A person chooses between leisure and consumption. All of their consumption comes

Question

A person chooses between leisure and consumption. All of their consumption comes from current income. The utility derived from any combination of leisure and consumption is given by: u=YL-88Y where u is utility, L is hours of leisure per week and Y is the number of dollars of income all of which will be spent on consumption. The person can work as many hours as they wish during the week at a constant wage of $4 per hour. There is no other source of income.

i. Identify the equation for this person's budget constraint.

ii. Draw this person's budget constraint (I will be able to do this with a confirmation on my answer for part i.)

iii. Draw on the same graph the indifference curves associated with u=6000, u=6400 and u=6800 (do I just plug in random points in the utility function above to find these indifference curves?)

iv. Find the utility maximizing combination of income and leisure. How many hours will this person work?

v. Imagine the wage rate increases to $8 per hour. Will this person work more hours?

Explanation / Answer

Hey! Plot a graph with Leisure(no. of hours in a week) on y axis and Consumption(no. of $) on x axis. (i) & (ii) The maximum amount of leisure possible is 24 * 7 = 168. There are 24 hrs in a day & 7 days in a week. At this point income = 0. Now minimum amount of leisure possible = 0. At this point income = 24 * 7 * 4 = $672 given that wages are $4 and see all time is being spent in working. So in the graph draw a straight line with y intercept = 168 and x intercept 672. Remember we had Leisure(no. of hours in a week) on y axis and Consumption(no. of $) on x axis. There you have your budget constraint. It passes through (0,168) and (672,0) so its equation becomes Y + 4L = 672 (iii)Yes, you have to just plug in random points in the utility function above to find these indifference curves. Because that is what the utility curve essentially means, for several combinations of Y and L you will achieve U = given. (iv) You have to find combination for maximum utility. Given function U = YL - 88Y Partial Derivative of U wrt Y = L - 88 Set these to zero This gives L = 88 This means he'll work for (24*7)-88 hours ie 80 hours and thus earn $320. Notice that (320,88) passes through budget constraint which should be the case for maximum utility. (v) If wage increases to $8, his utility curve is likely to change. In that possibility he may or may not work more depending on the new utility curve.

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