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Clark gains utility from consumption c and leisure l and his preferences for consumption and leisure...

Clark gains utility from consumption c and leisure l and his preferences for consumption and leisure can be expressed as U(c, l) = 2(√ c)(l). This utility function implies that Clark’s marginal utility of leisure is 2√ c and his marginal utility of consumption is l √ c . He has 16 hours per day to allocate between leisure (l) and work (h). His hourly wage is $12 after taxes. Clark also receives a daily check of $30 from the government no matter how much he works.

1. Graph Clark’s budget constraint with leisure on the x-axis and consumption on the y-axis.

2. What is Clark’s marginal rate of substitution (MRS) when l = 3 and he is on his budget line?

3. At what wage rate would Clark be indifferent between working his first hour and being unemployed (his “reservation wage”)?

4. Find Clark’s optimal amount of consumption and leisure (the values that maximizes his utility).

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Answer #1

PART-1) Clark’s consumption will be: Labor income + Unearned income i.e.

Clarks consumption will be: Labor income Unearned income i.e. c- wh+v The equality in the budget constraint indicates two points 1) Clark cannot consume more than he earns (cメwh + u) and 2) 2) Clark wants to consume all his income (ct wh+v) Another constraint will be the physical time constraint. It is given as: h 16 -I Substituting h, we have c- w(16 -) v Putting w = 12 and v-30, the budget constraint as c = 12(16-1) + 30 c-121222 slope intercept GRAPEH 30 16 PART-2) When Clark enjoys leisure of 3 hours, the consumption should be 186 30+ 12 *(16-3)) Thus, the MRS when l = 3 is MRS18 MUc (8186) MUL 2 x 186 124. = l (3,186) c(3,186) (3,186)VeTPART-3) IND W1 > W2 > W3 slope -W1 slope-W2 slope - -W3 A-(16,30) wres MRSat A-(16,30) MUL WresMUc at A-(16,30) 2 x 30 16 $3.75 reslat A-(16,30) PART-4)budget constraint: c-121222 wMRS c 121 + 222. 20-12222) 12- 121 444-241 12.33 Hence, Clark will prefer to leisure 12.33 hours, work 3.67 hours, and thus consume $30 + (S12 * 3.67) -$74.04 each week

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