Wednesday, February 25, 2015

The cobweb model


In writing the posts about expectations I stumbled across the cobweb model of how expectations can impact prices in the short run, creating periodic fluctuations and volatility. The basic idea is that there are two possibilities for a series of price adjustments, convergent and divergent [from Wikipedia]


If we take the price elasticity conditions $e^{s} < |e^{d}|$ for convergent and $e^{s} > |e^{d}|$ and use the information transfer model for the elasticities (convergent case):

$$
\frac{\kappa Q_{0}^{s}}{Q_{ref}^{s}} \lt \left| - \frac{Q_{0}^{d}}{\kappa Q_{ref}^{d}} \right|
$$

If we assume an equilibrium market price $P_{0}$, then we can say the information source $Q_{0}^{d}$ and destination $Q_{0}^{s}$ are equivalent as well as the initial conditions ($Q_{ref}^{x}$)

$$
Q_{0}^{d} = \kappa P_{0} Q_{0}^{s}
$$

$$
Q_{ref}^{d} = \kappa P_{0} Q_{ref}^{s}
$$

we obtain the conditions:

$$
\frac{\kappa Q_{0}^{s}}{Q_{ref}^{s}} \lt \frac{\kappa P_{0} Q_{0}^{s}}{\kappa^{2} P_{0} Q_{ref}^{s}}
$$

$$
\rightarrow \;\;\; \kappa \lt \frac{1}{\kappa}
$$

$$
\rightarrow \;\;\; \kappa^{2} \lt 1
$$

for the convergent case, and analogously

$$
\kappa^{2} \gt 1
$$

for the divergent case. Basically the convergence and divergence is set by the value of $\kappa$.

11 comments:

  1. This comment has been removed by a blog administrator.

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    1. I don't know what happened with the formatting there. You used to be able to delete comments in BlogSpot but I can't see how to delete this one. I presume that you have the power to delete comments. I will reformat my comment source and then try to post again.

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    2. I've had some trouble with Firefox lately not recognizing that I was the author of a comment for blogspot (and hence I can't delete or even post under my name) ... other browsers (chrome, safari and IE) seem to work fine.

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  2. This comment has been removed by a blog administrator.

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    1. Sorry same problem again

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    2. The issue is that Mathjax (the math rendering javascript) is set up to use dollar signs as LaTeX delimiters (weird for an economics blog, I know). I fished your comment out and replaced the dollars with Euros ...

      Begin quote:

      ...

      This is very interesting (although I don’t understand the mathematics). I had not heard of the cobweb model. Here are a couple of comments on how I think about this at the micro level.

      First, economists are obsessed with price. Businesses are not obsessed with price. They are obsessed with

      Profit = Revenue - Expenditure

      Both revenue and expenditure are measured in terms of volume * price.

      Suppose my kettle manufacturing business (see previous comment on Expectations post) forecasts that there will be a demand for 50 kettles at a unit price of €30. It also expects that it can manufacture 50 kettles at a unit cost of €20.

      Expected profit = (50 * €30) - (50 * €20) = €1500 - €1000 = €500

      Now suppose that it commits to its plan but then finds that there is demand for only 40 kettles but also that the actual price is €40. Perhaps, people think that the business’s kettles are higher quality than the business expected so are willing to pay a higher price.

      Actual profit = (40 * €40) - (50 * €20) = €1600 - €1000 = €600

      This is a better outcome than the plan even though the expected price and expected volume were both wrong. The business also has an inventory of 10 unsold kettles in which it has invested €20 each, a total of €200. It can then sell these in the following period raising further revenue with no further expenditure. It will also adjust its expected price for the following period to €40.

      I’m not an expert in supermarket business models but, as far as I can see, their model is to supply more food that will be demanded and then throw away the remainder. They do this by increasing the price.

      For example, suppose a supermarket buys 100 bananas at €1 each and then sells 90 bananas at €1.2 and throws away the other 10 bananas.

      Actual profit = (90 * €1.2) - (100 * €1) = €108 - €100 = €8

      This is a good outcome (although, of course, selling 100 bananas at €1.2 would be better). The supermarket’s expectations are fulfilled even though 10 bananas are wasted. This demonstrates another aspect of expectations. The supermarket expects that 10% wastage will still allow it to compete with other supermarkets. Economists never talk about this type of expectations but that’s because economists don’t bother to observe the real world.

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    3. I would agree -- businesses are trying to optimize profits, which makes their utility decisions a lot more complicated than is typically modelled ... hence why I try to stay away from looking at the decisions of individual agents and firms!

      Also, according to economists, profit is zero in perfect competition with no barriers to entry ... the price of a good becomes its marginal unit cost of production :)

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  3. Second, I have been thinking about the accounting identity

    Consumption (C) + Investment (I) = Consumption (C) + Saving (S)

    This identity is true only at the macro level. However, we can discuss the terms in the identity at the micro level.

    Suppose that my kettle business makes a forecast based on its expectations. It then engages with the world. If we assume that expectations of unit costs and unit prices are accurate, then there are three possible outcomes.

    Case 1: The actual demand is consistent with the forecast. Then we have

    C (sales by business) = C (purchases by customers) = Expected demand

    Case 2: The actual demand is less than the expected demand. We still have C = C for those kettles which were sold. However, as the demand is less than the supply, the business will be left with a surplus inventory of kettles. In terms of the identity, this is a surplus in Investment.

    Case 3: The actual demand is more than the expected demand. We still have C = C for those kettles which were sold. However, the demand is more than the supply. As a result, customers will have set aside an amount of funds from their income to pay for the demanded number of kettles but, as some of this demand was unfulfilled, some of the customers will be left will unspent funds. In terms, of the identity, this is a surplus in Saving.

    Only one of these cases will apply for my kettle business in each period. However, the identity holds at the macro level. This tells us that, in the absence of an imbalance between voluntary business investment and voluntary customer saving, total unintended business investment (excess inventory) in some markets matches total unintended customer saving (unspent funds) due to an inability to buy products in other markets.

    Based on this example, from what I can see, it is in this sense that I = S is important. I = S here is a problem with matching unsold inventory with unspent money. If businesses are holding surplus inventory then they will not manufacture more product until the inventory is reduced so they will cut back. If many businesses cut back at the same time then we get a recession. Customers then panic by saving voluntarily which makes things worse.

    Note that, when economists discuss this they talk as though I causes S or S causes I. However, in my example, it is the mismatches in expectations which cause both I and S.

    Note also that, just to be perfectly clear, I am talking ONLY about UNINTENDED I and S due to mismatches in supply and demand. Businesses can also invest intentionally and customers can also save intentionally.

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    1. That is interesting -- a kind of forced investment/saving from expectations not meeting reality.

      You've probably seen David Glasner's take on the accounting identity:

      http://uneasymoney.com/2015/02/17/savings-and-investment-arent-the-same-thing-and-theres-no-good-reason-to-define-them-as-such/

      I've never really stuck my head into the S = I discussion or how each works ... I am not sure of how one should partition NGDP, besides a piece that unequivocably could be called G (all NGDP transactions with the government on one end).

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  4. Finally, I previously mentioned Post Keynesian economics and promised to provide some background and references from my non-economist perspective. I still owe you that. However, note here that the cobweb model was developed by Nicholas Kaldor who was a Post Keynesian. I don’t think this is an accident. It’s caused by the (mostly unspoken) fundamental mental models used by different economists in thinking about economics. It’s the different fundamental mental models which cause the incompatible schools of thought. Even though I’m not an economist, I know that, if I were, I would be a Post Keynesian.

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    1. I am pretty sure I wouldn't fit in any of the schools -- I effectively say a low inflation economy is "Keynesian", while a high inflation country is "Monetarist" ...

      I think part of the reason macro schools exist is because the macroeconomy doesn't nicely project onto the different school bases:

      http://informationtransfereconomics.blogspot.com/2013/09/an-information-transfer-history-of.html

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Also, try to avoid the use of dollar signs as they interfere with my setup of mathjax. I left it set up that way because I think this is funny for an economics blog. You can use € or £ instead.