Saturday, June 28, 2014

Is the supply curve flat?


Answering the question in the title, no, but during the course of writing the past few posts, I'd looked at the wikipedia article on general equilibrium. I saw this random bit about Sraffa:
Anglo-American economists became more interested in general equilibrium in the late 1920s and 1930s after Piero Sraffa's demonstration that Marshallian economists cannot account for the forces thought to account for the upward-slope of the supply curve for a consumer good.
What follows is a just-so story mechanism about how changes in output should affect the price. It appears as though Sraffa's argument entirely ignores the premise of Marshallian supply and demand diagrams (there is a single good in a single market) by asserting that there are other goods and factors of production. The use of "first order" in the wikipedia explanation of the mechanism is also pretty laughable since Sraffa didn't include a single equation much less any scales which we could use to say that anything is "first order". I tried to read Sraffa but was instantly filled with a grim sense of philosophers talking about what change is. I then Googled around and found this, which helped a bit. I concluded that information theory is the proper way to deal with the entire situation.

What are we talking about when we draw supply and demand curves anyway? Let's go back to the beginnings of this blog. Supply (S) and demand (D) are related by the equation:

$$\text{(1) }P = \frac{dD}{dS} = \frac{1}{\kappa} \; \frac{D}{S} $$

where P is the price. The general solution (general equilibrium) to this is:

$$\text{(2) } \frac{D}{D_{ref}} = \left( \frac{S}{S_{ref}} \right)^{1/\kappa} $$

A demand curve is what you get when you look at the partial equilibrium by holding demand constant $D = D_{0}$ (an "exogenous" constant information source) and relating it back to the price to get a pair of equations:

$$ P = \frac{1}{\kappa} \; \frac{D_{0}}{\langle S \rangle}$$

$$ \Delta D \equiv D - D_{ref} = \frac{D_{0}}{\kappa}  \log \frac{\langle S \rangle}{S_{ref}} $$

Symmetrically, a supply curve is what you get when you look at the partial equilibrium by holding supply constant (an "exogenous" constant information destination)

$$ P = \frac{1}{\kappa} \; \frac{\langle D \rangle}{S_{0}}$$

$$ \Delta S \equiv S - S_{ref} = \kappa S_{0}  \log \frac{\langle D \rangle}{D_{ref}} $$

What are the angle brackets for? They're there to remind us that the variable is "endogenous" (the expected value inside the model) while the other variable is exogenous. These angle bracket variables are important to the discussion above because they parameterize our position along a supply or demand curve. Here are the supply (red) and demand (blue) curves along with the "fully endogenous" general equilibrium solution (gray dotted):


The demand curve seems to make intuitive sense -- at least at first. If the price goes up, the quantity of a good demanded goes down. But you get into trouble if you try this reasoning the other way: if the price goes down, the quantity demanded goes up? Maybe. Maybe not. If you weren't getting enough at the current price, it might. That depends on your utility, though. Sometimes this is referred to as the diminishing marginal utility of a good: the price you are willing to pay goes down the more widely available a good is. But there we've gone and switched up the independent variable again. The first half (effect of a price change) looks at price as independent, while the second half (diminishing marginal utility) looks at $\langle S \rangle$ as the independent variable [1]. 

Both of these get the partial equilibrium analysis in the wrong order mathematically. What we have is an exogenous (independent) change in demand. If demand increases and is satisfied (which we assume by taking $\langle S \rangle$ as endogenous/dependent), the price goes down.

This seems like a totally non-intuitive way to think about it. What is really going on here?

What we have is a system in contact with a "demand bath" [2] (or better yet, an "aggregate demand bath"). You could also call it an "information bath". If that bath didn't exist, adding (satisfied) demand would make the price go up, per equation (2) above (and it would move along the gray dotted curve in the figure above). What the bath is doing is sucking up any extra demand (source information) that we are creating by moving along the demand curve, so that "demand" is the same before and after the shift. Since there is "no change" in demand (any change is mitigated by the bath), the next variable in the chain, $\langle S \rangle$ effectively becomes the changing independent variable. This means the explanation is that decreasing/increasing the supply increases/decreases the price at constant demand (in the presence of a demand bath).

So why does the supply curve slope upwards? This time we're in contact with a "supply bath", so "the supply" is basically the same after we move along the curve. Moving along the supply curve is a change in $\langle D \rangle$. This means the explanation is that decreasing/increasing the demand decreases/increases the price at constant supply (in the presence of a supply bath).

Therefore there is no actual change in the supply along a supply curve so there is no bidding up factors of production or lack thereof per Sraffa. What we're doing is increasing the demand, so the price goes up.

That is to say supply and demand curves are kind of misnomers. A supply curve is the behavior of the price at constant supply, but is parameterized by increasing or decreasing demand. A demand curve is the behavior of the price at constant demand, but is parameterized by increasing or decreasing supply. [3]

[1] Yes, economists take price to be the independent variable, but in the formulation above it is more natural that either supply or demand (or both) are the independent variable(s). The price is the derivative of the demand with respect to the supply (the marginal change in demand for a marginal change in supply).

[2] This whole description is based on an isothermal expansion/compression of an ideal gas in contact with a thermal bath.

[3] Shifts "of" the supply and demand curves (as opposed to shift "along") are effectively changes in the information bath

18 comments:

  1. What's the difference between p and P? (from the link to the post you mention is from the beginning of this blog)

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    1. There is no particular difference. I tend to use P for the price level and p for a generic price but sometimes, like in this post, P is a generic price.

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  2. Jason, this was helpful, but I'm not there yet. I'll have to look up your thermodynamic references I think.

    Let me ask you about this sentence about the downward sloping demand curve:

    "This means the explanation is that decreasing/increasing the supply increases/decreases the price at constant demand (in the presence of a demand bath)."

    Could we rewrite that as:

    "This means the explanation is that decreasing/increasing < S > increases/decreases the price at constant demand (in the presence of a demand bath)."

    or again as:

    "This means the explanation is that decreasing/increasing the expected quantity supplied increases/decreases the price at constant demand (i.e along a fixed demand curve) (in the presence of a demand bath)."

    And of course this implies similar rewrites for the similar sentence describing the upward sloping supply curve.

    I remember that Nick Rowe taught me to always think of demand and supply as curves, NOT quantities. So he always corrected me when I used "supply" when "quantity supplied" was what I really meant, or vice versa (and similarly for demand).

    So do you see where I inserted "quantity supplied" above? Do you agree that's an acceptable way to phrase it, or not?

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  3. Also, what's the significance of the point on the red upward sloping supply curve labeled < D > (expected value of quantity demanded?) at delta_S = -0.1 and the corresponding point labeled < S > (expected value of the quantity supplied?) on the blue downward sloping demand curve at delta_D = -0.1? These values cannot be read off the y-axis (P), right? So what do they signify?

    Also, for the gray general equilibrium curve, you have (in black lettering) written next to it:

    P = (1/kappa) * ((delta_S + Sref)/Sref)^(1/kappa - 1)

    Comparing Eq. 1 and 2. though, that's not what I get. Starting with Eq. 2:

    D/Dref = (S/Sref)^(1/kappa)

    P = (1/kappa)*D/S = (Dref/kappa)*(S^(1/kappa - 1))/(Sref^(1/kappa))
    = (Dref/kappa)*((delta_S + S_ref)^(1/kappa - 1))/(Sref^(1/kappa))

    Or alternatively:

    P = (1/kappa)*D/S = (1/kappa)*D/(((D/Dref)^k)*Sref)
    = ((Dref^k)/(kappa*Sref)*(D^(1-k)) = ((Dref^k)/(kappa*Sref)*((delta_D + Dref)^(1-k))

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  4. it seems that you just assumed constant supply and constant demand that get parametarised and changed the meaning of demand curves supply curves to ''That is to say supply and demand curves are kind of misnomers. A supply curve is the behavior of the price at constant supply, but is parameterized by increasing or decreasing demand. A demand curve is the behavior of the price at constant demand, but is parameterized by increasing or decreasing supply. '' and you say ''Therefore there is no actual change in the supply along a supply curve so there is no bidding up factors of production or lack thereof per Sraffa.'' YES becuase you assumed a supply curve to be just constant supply parametarised by demand '' the behavior of the price at constant supply''why would you assume constant supply in the first place and justify it as an ''information bath'' you didnt refute sraffas argument you just literally redefined the meaning of demand and supply curves and assume constant supply and constant demand and their parametirization of them seperatly be demand and supply as the supply and demand curves which is not what demand and supply curves ARE IN THE FIRST PLACE you just assume them to be that way by redifining their meaning so that when talking about demand and supply curves sraffas argument wont exist but you havent justified why you would assume the supply curve to be a ''supply bath'' constant supply parametirised by demand in the first place and change the entire meaning of the supply and demand curve and talking about what actually upward slopping market demand curve was originally meant marshall didnt stay faithfull to the original use either so you accusing sraffa of misrpresenting marshalls original use makes no sense since its based on misrepresentation in the first place

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    1. The entire exercise here was effectively redefining supply and demand curves using information theory, so I don't understand your point. I know I'm defining a new theory therefore "you're defining a new theory" is not really a criticism but rather a statement of fact.

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    2. well the thing is a problem arises cause you are in asituation where you have to say that constant supply or a supply bath is parameterized the constant supply is a point of stability however in real life there mist demand curve sin hevay industry are already flat with constant costs and constant supply short term and they are already parametrised by demand so ultimately sraffas theory could be trslated to that demand parametriation resulted in constant supply and constant costs through compettition the demand parametrisation in that case isnt zero yet the supply is still constant and there are still constant costs unlearning economics linked a report which shows exactly that which would make that statement . ''A supply curve is the behavior of the price at constant supply, but is parameterized by increasing or decreasing demand. A demand curve is the behavior of the price at constant demand, but is parameterized by increasing or decreasing supply'' how themn in real life demand parametrisation that wasnt zero resulted in constant supply and constant costs when demand parametrisation of constant supply or supply bath that isnt zero shouldnt result in constant supply but besides that the point is at this point you are talking about demand and supply curves at all constant supply or demand information baths getting parametrsised by demand or supply are not demand curves or supply curves at all so its not a reinterpretation or a new theory its a theory taht literally doesnt have suuply or demand curves but is literally something else so why would you define it as demand curves or supply curves and hwy would you calima that somehow sraffa ignores the premise of marshalian digrams when you dont use supply curves or demand curves at all that appears as a marshalian diagram but its not cuase parametrsied information baths arent demand and supply curves they are just information baths so why even name them that and there is also the problem i mentioned which i mentioned in the original post as well i mean when i hear you calim at the beggining of your article that sraffa misrepresents the marshalian diagarm i would expect you to talk about the supply curves and demand curves they were talking about

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    3. : “The overwhelmingly bad news here (for economic theory) is that,
      apparently, only 11 percent of GDP is produced under conditions of rising marginal cost…
      Firms report having very high fixed costs—roughly 40 percent of total costs on average. And many more
      companies state that they have falling, rather than rising, marginal cost curves. While there are reasons to
      wonder whether respondents interpreted these questions about costs correctly, their answers paint an image
      of the cost structure of the typical firm that is very different from the one immortalized in textbooks.” (105)
      (Blinder 1998, pp. 102, 105). See also Reynolds (1987, pp. 53-62).

      Delete
    4. In the information equilibrium model, the supply curve can have almost any slope (it can be flat, but doesn't have to be).

      Any description of supply and demand has to be consistent with information theory: there are two distributions of demand and supply which have information entropy. And that information entropy must be equal if demand meets supply.

      Sraffa' arguments represent additional restrictions on the form of supply and demand beyond these information-theoretic ones that do not appear to create a theory that matches empirical data and are therefore unnecessary.

      Delete
    5. “The overwhelmingly bad news here (for economic theory) is that,
      apparently, only 11 percent of GDP is produced under conditions of rising marginal cost…
      Firms report having very high fixed costs—roughly 40 percent of total costs on average. And many more
      companies state that they have falling, rather than rising, marginal cost curves. While there are reasons to
      wonder whether respondents interpreted these questions about costs correctly, their answers paint an image
      of the cost structure of the typical firm that is very different from the one immortalized in textbooks.” (105)
      (Blinder 1998, pp. 102, 105). See also Reynolds (1987, pp. 53-62). it can actually 40precent flat supply curves and sraffaa didnt say that all supply curves are flat he said given industries that have spare capacity that is the case and if anything sraffa wanted to show exactly that supply curves arent mostly upwards from blinders data only 11 precent rising marginal cost and 40 precent fixed constant cost 40 precent flat supply curves so actually it seems that it matches blinders data under conditions of price wars law of one price profit rate equlisation which is the 11 precent rising marginal cost that is the case as sraffa himslef says but for the most firms taht have big enough market share to have spare capacity and not participate in price wars as much there is a 40 precent fuxed costs constant costs flat supply curve so it seems that it does muatch the data thats why im saying this

      Delete
  5. https://books.google.gr/books?id=CMffCgAAQBAJ&pg=PA69&lpg=PA69&dq=smith+supply+curve+sraffa&source=bl&ots=HhNjAgYA9N&sig=F3cvc1LUKURGKCi9gAgWYrq35NE&hl=el&sa=X&ved=0ahUKEwjVmefUu6zWAhWHvBQKHTDcAQoQ6AEIOjAC#v=onepage&q=smith%20supply%20curve%20sraffa&f=false sraffas argument is that constant costs that give the falt curve is a supply curve that is already parametarised by demand in the real world there exist flat supply curves in heavy industry empirically so that the flat supply curve is a supply curve parmeterised by demand at constant costs sraffa does not assume constant supply he says in the case of constant costs this will be the supply curve which results in cosntant supply in the real world there are flat supply curves which they have been already parametirised by demand in heavy industry they still result in constant costs giving the falt supply curve so defining the supply curve that way wouldnt amke any sense but you see it becomes that way when you have already ssumed this ''Therefore there is no actual change in the supply along a supply curve so there is no bidding up factors of production or lack thereof per Sraffa. What we're doing is increasing the demand, so the price goes up.''see the upward slopping supply curve is literally the factors of production causing increasing marginal cost resulting in an increasing supply curve thats what the upward slopping supply curve is what you have assumed is that the supply curve is not that but rather a ''supply bath'' which is parametarised by demand but that is not a supply curve as i have already described since a supply curve is not parametarised by demand but is caused by the rising marginal cost not demand influencing the price
    thats not the definition of the supply curve that sraffa is objecting to infact that ''supply bath'' couldnt even be called supply in the marshallian sense since in the marshalian sense or any definition of the supply curve the the marginal cost made by the bidding factors of production make the curve in the first place NOT THE PARAMETIRISATION OF IT BY DEMAND you literally redifined the supply curve in a way that srffa or marshallk never refered to and made this argument

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    Replies
    1. I know I literally redefined supply and demand curves. That was the point: to redefine them using information theory. The interpretation turns out to be different from Sraffa or Marshall, but given both were operating before the existence of information theory, that's to be expected, no?

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    2. well the thing is a problem arises cause you are in asituation where you have to say that constant supply or a supply bath is parameterized the constant supply is a point of stability however in real life there mist demand curve sin hevay industry are already flat with constant costs and constant supply short term and they are already parametrised by demand so ultimately sraffas theory could be trslated to that demand parametriation resulted in constant supply and constant costs through compettition the demand parametrisation in that case isnt zero yet the supply is still constant and there are still constant costs unlearning economics linked a report which shows exactly that which would make that statement . ''A supply curve is the behavior of the price at constant supply, but is parameterized by increasing or decreasing demand. A demand curve is the behavior of the price at constant demand, but is parameterized by increasing or decreasing supply'' how themn in real life demand parametrisation that wasnt zero resulted in constant supply and constant costs when demand parametrisation of constant supply or supply bath that isnt zero shouldnt result in constant supply but besides that the point is at this point you are talking about demand and supply curves at all constant supply or demand information baths getting parametrsised by demand or supply are not demand curves or supply curves at all so its not a reinterpretation or a new theory its a theory taht literally doesnt have suuply or demand curves but is literally something else so why would you define it as demand curves or supply curves and hwy would you calima that somehow sraffa ignores the premise of marshalian digrams when you dont use supply curves or demand curves at all that appears as a marshalian diagram but its not cuase parametrsied information baths arent demand and supply curves they are just information baths so why even name them that and there is also the problem i mentioned which i mentioned in the original post as well i mean when i hear you calim at the beggining of your article that sraffa misrepresents the marshalian diagarm i would expect you to talk about the supply curves and demand curves they were talking about

      Delete
    3. : “The overwhelmingly bad news here (for economic theory) is that,
      apparently, only 11 percent of GDP is produced under conditions of rising marginal cost…
      Firms report having very high fixed costs—roughly 40 percent of total costs on average. And many more
      companies state that they have falling, rather than rising, marginal cost curves. While there are reasons to
      wonder whether respondents interpreted these questions about costs correctly, their answers paint an image
      of the cost structure of the typical firm that is very different from the one immortalized in textbooks.” (105)
      (Blinder 1998, pp. 102, 105). See also Reynolds (1987, pp. 53-62).

      Delete
  6. my main disgreement would be that thata not a refutation to sraffa its just a new theory of parametrise information baths which wouldnt be supply curves or demand curve sin any sense but you would just define them that way for consistency

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    Replies
    1. On its own, it's not a refutation of Sraffa, but in combination with empirical data it becomes one. If Sraffa is inconsistent with information equilibrium and information equilibrium is consistent with data, then Sraffa must be incorrect.

      And as I've shown elsewhere on the rest of this blog, the information equilibrium approach is a good explanation of empirical data (whereas Sraffa is not, and mostly can't even be compared to data because it's prose, not equations)

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    2. This comment has been removed by the author.

      Delete
    3. “The overwhelmingly bad news here (for economic theory) is that,
      apparently, only 11 percent of GDP is produced under conditions of rising marginal cost…
      Firms report having very high fixed costs—roughly 40 percent of total costs on average. And many more
      companies state that they have falling, rather than rising, marginal cost curves. While there are reasons to
      wonder whether respondents interpreted these questions about costs correctly, their answers paint an image
      of the cost structure of the typical firm that is very different from the one immortalized in textbooks.” (105)
      (Blinder 1998, pp. 102, 105). See also Reynolds (1987, pp. 53-62). it can actually 40precent flat supply curves and sraffaa didnt say that all supply curves are flat he said given industries that have spare capacity that is the case and if anything sraffa wanted to show exactly that supply curves arent mostly upwards from blinders data only 11 precent rising marginal cost and 40 precent fixed constant cost 40 precent flat supply curves so actually it seems that it matches blinders data under conditions of price wars law of one price profit rate equlisation which is the 11 precent rising marginal cost that is the case as sraffa himslef says but for the most firms taht have big enough market share to have spare capacity and not participate in price wars as much there is a 40 precent fuxed costs constant costs flat supply curve so it seems that it does muatch the data thats why im saying this thanks for the responce though

      Delete

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