# Some properties of price-consumption curves and income-consumption curves

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## Summary

The theory of indifference maps is a static theory for consumer’s behaviour. As an illustration of this theory we examine in this note the demand reactions of an individual consumer in three situations. Given the prices*p*_{1} …,*p*_{n} and his income μ, let Q=(*q*_{1}, …,*q*_{n}) be the consumer’s optimal budget. Now in the first place, if one price is allowed to vary, say*p*_{i}, while the income and the other prices are held fixed, the optimal budget will describe a curve known as a*price-consumption curve*. Making*i*=1, …,*n* we obtain*n* such curves passing through Q, say A_{1}, …, A_{n}. Second, consider an inflatory situation where all prices vary in proportion, while income μ is constant. In the staic theory of indifference maps this is equivalent to allowing μ to vary while all prices are held fixed. Thus the optimal budget will describe what is known as the*income-consumption curve* which passes through Q, say C. Third, suppose in the inflatory situation that one of the prices, say p_{i}, is held fixed. The optimal budget will then describe a certain curve, say B_{1}; this will be called a*price-consumption curve of type* B. There are*n* such curves which pass through Q, say B_{1}, …, B_{n}.

The price-consumption curves A_{1} and B_{i} and the income-consumption curve C form a convenient geometric representation of the demand functions of the consumer.

The equations of the curves A_{i}, B_{i}, C are given in no. 1. Certain simple interrelations between these curves are pointed out in no. 2. In no. 3, finally, we examine the orientation of the curves A_{i}; anaiogous results are obtained for the curves B_{i}.

The results of no. 2 are as follows. In the case of two goods,*n*=2, the*price-consumption curves* A_{1} and B_{2} will coincide, and likewise A_{2} and B_{1}. Considering the passage through Q for falling (rising) prices, the three curves A_{1}=B_{2}, A_{2}=B_{1} and C will pass through the budget line*p*_{1}q_{r}+p_{2}q_{2}=μ from below (above). In this passage, C is intermediate between A_{1}=B_{2} and A_{2}=B_{1}. In the case of*n* goods, all of the*2n* curves A_{i}, B_{i} will in general be different. It is a rather typical situation that a curve A_{i} will be more or less perpendicular to the corresponding curve B_{i}. In the passage through Q for falling (rising) prices, the 2n+1 curves A_{i}, B_{i}, C will pass through the budget plane σ*p*_{i}q_{i}=μ from below (above). On the upper side of the budget plane, the curves A_{i} will form the edges of a corner at Q, say Λ. Considering the passage of the curve C through the point Q. it is shown that this passage will always take place within the corner Λ. A similar theorem holds for the curves B_{i} and C.

In no. 3 the geometric interpratation of demand functions is carried further; this leads to a theorem on what is known as the substitution effect. As an immediate corollary, the theorem implies that there is the following similarity between the corner Λ and the corner which the positive coordinate axes*q*_{1} form at the origin: The consecutive order of the edges A_{i} at Λ is the same as the order of the axes*q*_{i} at the origin; moreover, if we use the direction of the curve C at Q to project the curves A_{i} into the budget plane, the projection of A_{i} will form a smaller angle with the*q*_{i}-axis than with any other coordinate axes. Again, statements of the same type hold true for the curves B_{i}. The theorem in question is proved by an intuitive argument. An alternative, purely analytic proof of the theorem, ends the paper.

## Keywords

Substitution Effect Direction Cosine Relative Prex Straight Line Parallel Budget Plane## Preview

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## References

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