Final answer:
To determine the slope of the indifference curve in a Crusoe economy, calculate the marginal rate of substitution by taking the ratio of the marginal utilities regarding consumption and leisure. This slope represents the trade-off between leisure and consumption, and the substitution effect illustrates changes in labor-leisure choices in response to wage changes to maintain utility.
Step-by-step explanation:
The student is asking how to derive the slope of the indifference curve in a standard Crusoe economy, specifically in a context where preferences are of a particular logarithmic form, involving consumption (C) and labor (L). To derive the slope of the indifference curve, we need to calculate the marginal rate of substitution (MRS) which equates to the rate at which a consumer is ready to give up one good in exchange for another, while maintaining the same level of utility. The slope of the indifference curve at any point is given by the negative of the MRS. In our particular case, MRS can be found by taking the ratio of the marginal utilities concerning consumption and leisure, which are -θ/(C) and (1-θ)/(1-L) respectively.
The economic interpretation of the slope of the indifference curve, or the MRS, is the trade-off a consumer is willing to make between two goods. In the context of the labor-leisure trade-off, it represents the trade-off between leisure and consumption: how many units of consumption a person is willing to forego to gain an additional unit of leisure, while keeping their utility constant.
Using the Petunia labor-leisure example, after receiving a raise, the slope of Petunia's budget constraint changes, indicating a different rate of trade-off between work and leisure hours due to changes in wage rate. The substitution effect describes how Petunia's choice of labor and leisure changes to maintain the same level of utility when her wage rate changes, making income relatively cheaper and leisure relatively more expensive.