Abstract
Demand and supply are the fundament for almost all economic reasoning. The concepts are based on the idea that utility and scarcity jointly determine the value of commodities. The idea may go back as far as to Greek antiquity and Plato who wrote about almost everything. In this Chapter we attempt to collect utility functions which are useful for global dynamic systems. Particularly, we emphasize the usefulness of Lancaster’s theory where it is not the marketed commodities that are entered in the utility functions, but their “properties” assumed to be measurable scores. This is a seldom used approach, which, however, gives us a unique possibility to model commodities that are close substitutes.
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Notes
- 1.
This posthumous work is a real treasure, though very extensive with its 1260 pages in small print. It is also a bit fragmentary, which is explained by the fact that Schumpeter worked with it almost until his death, after which his collaborators continued using his notes. We will refer to this great work repeatedly, though reading from cover to cover takes a bit of patience. Fortunately, the book has an excellent index for topics and authors, so it is easy to find specific information.
- 2.
By the way, today’s inhabitants of metropolis hardly agree that fresh air is a free commodity, and poor farmers in their daily fight for fresh water hardly agree about fresh water either. The latter is not historically new. European cities still around 1700 were deficient in water supply, which was replaced by an enormous consumption of alcohol.
- 3.
One can still say hello to Bentham at University College in London, where, according to his will, his skeleton sits on a chair in a special wooden cabinet, in his own clothing stuffed with hay. Unfortunately, the preservation of his head went wrong, so it is replaced by wax.
- 4.
Menger (1871).
- 5.
Jevons (1871).
- 6.
For instance considering complements and substitutes among consumers’ goods, one looks at the sign of the derivative of demand with respect to changes of price of another good. However, if we want to find a formula for a demand function that holds while circumstances change during a dynamic process, we find very little, especially if we want to portray groups of commodities as substitutes (goods that more or less satisfy the same needs) or complements (such that preferrably are consumed in combination).
- 7.
Economics has a generally accepted code of symbols which greatly simplifies understanding of a model: p for prices, q for quantities, k for capital, l for labour, r for capital rent, w for wage rate, i for interest rate, Y for income, I for investment, S for saving, etc. This is very convenient for fast recognition of a problem.
- 8.
This is totally symmetric. It can easily be given a more general shape through linear transformations q i = a i x i
$$\displaystyle \begin{aligned} U=2a_{1}x_{1}+2a_{2}x_{2}-a_{1}^{2}x_{1}^{2}-a_{2}^{2}x_{2}^{2} \end{aligned}$$though there is no point in complicating things. Economists some times have the bad habit of complicating things to make a general appearance, instead of stripping cases one wants to make to the essentials.
- 9.
It would be more appropriate to write
$$\displaystyle \begin{aligned} p_{1}q_{1}+p_{2}q_{2}\leq y \end{aligned}$$i.e., expenditures may not exceed the available budget, but as long as consumption needs are not saturated we may use the equality. Though, particularly with the example of the paraboloid, it has a global maximum, beyond which greater expenditure decreases utility.
- 10.
Cobb and Douglas (1928).
- 11.
This is due to the assumed symmetry as we took α 1 = α 2 = 0.5 . Other exponents result in other income shares, for instance α 1 = 1, α 2 = 1.5 in the shares \(\frac {1}{1+1.5}=0.4,\frac {1.5}{1+1.5} =0.6\), so \(q_{1}=0.4\frac {y}{p_{1}}\), \(q_{1}=0.6\frac {y}{p_{1}}\).
- 12.
Later we will see how messy such group aggregation becomes if the demand functions were linear.
- 13.
This is an idea the present author once got when supervising a business economics student. It seems to be easy to set up oligopoly models, though, admittedly, it is difficult to assume nonzero production costs as well. One may argue that in oligopoly theory variable production costs were put equal to zero in order to keep the model sufficiently simple, but the defence is not entirely convincing. The idea remains largely unexplored.
- 14.
- 15.
Different brands of, say, cars can be characterized through their properties: Engine power, safety, comfort. If we can quantify these characteristics, the marketed goods can be represented by vectors of such properties. Just these vectors can be used to identify similar commodities, i.e., close substitutes. Once we can put value tags on the properties, we can even compare different designs.
- 16.
This also opens a wider perspective—as we get shadow prices for the properties, we can also extend models to choose design for the competing products. Hopefully, in a global dynamics, we could then not only find out the orbits of changing transaction and commodity prices, but also study the design of the competing products. It would be interesting to know if the competitors would asymptotically converge upon the same design, or, rather choose very different designs. Other attractors than fixed points, for instance periodic, would be even more interesting, as one could then model fashion cycles.
- 17.
Economists as a rule have the habit of including as many parameters as possible, probably to attain some kind of “generality”. In this book, we rather use the custom from physics to only include parameters that individually influence something in a model. For instance the budget constraint
$$\displaystyle \begin{aligned} p_{1}q_{1}+p_{2}q_{2}=y \end{aligned}$$can be written
$$\displaystyle \begin{aligned} \frac{p_{1}}{y}q_{1}+\frac{p_{2}}{y}q_{2}=1 \end{aligned}$$so income is just a scaling factor for prices.
- 18.
The budget constraint also needs a Lagrange multiplier, but it has no significance for the model and can be absorbed as a factor in λ 1, λ 2.
- 19.
They represent, for each property, the ratios to which the two products of marketed goods contain these in their design. Note that b 1, b 2 depend on design decisions by both competitors, and also that they are defined in an asymmetric way.
- 20.
Of course, only restricted price ratios provide for positive demand of both commodities. Most succinctly the condition can be expressed in only two compound variables, \(b_{1}\frac {p_{2}}{p_{1}}\) and \(b_{2}\frac {p_{1}}{p_{2}}\) ,
$$\displaystyle \begin{aligned} -2<\frac{\left( b_{1}\frac{p_{2}}{p_{1}}-1\right) +\left( \frac{1}{b_{2}} \frac{p_{2}}{p_{1}}-1\right) }{(b_{1}\frac{p_{2}}{p_{1}}-1)\text{ \textperiodcentered }(\frac{1}{b_{2}}\frac{p_{2}}{p_{1}}-1)}<0 \end{aligned}$$In this interval the commodities are substitutes. If any of the demand equations is negative, then only one commodity is demanded, so \(q_{1}=\frac {1 }{p_{1}},q_{2}=0,\) or \(q_{1}=0,q_{2}=\frac {1}{p_{2}}\), the awkward case commented above.
References
Benthnam J (1789) An introduction to the principles of morals and legislation. Prometheus Books, Amherst
Cobb CW, Douglas PH (1928) A theory of production. Am Econ Rev 18:139–165
Jevons WS (1871) The theory of political economy. Palgrave Macmillan, Basingstoke
Lancaster KJ (1966) A new approach to consumer demand. J Polit Econ 74:132–157
Lancaster KJ (1971) Consumer demand: a new approach. Columbia University Press, New York
Menger C (1871) Grundsätze der Volkswirtschaftslehre. Braumüller, Vienna
Schumpeter JA (1954) History of economic analysis. Allen & Unwin, London
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Puu, T. (2018). Utility and Demand. In: Disequilibrium Economics. Springer, Cham. https://doi.org/10.1007/978-3-319-74415-5_1
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DOI: https://doi.org/10.1007/978-3-319-74415-5_1
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