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I. The Robinson Crusoe Economy
Plays a dual role: as consumer and as producer.
He chooses between two goods: leisure and coconuts as a customer. If he sits on the beach watching the ocean, he is consuming leisure. If he spends his time gathering coconuts, he has less time for leisure but gets to eat the coconuts. We can depict Robinson production opportunities and preferences over the two goods.
At this point, the slope of the indifference curve must equal the slope ofthe production function by the standard argument: if they crossed, therewould be some other feasible point that was preferred.
The utility maximizing choice for Robinson must be the point at which the highest indifference curve just touches the production function. Why? At any point inside the production function, Robinson could choose a different point that involved less labor and/or more coconuts. Given that he prefers to be on higher indifference curves, he should choose the highest one that is possible. This means an indifference curve that just touches the production function.
Assuming the production function and the indifference curves are both differ¬entiable at this point, we can conclude that at the optimum choices for labor and coconuts, the marginal product of labor equals the marginal rate of substitution between leisure and coconuts.
This makes sense. The marginal product of labor is the extra amount of coconuts Robinson would get from giving up one unit of leisure. The MRS is the marginal utility gets from coconuts per unit of marginal utility from leisure. So, imagine that initially the MRS waw greater than the MP . Then think about what happens if Robinson spends one less hour gathering coconuts. His consumption of coconuts goes down by the MP of labor. His utility from leisure goes up by the marginal utility of leisure. His utility from coconuts goes down by the marginal utility of coconuts divided by the marginal utility of leisure. Under the assumption that the MRS
Plays a dual role: as consumer and as producer.
He chooses between two goods: leisure and coconuts as a customer. If he sits on the beach watching the ocean, he is consuming leisure. If he spends his time gathering coconuts, he has less time for leisure but gets to eat the coconuts. We can depict Robinson production opportunities and preferences over the two goods.
At this point, the slope of the indifference curve must equal the slope ofthe production function by the standard argument: if they crossed, therewould be some other feasible point that was preferred.
The utility maximizing choice for Robinson must be the point at which the highest indifference curve just touches the production function. Why? At any point inside the production function, Robinson could choose a different point that involved less labor and/or more coconuts. Given that he prefers to be on higher indifference curves, he should choose the highest one that is possible. This means an indifference curve that just touches the production function.
Assuming the production function and the indifference curves are both differ¬entiable at this point, we can conclude that at the optimum choices for labor and coconuts, the marginal product of labor equals the marginal rate of substitution between leisure and coconuts.
This makes sense. The marginal product of labor is the extra amount of coconuts Robinson would get from giving up one unit of leisure. The MRS is the marginal utility gets from coconuts per unit of marginal utility from leisure. So, imagine that initially the MRS waw greater than the MP . Then think about what happens if Robinson spends one less hour gathering coconuts. His consumption of coconuts goes down by the MP of labor. His utility from leisure goes up by the marginal utility of leisure. His utility from coconuts goes down by the marginal utility of coconuts divided by the marginal utility of leisure. Under the assumption that the MRS