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Summary

This chapter introduces the important idea of a vector through the example of bundles of goods. The dot product of two vectors is defined and it is shown how a budget constraint can be expressed in terms of dot product. It is explained how, in order to rank bundles according to a particular consumer's preference, we can use a utility function. Indifference curves are defined as the contours of the utility function. Linear and convex combinations and the concept of a convex set are explained. The utility maximisation problem -- to maximise utility subject to a budget constraint -- is explored and the relevance of convexity is emphasised.

The family ${\mathcal{F}}_{\unicode[STIX]{x1D706}}$ of orientation-preserving harmonic functions $f=h+\overline{g}$ in the unit disc $\mathbb{D}$ (normalised in the standard way) satisfying

$$\begin{eqnarray}h^{\prime }(z)+g^{\prime }(z)=\frac{1}{(1+\unicode[STIX]{x1D706}z)(1+\overline{\unicode[STIX]{x1D706}}z)},\quad z\in \mathbb{D},\end{eqnarray}$$

for some

$\unicode[STIX]{x1D706}\in \unicode[STIX]{x2202}\mathbb{D}$, along with their rotations, play an important role among those functions that are harmonic and orientation-preserving and map the unit disc onto a convex domain. The main theorem in this paper generalises results in recent literature by showing that convex combinations of functions in

${\mathcal{F}}_{\unicode[STIX]{x1D706}}$ are convex.

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14 - Vectors, Preferences and Convexity

Summary

ON CONVEX COMBINATIONS OF CONVEX HARMONIC MAPPINGS

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14 - Vectors, Preferences and Convexity

Summary

ON CONVEX COMBINATIONS OF CONVEX HARMONIC MAPPINGS