Basic model

In the case of the basic model, the analysis indicates that it is appropriate to define the following fivev-cutoffs:

• vˆ01, where 0≤ˆv01≤1:

Individuals prefer one skill to none if v >vˆ01. Otherwise, they prefer zero skills to one.

• vˆ12 and ˆv21, where ₂1_π ≤ˆv12≤vˆ21≤1:

Individuals prefer two skills to one if ˆv12 < v < vˆ21 Otherwise, they prefer one skill to

1

Formal results and proofs are provided in Appendix I.

2_{Though it is straightforward to determine the behavior of these cutoffs, it is possible to provide a closed-form}

two.

• vˆ20 and ˆv02, where ₂1_π ≤ˆv20≤vˆ02≤1:

Individuals withv≥ 1

2π prefer two skills to none

3 _{if eitherv <vˆ}

20orv >ˆv02. Otherwise,

they prefer zero skills to two.

The first of these cutoffs indicates that less balanced (higher-v) individuals are more likely to prefer one skill to none. The second and third cutoffs (second bullet point) above indicate that individuals of intermediate to low specific-ability balance are most likely to prefer two skills to one, while the third and fourth cutoffs (third bullet point) above suggest that those of lowest and highest specific-ability balance are most likely to prefer two skills to none, given that they may at any time prefer two to one. Possible relative placements of these cutoffs along the v-continuum are illustrated in Figure3.1.1.

It is noteworthy that the most balanced individuals, namely those with v < ₂1_π, never

acquire two skills, as they never prefer two skills to one. Intuitively, this result obtains because the marginal surplus from level in the acquisition of a second skill is always negative in the basic model. For this reason, two skills are preferred to one only when the marginal surplus from diversity is sufficiently large to make the total marginal surplus positive, which never occurs for the most balanced individuals.

Furthermore, the behavior of these five cutoffs depends on g andσ as follows: • vˆ01 decreases ing and increases inσ:

∂vˆ01

∂g <0, ∂ˆv01

∂σ >0

• vˆ12and ˆv21converge asgincreases andσ decreases but diverge when the reverse is true:

∂ˆv12 ∂g >0, ∂vˆ21 ∂g <0, ∂ˆv12 ∂σ <0, ∂vˆ21 ∂σ >0 3_{Individuals withv <} 1

2π always prefer one skill to two. Therefore, they always acquire either one skill or none. For this reason, I consider a choice between zero and two skills only for those individuals who may at some time find it desirable to acquire two skills.

• vˆ20and ˆv02converge asgincreases andσ decreases but diverge when the reverse is true: ∂ˆv20 ∂g >0, ∂vˆ02 ∂g <0, ∂ˆv20 ∂σ <0, ∂vˆ02 ∂σ >0

In consequence, the fraction of individuals who prefer one skill to none is larger when general ability is higher and perceived uncertainty is lower. Similarly, the fraction of individuals who prefer two skills to one is larger whengis lower andσis higher, while the fraction of individuals who prefer two skills to none, given that they may at some time prefer two skills to one, is larger wheng is higher andσ is lower.

In particular, the marginal surplus from the level of a second skill decreases in g and increases in σ.For this reason, two skills are less likely to be attractive, relative to one, when g is high andσ is low.

The relative positions of these five cutoffs are of interest in determining the overall skill acquisition pattern across individuals. Moreover, sinceg is normalized between 0 and 1, these relative positions depend primarily on the value of σ.At extremely low uncertainty (σ →0), it follows that

0 = ˆv01<vˆ20= ˆv02<ˆv12= ˆv21<1

In this case, all individuals find it worthwhile to acquire one skill, though none face sufficient incentive to acquire two. The absolute value of the negative marginal surplus from level investment in a second skill is particularly large for low uncertainty, since individuals optimally invest in a higher level of skill when uncertainty is low, and it thus swamps the potential gain that workers would derive from a greater breadth of competence. Therefore, they all opt for full specialization.

In contrast, at extremely high uncertainty (σ→ ∞), it follows that

0<vˆ20<vˆ12<vˆ02<ˆv21= ˆv01= 1

Therefore, at the highest levels of uncertainty, individuals of high specific-ability balance ac- quire no skills, while those of lowest specific-ability balance all acquire two. Those who do

choose to invest in human capital prefer two skills to none but would find investment in any one skill insufficiently rewarding. Acquiring one skill at such high uncertainty is costly but provides no benefit. In fact, no individual acquires exactly one skill in this case.

A variety of intermediate skill acquisition patterns exist relative to these two limiting cases. In particular, simulations indicate that the skill acquisition pattern, from individuals of highest to lowest specific-ability balance, generally does proceed from zero skills, to one, to two, for intermediate values of uncertainty. These simulation results are presented in Appendix II.

Thus, the primary qualitative predictions of the basic model in the case of high perceived uncertainty4 may be summarized as follows:

• Individuals of greater specific-ability balance tend to acquire fewer skills.

• Individuals of higher general ability are more likely to acquire a positive number of skills but are also more likely to specialize fully.

• Therefore, those individuals who are observed to acquire multiple skills should be those

who are less balanced and less generally able.