Question

Show your work! Answers without work will not get any points. You should submit your own work. Cases of cheating will be sent to disciplinary committee for disciplinary action. 1. (35 points) Consider a standard Crusoe economy. There is a household/consumer with the following preferences: U(CL) = Ca (1 - L) where C is consum ption and L is hours worked and a,ß are positive. The household collects coconuts using labor and technology. The production function of the firm is given by: Y = AL

(a) Derive the slope of the indifference curve. What is the economic interpretation of the slope?

Answer 1

The slope of the indifference curve, measured by the marginal rate of substitution (MRS), reflects the trade-off between leisure and labor for constant utility in a Crusoe economy. It is derived by differentiating the utility function and indicates how much consumption a consumer needs to compensate for an extra hour of labor.

Step-by-step explanation:

The slope of the indifference curve in this standard Crusoe economy represents the trade-off between leisure and labor that leaves the household's utility unchanged. To derive the slope, we differentiate the given utility function U(CL) = Ca (1 - L) with respect to L, treating consumption (C) as a constant. This gives us the marginal rate of substitution (MRS) between labor and consumption. Mathematically, the MRS is the negative of the derivative of consumption with respect to labor, holding utility constant.

dU/dL = aC(-1) = -aC

The slope of the indifference curve, or MRS, is therefore -aC. This slope shows how much additional consumption is needed to compensate for the disutility of one extra hour worked, keeping utility constant. The economic interpretation of the slope is that it measures the rate at which the consumer is willing to substitute leisure for consumption while maintaining the same level of utility.