A Theory of Inefficient College Entry and Excessive Student Debt

Lemma 1.

There existsθ∗, such thatμ\gtθ∗\gt0, and:

1. If workers with abilities fromθ∗toμacquire education, then all workers are better off than if no worker acquires education.

2. Any worker with ability belowθ∗is better off by remaining unskilled.

Proof.

Part (i). Define θ∗ in the following way:

where (μ−θ∗)/θ∗ is the amount of skilled labor when all workers with abilities above θ∗ acquire education; β((μ−θ∗)/θ∗) is the marginal product of skilled labor when all workers with abilities above θ∗ acquire education; p(θ∗) is the per-period cost of education for the worker of ability θ∗; γ((μ−θ∗)/θ∗) is the marginal product of unskilled labor when all workers with abilities above θ∗ acquire education.

Equation (14) states that the worker of ability θ∗ is indifferent between remaining unskilled or becoming skilled, conditional on all workers with abilities above θ∗ acquiring education. Because U(⋅) is a monotonically increasing function, we can rewrite eq. (14) as

We will now show that θ∗ exists and μ\gtθ∗\gt0. First notice that the left-hand side of eq. (15) is monotonically increasing in θ∗, while the right-hand side of eq. (15) is monotonically decreasing in θ∗. Then, notice that if we let θ∗=0, then the left-hand side of eq. (15) is smaller than its right-hand side by restriction (6). If we let θ∗=μ, then the left-hand side of eq. (15) is larger than its right-hand side by restriction (7) because p(μ)\ltp(0). Hence, eq. (15) must hold with equality for some θ∗∈(0,μ).

We will now show that all workers are better off if all workers with abilities from θ∗ to μ acquire education than if no worker acquires education. Assume that all workers with abilities from θ∗ to μ acquire education and consider any worker of ability θ0, such that θ0\gtθ∗. This worker’s expected utility is given by

Had no worker acquired education, this worker’s expected utility would have been given by

Since θ0\gtθ∗, we have that p(θ0)\ltp(θ∗), and it follows from eq. (15) that β((μ−θ∗)/μ)−p(θ0)\gtγ((μ−θ∗)/μ). Moreover, γ((μ−θ∗)/μ)\gtγ(0) (the marginal product of unskilled labor when all workers are unskilled is lower than the marginal product of unskilled labor when workers with abilities above θ∗ are skilled). Therefore, β((μ−θ∗)/μ)−p(θ0)\gtγ(0) and hence eqs (16)\gt(17).

Last, consider any θ1 such that θ1\ltθ∗. Observe, as before, that γ((μ−θ∗)/μ)\gtγ(0), and hence this worker’s consumption is higher when workers with abilities from θ∗ to μ acquire education than when no worker acquires education.

Part (ii). Consider any worker with ability θ′ such that θ′\ltθ∗. If this worker acquires education, then all workers with abilities above θ′ also acquire education. Therefore, the minimum share of skilled workers if the worker with ability θ′ acquires education is (μ−θ′)/μ. It follows that the marginal product of skilled labor cannot exceed β(μ−θ′)/μ if the worker with ability θ′ acquires education. Thus, if the worker with ability θ′ becomes skilled, his or her expected utility is no greater than

If he or she remains unskilled, however, her expected utility is no less than

Since θ′\ltθ∗, we have that β(μ−θ′μ)−p(θ′)\ltγ(μ−θ′μ) and hence eqs (19)\gt(18).

Proposition 1:

Assume that all workers are rational and have perfect knowledge ofβ(⋅), γ(⋅), cost(θ), and the distribution ofθ. Also assume that the amount of loans that the government is ready to provide overTperiods, L, satisfiesL=∫θTμcost(θ)dθ, whereθT\ltθ∗. Then, only workers with abilities fromθ∗ to μacquire education in equilibrium.

Proof.

Consider any worker with ability θ0 such that θ∗\gtθ0. By Lemma 1, this worker is better off remaining unskilled. He or she understands it and decides not to become skilled. Hence, all workers with abilities below θ∗ decide not to acquire education.

Now consider any worker with ability θ1 such that θ∗\ltθ1. By Lemma 1, all workers (including this one) are better off if this worker acquires education. Hence, all workers with abilities above θ∗ decide to acquire education.

Also notice that β((μ−θ∗)/μ)−p(θ1)\gtγ((μ−θ∗)/μ)\gt0, which follows from eq. (15) and the fact that p(θ1)\ltp(θ∗) since θ1\gtθ∗. Thus, no worker defaults on her debt and the government does not impose any taxes.

Proposition 2:

Assume that workers have perfect knowledge ofβ(⋅), γ(⋅), cost(θ), and the distribution ofθ. However, they are myopic. Also assume that the amount of loans that the government is ready to provide overTperiods, L, satisfies L=∫θTμcost(θ)dθ, where θT\ltθ∗. Then,

1. There is inefficient college entry: Some workers with abilities belowθ∗acquire education in equilibrium.

2. There existsθ∗∗\gtθ∗such that all workers with abilities betweenθ∗andθ∗∗would have been better off if they had remained unskilled.

Proof.

Part (i). Recall that θt is the lowest level of ability that can obtain loans from the government in period t. Since θT\ltθ∗, there exists i∗∈{1,…,T} such that θi∗≤θ∗\ltθi∗−1. For any j≤i∗, θj−1\gtθ∗, and therefore

It follows from eq. (22) that all workers with abilities between θ∗ and θi∗−1 want to become skilled. At the same time, the government is willing to provide loans to all workers with abilities above θT, where θT≤θi∗\ltθi∗−1. Therefore, all workers with abilities between θ∗ and θi∗−1 acquire education.

Since ∑0∞δtU(β(μ−θi∗−1μ)−p(θ∗))\gt∑0∞δtU(γ(μ−θi∗−1μ)) by eq. (22), there exists θ′′\ltθ∗ such that ∑0∞δtU(β(μ−θi∗−1μ)−p(θ′′))=∑0∞δtU(γ(μ−θi∗−1μ)), and therefore all workers with abilities above θ′′ acquire education. Since θ′′\ltθ∗, there is inefficient college entry.

Since workers with abilities below θ∗ acquire education, some of them may default. In this case, the government will have to impose a tax to balance its budget. However, since workers’ beliefs about future wages are such that they never default, this tax does not affect workers’ decisions whether to acquire education or not.

Part (ii). We showed above that there exists θ′′\ltθ∗ such that all workers with abilities above θ′′ acquire education. Define θ∗∗ in the following way:

The proof that θ∗∗ exists is analogous to the proof that θ∗ exists in Lemma 1. It follows from eq. (21) that all workers with abilities lower than θ∗∗ would have been better off by remaining unskilled. Since ∑0∞δtU(β(μ−θ′′μ)−p(θ∗))\lt∑0∞δtU(γ(μ−θ′′μ)), it follows that θ∗∗\gtθ∗.

Proposition 3:

Assume that workers have perfect knowledge ofβ(⋅), γ(⋅), cost(θ), and the distribution ofθ. However, they are myopic. Also assume that the amount of loans that the government is ready to provide overTperiods, L, satisfiesL=∫θTμcost(θ)dθ, whereθT\ltθ∗. Then, the maximum difference between θ∗and the ability of the lowest ability worker that acquires education is decreasing inT.

Proof.

As we showed in the proof of part (i) of Proposition 2, all workers with abilities between θ∗ and θi∗−1 acquire education. Notice that since θ∗\gtθi∗, we have

Therefore, no worker with ability below θi∗ acquires education. It follows that the last worker that acquires education has ability between θi∗ and θ∗. Denote this worker’s ability by θ′′, as in Proposition 2. Recall that ∫θi−1θicost(θ)dθ=L/T. Since θi∗−1\gtθ∗\gtθ′′≥θi∗, it follows that

It follows from inequality eq. (23) that increasing T decreases the maximum possible value of cost(θ′′) and hence increases the minimum possible value for θ′′, thus decreasing the distance between θ∗ and θ′′.

Proposition 4:

Assume that there exists an ability thresholdωsuch that all workers with abilities aboveωare rational and all workers with abilities belowωare myopic and that rational workers are aware of the presence of myopic workers and their distribution in the population. Also assume that the amount of loans that the government is ready to provide, L, satisfiesL=∫θTμcost(θ)dθ, whereθT\ltθ∗. Then, there will be inefficient college entry as long asωis sufficiently high.

Proof.

Note that ω is the ability of the lowest ability rational worker: All workers with abilities between ω and μ are rational and all workers with abilities between 0 and ω are myopic. As in the proof of Proposition 2, let θ′′ denote the lowest ability of workers that acquire education if all workers are myopic. Also as in Proposition 2, define θ∗∗ in the following way:

We will consider two cases: ω\ltθ∗∗ and ω≥θ∗∗.

First, let ω\ltθ∗∗ and consider the lowest ability rational worker (whose ability is ω). This worker will not become skilled since he or she realizes that if he or she becomes skilled then myopic workers will start acquiring education, until their ability level reaches θ′′. However, in that case, this worker is better off by remaining unskilled. Since the government stops extending loans for education once a worker decides not to become skilled, it follows that no myopic worker can become skilled. Rational workers will be acquiring education as long as it is beneficial for them (until their ability reaches max{θ∗,ω}). Since max{θ∗,ω}≥θ∗, there will be no inefficient college entry.

Now let ω\gtθ∗∗ and consider the lowest ability rational worker (whose ability is ω). This worker will become skilled since he or she is better off by becoming skilled even if all workers with abilities from θ′′ to μ become skilled. Since this worker acquires education, myopic workers also will start acquiring education, until their ability level reaches θ′′. Since θ′′\ltθ∗, there is inefficient college entry.

To summarize, there is inefficient college entry if the share of rational workers is below (μ−θ∗∗)/μ. Equivalently, there is inefficient college entry if the share of myopic workers is above θ∗∗/μ.

Proposition 5:

Assume that there exists an ability thresholdωsuch that all workers with abilities aboveωare rational and all workers with abilities belowωare myopic. However, rational workers are unaware of the presence of myopic workers and behave as if all workers in the population were rational. Also assume that the amount of loans that the government is ready to provide, L, satisfiesL=∫θTμcost(θ)dθ, where θT\ltθ∗. Then, some rational workers may acquire education even if it is inefficient for them to do so.

Proof.

As in Lemma 1, define θ∗ in the following way:

As in the proof of Proposition 2, let θ′′ denote the lowest ability of workers that acquire education if all workers are myopic. Also as in Proposition 2, define θ∗∗ in the following way:

Recall (from Proposition 2) that θ∗∗\gtθ∗\gtθ′′.

First consider the case in which ω≥θ∗∗. Consider the lowest ability rational worker (whose ability is ω). This worker conjectures that all other workers are rational and that they will therefore acquire education until their ability level reaches θ∗. Since this worker’s ability is above θ∗, it is optimal for his or her to acquire education, given his or her beliefs about other workers’ behavior. Since workers with ability below ω are myopic, they will keep obtaining education until their ability level reaches θ′′. Thus, there will be inefficient college entry. However, because workers with ability level θ∗∗ are indifferent between becoming skilled and remaining unskilled if all workers with ability level above θ′′ acquire education and because the lowest ability rational worker has ability level θ∗∗, this inefficient college entry affects only myopic workers.

Now consider the case in which θ∗∗\gtω≥θ∗. Similar to the first case above, the lowest ability rational worker will find it optimal, based on her belief, to obtain education. Also similar to the first case above, all workers with ability above θ′′ will acquire education. However, since the ability of the lowest ability rational worker is below θ∗∗, this implies that some of them acquire education even though it is inefficient for them.

Finally, consider the case in which θ∗\gtω. In this case, the lowest ability rational worker (whose ability ω is below the efficient level of college entry, θ∗) will not become skilled since his or her expected return from becoming skilled is lower than her expected return from remaining unskilled. Since the government stops extending loans for education once a worker decides not to become skilled, it follows that no myopic worker can become skilled. Rational workers will be acquiring education as long as it is beneficial for them (until their ability reaches max{θ∗,ω}). Since max{θ∗,ω}=θ∗, there will be no inefficient college entry.

To summarize, there is inefficient college entry if the share of rational workers is below (μ−θ∗)/μ, which is higher than (μ−θ∗∗)/μ since θ∗∗\gtθ∗. This implies that, to prevent inefficient college entry, the share of rational workers must be higher under the assumptions of this proposition than under the assumptions of Proposition 4. Furthermore, when θ∗∗\gtω≥θ∗, some rational workers will acquire education even though it is inefficient for them to do so.

Proposition 6:

Assume that there exists an ability thresholdωsuch that all workers with abilities belowωare rational and all workers with abilities aboveωare myopic. Also assume that the amount of loans that the government is ready to provide, L, satisfiesL=∫θTμcost(θ)dθ, whereθT\ltθ∗. Then, there is no inefficient college entry if the share of myopic workers in the population is below(μ−θ∗)/μ. If the share of myopic workers in the population is above(μ−θ∗)/μ, then there is inefficient college entry; however, no rational worker acquires education in this case.

Proof.

First consider the case in which ω≥θ∗ (i.e., all myopic workers have ability above the efficient threshold level). Consider the highest ability rational worker (whose ability is ω). Since ω≥θ∗, this worker will correctly conjecture that his or her returns to acquiring education exceed the costs and will therefore be willing to acquire education. Similarly, all rational workers with ability level up to θ∗ will acquire education. Since there are no myopic workers with ability below θ∗, no worker with ability below θ∗ will acquire education and therefore there will be no inefficient college entry.

Now consider the case in which ω\ltθ∗ (i.e., some myopic workers have ability below the efficient threshold level). Since some myopic workers have ability below the efficient threshold level and since they receive access to financing before rational workers do (in accordance with Algorithm 1), there will be inefficient college entry. Consider the highest ability rational worker (whose ability is ω). Since ω\ltθ∗, this worker will correctly conjecture that his or her returns to acquiring education are lower than the costs and will therefore be unwilling to acquire education. Hence, no rational worker will acquire education in this case.