Economics of Talent: Dynamics and Multiplicity of Equilibria

Abstract

The economics of art and science differs from other branches by the small role of material inputs and the large role of given talent and access to markets. E.g., an African violinist lacks the audience ( = market) to appreciate her talent unless it is so large that it transgresses regional constraints; conversely, a European violinist of equal talent may be happy to end up as a member of one of the regional orchestras. This paper draws attention to this second aspect and models dynamic interactions between investments into two stocks, productive capital and access (or bargaining power). It is shown that there exists multiple equilibria. The separation between pursuing an artistic career or quitting depends on both idiosyncracies, individual talent and individual market access (including or depending on market size), which explains the large international variation in the number of people choosing a career in arts as market access is affected by geographic, linguistic, and aesthetic dimensions.

Appendix

1.1 Vector Field Structure and Jacobian

The vector field $\vec{F}(\vec{y})$ associated with the dynamic system (2) is smooth and regular in all the points except for the discrete set of critical points, where $\vec{F} =\vec{ 0}$, for which we get the polynomial equation. Local behavior in these critical points (also named as nodes, or steady states) is determined by the set of eigenvalues. If there are two negative and two positive eigenvalues, the node is “stable” in the sense that all flows within the associated stable 2-dimensional manifold converge to this steady state; similar stability properties apply for a pair of complex eigenvalues with negative real parts except that the steady state is a focus and attained by transient oscillations. The crucial case for a threshold is that where only one eigenvalue is negative.

The dynamic system (6)–(9) has the following matrix of partial derivatives J:

$$\displaystyle{ J = \left (\begin{array}{cccc} -\delta & 0 &1/2b_{2} & 0 \\ 0 & -\delta & 0 &1/2c_{2} \\ 0 & - a/(1 + G)^{2} & r+\delta & 0 \\ - a/(1 + G)^{2} & 2aK/(1 + G)^{3} & 0 & r+\delta \end{array} \right ). }$$

(12)

The determinant of the Jacobian $\left (J\right )$ is then given by:

$$\displaystyle{ \det J = (r+\delta )^{2}\delta ^{2} + \frac{a(r+\delta )K} {c_{2}(1 + G)^{3}} - \frac{a^{2}} {4b_{2}c_{2}(1 + G)^{4}}. }$$

(13)

The theorem of Dockner (1985) says that detJ < 0 is equivalent to one negative eigenvalue of J, such that stability is restricted to a one-dimensional manifold in the state space (and thus of zero probability). Since the sum of principal minors of dimension 2 in (12) is negative (more precisely, $-2(r+\delta )\delta$), detJ > 0 implies saddlepoint stability, i.e., two eigenvalues are either negative or have negative real parts. Therefore, a higher b ₂ and c ₂ reduce the negative term in detJ and thus the chances for an unstable steady state.

Lemma 1

The trivial equilibrium $K = 0,G = 0$ is stable if a ² < 2b ₂ c ₂ δ ² (r + δ) ² . Thus, for sufficiently low talent and sufficiently high values of discount and depreciation there is convergence to zero equilibrium for at least a subset of initial values (K(0),G(0)).

The proof is trivial, since K = 0 implies zero in the second term of expression (3). In the opposite case of inequality this equilibrium is unstable,¹³ and thus sufficiently high talent gives an escape from such an outcome.

The condition for an unstable steady state $\left (\det J < 0\right )$ is:

$$\displaystyle{ a^{2} > (r+\delta )^{2}\delta ^{2}(1 + G)^{4} + 2a\delta (r+\delta )K(1 + G). }$$

(14)

Since G > 0, K > 0 at an interior steady state, a sufficiently high talent is necessary to unsettle the origin as the only equilibrium. Two positive steady states, one stable the other unstable, is then typical.

1.2 Network Effect on Market: Social Influence Versus Normal Preferences

We consider the case of heterogeneous consumers, who define the market for each author, that can be also of many types, A. We want to show that market structure (with or without network influence on preferences) has a pronounced effect on the return to talents. In particular, network effect raises the threshold for market entry, but also makes returns for more talented authors higher.

For the sake of mathematical transparency, assume a continuum of individuals (consumers) of measure one, uniformly distributed over interval [0, 1] and indexed by i. We start from the model of “normal market”, with individual valuation of a book without social network effect given by V _i  = Ai. Let the price of a book, p, be given exogenously. Each individual of type i decides whether to buy this book or not, comparing his valuation with the price, i.e. one book is purchased by those who have Ai > p. It is easy to find that the threshold agent is characterized by $i^{{\ast}} = p/A$, and that the measure of such agents is 1 − i ^∗. Hence, in the normal case the demand is given by the formula $d1 = 1 - p/A$. Normalizing price to one, we get the dependence of market size on talent. For A < p, nobody will buy a book from this author, while for A > p the demand grows as $d1 = 1 - 1/A$, approaching in the limit A → ∞ the maximal demand, equal to one in our case.

There exist many possibilities to model social influence. We assume that social influence transforms the individual preferences into $W_{i} = V _{i}\,f\left (M\right )$. Let us consider mathematically simple and transparent case W _i  = 2MV _i , where M denotes the measure of consumers who like this book. Again, there is a uniform distribution of individual valuations V _i  = Ai. Since the total measure of consumers is one, the average measure is 1/2 (when only half like), and factor 2 is taken for comparative transparency: when half of agents like an author, the social effect does not change the valuation of the mean individual. When we account for social influence and look for a threshold consumer, $\bar{\imath}$, we need to equate his valuation, $W\left (\bar{\imath}\right )$ with price. This leads to a quadratic equation $2A\bar{\imath}(1-\bar{\imath}) = p$ with two solutions: $\bar{\imath}_{1,2} = 1/2(1 \pm \sqrt{1 - 2p/A})$. Note that the function W has its maximum for a medium consumer: as soon as he buys, a cascade of purchases will occur, and a finite measure will buy. Now the threshold becomes different: for A < 2p nobody will buy a book, and the demand is thus zero. The we have a discontinuity: demand jumps immediately to 1/2, and after increases slower: $d2 = 1/2(1 + \sqrt{1 - 2p/A})$. Figure 6 presents both curves for comparison.

Fig. 6

Demand for creative work as a function of talent for unit price. Normal case corresponds to uniform distribution of valuations by consumers. In the case of social network the threshold grows, but later demand rises sharply