The Price of a Pooch — Part Two: The Cobweb Model
A little while ago we reflected on the rapid rise in the price of dogs during the lockdown — a development explained in terms of an outward shift in the demand curve for dogs coming up against an inelastic supply of dogs, largely determined by the existing supply of puppies. At the end of the piece we wondered what would happen to the price of dogs next year? There is an aspect of this question which especially interests economists: the impact of TIME LAGS in the production of puppies. Usually in basic economics we do not include time as a variable in our system. But when we do the effect can be significant — even explosive!
The supply of puppies cannot be varied at the whim of breeders. There is a biological cycle that cannot be avoided. This means that the number of puppies to buy in the market NOW is determined by decisions taken by breeders six months or so before. Hence the supply now is determined by the price and profitability of dog breeding six months ago. To model this we assume that the supply of puppies now, at time t, is Perfectly Inelastic and determined by the supply decision taken six months ago, that is t-1. The result, as we have seen, is that when demand shifts outwards, supply cannot respond at time t and the result is a big increase in the price of puppies. Dog breeders, seeing how high prices have risen for dogs, will take steps to increase the supply of puppies as they will expect high prices and profits. As a result, from six months onwards there will be a big surge in the supply of puppies in the market at the existing market price. When supply exceeds demand price can only FALL — the result is a collapse in puppy prices due to the surge in supply. Now breeders find themselves struggling to shift puppies at low prices. How do they respond? Obviously they will cut back the supply of puppies in the next breeding period. So over the next year supply suddenly falls and there will be a shortage of puppies for sale. So prices will rise! And as prices rise breeder will increase supply and after half a year or so the supply of puppies will exceed supply and price will fall and back and forth we seem set to go, with the price of dogs swinging widely up and down during alternate breeding cycles. It was exactly this kind of phenomenon that economists detected in the market for pigs and why this theory of fluctuating prices became known as the hog cycle.
Will it go on forever? This depends on the Elasticity of Supply of the product. If the supply is relatively INELASTIC the price changes will get smaller and smaller and converge to a stable equilibrium. But if the supply of dogs is ELASTIC the price swings can explode to infinity. These two cases of COBWEB THEORY are illustrated below.
First, a CONVERGING COBWEB. In the diagram below the Demand for dogs depends upon the current price of dogs at time t, i.e. Dt = f(Pt). However the Supply of dogs depends on the price in the last time period when breeding decisions were taken, i.e. St = f(Pt-1). Supply is then fixed at the current time (time t). In the diagram this is shown by the initial quantity supplied Q and the initial demand curve Dt, yielding an equilibrium price for dogs of OP. Demand then shifts out from D to D1, e.g. due to the lockdown pet effect. In the immediate period the supply of dogs is fixed at OQ. The supply of dogs is therefore Perfectly Inelastic at OQ and the effect of the increase in demand is to cause prices to increase rapidly to P1. This increase in price encourages dog breeders to rear more dogs, so in six months or so time the supply expands to OQ1, as shown by the supply curve St. Supply now exceeds demand at price P1 and the price will fall. But supply is now fixed in the short run at OQ1, and hence price has to fall to P2 to clear the market. In the next time period dog breeders react to the lower price by cutting supply to Q2. Excess demand for dogs now exists and prices rise to OP3. Price continues to adjust up and down in this way until a new long-run equilibrium is achieved at OPe and OQe. Tracing the path to equilibrium yields a cobweb pattern. In this example the price fluctuations get steadily smaller until the new equilibrium is reached. It is therefore described as a converging cobweb model.
However, if we assume that supply adjustments are more price-elastic then we obtain a diverging cobweb model in which price-swings get larger and larger and no long-term equilibrium is ever reached.
In the above diagram the increase in demand from Dt to D1 causes price to rise from OP to OP1 initially, so that supply expands significantly Oq1 in the following time period. This large increase in supply causes price to then drop all the way down to OP2, with the result that dog breeders scale back production significantly in the next period to just Oq2. Supply now falls well short of demand at price P2, causing price to leap up to P3, which only causes producers to supply even more dogs in the next period. The fluctuations in price and quantity get larger and larger and, as can be seen, a stable equilibrium price never emerges.
Cobweb theory therefore suggests that prices in the dog and pet market are unlikely to be stable for some time to come due to the time-lags inherent in the supply of live animals. While price changes are unlikely to explode, we can predict that the ripple effect of this year’s surge in demand for dogs will be making itself felt in the pet market for some time to come.
We can see that this model applies to markets where production is subject to the passage of time such as in agricultural markets — and the model is used to explain why and how prices in these markets fluctuate, often to the determinant of farmers whose incomes are, as a result, highly unstable.
By Dr Ian St John.
