Question

A firm is owned by a risk averse individual and operates in a perfectly competitive market. Preferences are expressed using the mean-variance model, i.e.; EU (EY, varY) where EY is expected income and vary is the variance of income. The only risk that the firm faces is the future wage settlement she will negotiate with her work force. The owner has to commit now to a level of labour, L, (i.e., before the wage rate is determined). Let P be the price of the product and let (L) be the production function, where L is the amount of labour. Assume (L) satisfies the usual conditions (ie., f'(L) > 0, f" (L) < 0); ie., the production function exhibits diminishing marginal product of labour. Let the wage rate be w(1+0) where ◊ is a random variable with expected value of zero and variance of > 0. Amount of labour, L, is chosen before the value of the random variable is known. Show how to characterize the optimal choice of L for the firm. How does the extent (level) of risk aversion (parameter k) affect the optimal choice? Explain. How does this optimal choice compare to the choice of a risk neutral individual? Discuss using mathematics, economic intuition and a relevant graph that depicts the optimal choice for L.

Answer 1

The optimal choice of labor
`(\(L\))` for the firm in a perfectly competitive market, considering the mean-variance model and risk aversion, involves maximizing expected income while minimizing income variance. The optimal
`\(L\)` is determined by balancing the marginal productivity of labor with the marginal disutility of risk, accounting for the variance in wage rates.

Step-by-step explanation:

In the mean-variance model, the owner seeks to maximize expected income
`(\(EY\))` while minimizing the variance of income
`(\(varY\))`, expressing risk aversion. The firm's production function
`(\(f(L)\))` with diminishing marginal returns implies that as more labor is employed, the marginal productivity decreases. Introducing the wage rate
`(\(w(1+\delta)\))` with a random variable
`(\(\delta\))` representing wage uncertainty, the owner chooses
`\(L\)` before the realization of
`\(\delta\)`.

The optimal
`\(L\)` is determined by equating the marginal productivity of labor
`(\(f'(L)\))` with the marginal disutility of risk, which is the product of the risk aversion parameter
`(\(k\))` and the variance of the wage rate
`(\(var(\delta)\))`. Mathematically, this is expressed as
`\(f'(L) = k \cdot var(\delta)\)`. The level of risk aversion
`(\(k\))` influences the optimal choice, with higher
`\(k\)` leading to a more conservative
`\(L\)` as the owner places greater weight on risk reduction.

Comparing this to a risk-neutral individual, a risk-averse owner will choose a lower
`\(L\)` to hedge against income variance. The graph depicting this relationship would show the intersection of the marginal productivity curve and the increasing marginal disutility of risk curve, highlighting the trade-off between maximizing expected income and minimizing income variability.