Given that MU_Y = marginal utility of income, MU_LE = marginal utility of leisure, and W is the wage, utility is maximized by allocating time between work and leisure so that:

Question

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To maximize utility, we want to equalize the Marginal Utility of Income to the Marginal Utility of Leisure. We can do that by setting the ratio of the two to be equal to the ratio of their prices. In this case, the price of leisure is equal to the wage, so we get MU_Y/P_Y = MU_LE/W. Solving for the optimal allocation of time between work and leisure requires setting this equation equal to 1, because we want the marginal utility per dollar spent on each good to be equal. Therefore, the equation simplifies to MU_Y/P_Y = MU_LE/W.

Given That MUY = Marginal Utility of Income, MULE =

Given That MU_Y = Marginal Utility of Income, MU_LE =