Question

A worker can produce \(x\) units of output at a cost of \(c(x)=x^{2} / 2 .\) He can achieve a utility level of \(\bar{u}=0\) working elsewhere. What is the optimal wage-labor incentive scheme \(s(x)\) for this worker?

Short answer

Optimal scheme is \( s(x) = \frac{x^2}{2} \).

Step-by-step solution

Understanding the Objective

The goal is to determine the optimal incentive wage-labor scheme \( s(x) \) that the worker should receive for producing \( x \) units, given that their cost of production is \( c(x) = \frac{x^2}{2} \) and they have an alternative utility level \( \bar{u} = 0 \).

Setting Up Worker’s Utility

The worker's net utility from working is given by the reward \( s(x) \) minus the cost of production \( c(x) \). Thus, their utility function is: \[ u(x) = s(x) - c(x) \] Given that the alternative utility is \( \bar{u} = 0 \), for the worker to accept this scheme, their utility must satisfy \( u(x) \geq 0 \).

Determining Implication for Incentive Scheme

For the worker to be indifferent between working under this scheme and the alternative, we set \( u(x) = 0 \). Therefore:\[ s(x) - \frac{x^2}{2} = 0 \] Solving for \( s(x) \), we get:\[ s(x) = \frac{x^2}{2} \]

Concluding the Optimal Scheme

The derived incentive scheme \( s(x) = \frac{x^2}{2} \) implies that the worker is paid exactly the cost of producing \( x \) units. Thus, the worker's utility is just enough to match their alternative utility \( \bar{u} = 0 \).

Key concepts

Wage-Labor Incentive

Wage-labor incentives are mechanisms designed to motivate workers by linking their pay to the amount of output they produce. In our current problem, the incentive scheme needed is one where the pay or reward, denoted as \( s(x) \), incentivizes the worker to produce \( x \) units of output without being worse off than choosing alternative employment.
This translates into a need to balance between encouraging workers to produce more and ensuring they do not feel overburdened by production costs.
By setting \( s(x) = \frac{x^2}{2} \), we ensure that the worker receives exactly what it costs them to produce their output.
In essence, this scheme guarantees that workers have no advantage in choosing an alternative job with utility level \( \bar{u} = 0 \), thereby keeping them motivated under this wage-labor system.

Worker's Utility Function

Understanding the concept of utility is crucial in optimizing incentive schemes. A worker's utility function represents the balance between the rewards they receive \( s(x) \) and the inherent costs \( c(x) \) associated with their work.
In this problem, the utility function is represented as \( u(x) = s(x) - \frac{x^2}{2} \). Here, it highlights the fact that a worker's satisfaction or utility is derived by subtracting the costs of production from the wages received.

• To keep the worker indifferent between working in this job and considering the alternative with zero utility, \( u(x) \) must equal zero.
• This results in the equation: \( s(x) - \frac{x^2}{2} = 0 \).

When \( s(x) \) is set to \( \frac{x^2}{2} \), the worker's utility is precisely zero, meaning they neither gain nor lose by remaining in their current position.

Cost of Production

Cost of production is a critical element in determining the optimal incentive scheme. It refers to the expenses a worker incurs to produce a certain level of output. Here, it is described by the function \( c(x) = \frac{x^2}{2} \).

• This cost increases with the square of the output, making producing additional units progressively more expensive.
• It reflects not only the physical inputs needed but also the effort and resources expended by the worker.

By equating \( s(x) \) with \( c(x) \), we ensure that the worker is compensated exactly for what they expend in producing \( x \) units. This equilibrium keeps the worker's net utility at the alternative benchmark, ensuring optimal motivation without financial loss.