The Intro textbook also only imposes a cost on switching, rather than banning it. The apple trees are a sunk cost. But if P falls far enough below ATC, so it's below AVC, you shut down as an apple producer, and plant banana trees instead.

And switching costs are real costs, that a central planner would take into account too, if the future is different from what he had expected. You need some sort of network effects to make your point that the market might get stuck in a bad equilibrium.

[Tel]

Suppose we start with the convex economy of Figure 2. and then the largest of the apple farmers lobbies government to introduce a licensing scheme where a hefty annual sum of 50 bananas is required in order to be allowed to farm apples. The largest of the apple farmers can pay for a license, the rest of them are forced to become banana farmers. Needless to say, the market price of apples is now going to rise relative to the market price of bananas (and there will be less fruit on the open market for everyone).

Let's say Bmax was at 200, but now we always have 50 bananas going to government (those ones are eaten by wife and friends of the chief of police and never get back onto the open market). Thus there's an effective new Bmax_1 moved to 150 (which is the short horizontal part of the curve and the price paid to produce 1 apple). Also, the maximum number of apples you can produce is also smaller, because in all cases you must at least produce the 50 bananas to cover license costs, so we have a new Amax_1 as well, and presumably a convex curve between the end-points based on multiple meshing production factors. The resulting PPF between Amax_1 and Bmax_1 will look pretty similar in shape to the original curve between Amax and Bmax, give or take a bit.

And monopolists do not, strictly speaking, have supply curves. Because the profit-maximising quantity of apples depends not just on price but also on the elasticity of demand.

The new apple monopoly does need to stay in business. So although the PPF will not provide sufficient information to calculate maximum profit, it will at least provide the production cost curve to the monopoly, and thus we get the *LOWEST PRICE* that the monopoly can ever be able to sell at (even if it makes zero profit). If you make only 1 apple, and the actual physical production cost of that apple is 1 banana, the apple must sell for at least 51 bananas (sufficient to cover government license costs, plus physical production costs).

If you make 2 apples, you can sell them each for 26 bananas and make zero profit.

Keep tracing that out and you get the sum of two curves, firstly the 1/x hyperbola as license costs are amortized over the revenue stream, and then the actual physical production cost which is an upward sloping curve as the marginal cost of each apple increases due to production factors. Remember this is the lowest price the monopoly can possibly sell for, we haven't even considered profit yet.

Clearly the hyperbolic curve will be dominant for small production volume, also we expect that as an elite luxury good, the price of apples will reduce fairly quickly as the production volume increases, because the exotic nature of this rare fruit is what makes an attractive table piece. Thus, it is highly unlikely the monopoly will ever produce enough to get away from that part of the curve dominated by license cost amortization. Eventually this lowest achievable price curve will bend around like the bottom of a bathtub and slope up again, probably meeting with some demand curve on the way down. At that point where it meets the demand curve we have the highest possible volume of apple production, given the market forces, because after that the monopoly apple farmer goes out of business. Clearly every monopoly will reduce production volume lower than that point, and yes I agree that profit maximization in the monopoly case requires trial and error feeling out the demand curve to know roughly where lack of demand starts to cause the price to drop.

We could put real numbers on it, and try drawing proper curves but when you sketch it quickly, for pretty much any plausible demand curve you end up on the left side of the bottom of the bathtub. Whether you want to call this a downward sloping supply curve, or perhaps insist that it isn't a "real" supply curve because of the monopoly profit I guess would depend on what you are trying to do with it. Come up with some way of deciding on a family of realistic demand curves and then look at how much the monopoly price and production volume move around for that family of demand curves... for the family of linear demand curves there's two parameters so you get a three dimensional supply chart. Shouldn't take too long to draw it up.

[Nick Rowe]

Tel: let me change your example slightly. If any apples are to be produced at all, 50 bananas (per year) must be destroyed as an offering to the apple god.

Bmax stays the same (because if you produce zero apples you don't need to destroy any bananas), but the rest of the curve shifts left by 50 bananas everywhere. Yep. Basically the same as you say.

"Keep tracing that out and you get the sum of two curves, firstly the 1/x hyperbola as license costs are amortized over the revenue stream, and then the actual physical production cost which is an upward sloping curve as the marginal cost of each apple increases due to production factors."

The first is the AFC curve, and the second is the AVC curve. Their sum is the ATC curve. But what matters for profit-maximisation is the MC curve. And yes, it will slope up in this example. And the other curve we need to figure out the monopolist's profit-maximising output is the MR curve. (For a linear demand curve it has twice the slope of the demand curve.) Get thee to an intro textbook!

[Odie]

Thanks for that intro, Nick. I really appreciate your teaching posts for non-economists. I am wondering about one thing: Is not one of the underlying assumptions that land (capital) and labor are fully utilized? Thus, in order to increase output in one commodity the production of the other has to be reduced? (Not necessarily in linear proportions, of course, as shown by your bowed out curve.)

[Nick Rowe]

Odie: Thanks!

To answer your question: Yes, exactly. And that is an assumption that may be false. And one of the things that would make it false is bad monetary policy. See my previous post for a very short introduction to that.

[Tom Brown]

Thanks for the post Nick.

[rsj]

Nick,

1) It's not a rectangle, with IRS -- to keep things simple, say Labor = L + N, allocated to producing the two IRS goods c and d. Total Labor = 10 (to avoid the weirdness where squaring a small number makes it even smaller instead of bigger).

c = L^2
d = N^2.

Then if all resources are producing c, we can produce 100 units. If all produce d, we can produce 100 units. A 5/5allocation of labor gives you (25, 25). So what is this production frontier? It has two spikes at the axes, a distance of 100 units from the origin, and it bends inwards. It is a concave shape. *Any* line joining two points on our spiky curve will always lie *above* the PPF, which is another way of saying that the PPF is concave (in 2 dimensions).

2a) To see getting stuck, just compute the marginal change in utility of trying to move away from where you are -- it will be negative if you move in all directions. So in a 1 period model, you stay where you are. What about a 2 period model -- say no time discount. You don't want to take all the pain in period 1 and then be optimal in period 2. You want to spread the pain out a bit over both periods, which means you move a little closer in period 1 and are a little away from the equilibrium in period 2.

2b) With infinitely many periods and a discount (so that utility is finite), you approach, but never reach the optimal point.

Interestingly, the amount of inertia is proportional to how close you are to the optimal point. You know that because at the optimal point the first derivative is zero, so it's very painful to move away. It's like trying to walk towards a goal but you are stepping in quick sand that gets deeper as you approach your optimal point. On the other hand, if you are far from the optimal point, say at -infinite marginal utility at the edges, then you will still move towards the goal even though there are losses.

[rsj]

To see some simple math of getting stuck, assume that when labor is repurposed to producing a different good, it is only half as productive. Take a simple linear production function (on labor). With normal love of variety utility, say c^.5 + d^.5, you want equal amounts, but suppose your labor is allocated at (4, 6) instead of at (5, 5).

Your current production at (4,6) is 4 apples and 6 bananas. If you want to move epsilon labor away from bananas and into apples, you new production function is

4 + 1/2e apples and 6 - e bananas, meaning your increase in utility as a result of making the change is (4 + 1/2e)^.5 + (6-e)^.5 - 4^.5 - 6^.5 ~ -0.08t. And in fact you *always* lose utility (even globally), for any t. You are stuck at 4 apples and 6 bananas in the one period model, and can only asymptotically approach the (5,5) best point over multiple periods (the quicksand gets deeper the closer you get to (5,5).

[rsj]

^^ In the above it should read:

when labor is repurposed to producing a different good, it is only half as productive in that period.

And the simple differential is
(4 + 1/2e)^.5 + (6-e)^.5 - 4^.5 - 6^.5 =~ -0.08t. (e.g. is negative).

Here, it is actually globally negative, e.g. (4 + 1/2e)^.5 + (6-e)^.5 - 4^.5 - 6^.5 < 0 for 6 > t > 0.

But your intuition should be able to figure out that this is a general principle: in a region around the optimal point, if you are in that region, then you wont move towards the optimal point in a 1 period model. This is because at the optimal point, the first derivative is zero, so utility is changing very slowly with a change in t. This means you feel the loss of the adjustment cost the most. But that means you feel a big pain in adjusting from being dt away to the optimal point as well. That pain is less than the pain of adjusting from being 2*dt away to being only dt away. The pain increases as you move towards the optimal point, and so if you don't want to make a very small change now, you aren't going to want to make a bigger change and move even closer. Then go to a 2 period model, and ask how much utility are you allowed to lose now in exchange for optimal utility in the second period, etc.

[Nick Rowe]

rsj: "1) It's not a rectangle, with IRS"

In your example, you assumed all labour was identical.

In my example, with the rectangle production set, I assumed: "Take an extreme example, where the South land won't grow apples and the North land won't grow bananas." So there are two types of land.

[rsj]

I was saying that with IRS, you don't have a convex PPF anymore, and in 2 dimensions, it's concave. If you are talking about imposing adjustment costs to a CRS system, then yes, it remains convex but it pinched, with the extreme example being a rectangle (in my 1/2 as productive it is still like the first quadrant of a pentagon).

So
IRS <--> breaks convexity
Switching costs <--> breaks convergence to optimal point

They are two separate effects.

But in general, you don't need all outputs to be IRS in order to break convexity, you lose convexity as long as a single output is not CRS or DRS.

[rsj]

-Breaking convexity means you lose the guarantee that an equilibrium exists.
-Adding adjustment costs means that whether or not there is an equilibrium, you lose the guarantee that you can reach it in finite time (even with a social planner).

If both are realistic, and dropping PC is realistic, then you also lose your supply curve (according your definition) or have badly behaved supply curves (according to my definition).

At which point, the common case is looking really different, and it seems that students are being inculcated with intuition that only applies to the uncommon case.

[Nick Rowe]

rsj: Intro Economics students are taught that with IRS (we call it "natural monopoly") you don't get *competitive* firms, so you won't get *competitive* equilibrium. Instead you get: monopoly equilibrium; monopolistic competition equilibrium; or oligopoly equilibrium. We do 4 chapters, for each of the 4 cases.

You are still not getting my point about heterogeneity/multiple factors.

Yes, with one homogenous factor, IRS means the PPF is bowed in. I know that. I teach my students that.

And with 2 factors, or with heterogenous factors, and with CRS, the PPF is bowed out.

So if you have IRS, and 2 factors, or heterogenous factors, the PPF could be bowed either in or out (or both). It depends which effect is bigger.

[rsj]

I'd say in general it's not *either* bowed in or bowed out. Convexity is very strong. CRS is also very strong. I think even a single IRS will screw you up, and I don't think that the number of factors matters.

if you intersect a convex shape with a hyperplane, it's still convex. This allows you to reduce the case of 2 factors to 1 factor.

E.g. take p1 = L^0.1*K^0.9. Take p2 = L^0.2*K^0.9. 0

[rsj]

part of my post dropped!

p1 = L^0.1*K^0.9
p2 = L^0.2*K^0.9.

Slice along L = K. (call that t):

P1(t) = t
p2(t) = t^1.1

The sliced set is not convex, so the original set could not have been convex. It's the 1.1 that's getting you, and you really needed both factors to be CRS.