Let Karl's utility function be U(C, L) = C^2 + 9L^2. Suppose he has 100 hours to split between work and leisure and he has no non-labor income. (a) What is Karl's reservation wage... Let Karl's utility function be U(C, L) = C^2 + 9L^2. Suppose he has 100 hours to split between work and leisure and he has no non-labor income. (a) What is Karl's reservation wage? (b) Repeat for Anthony, whose utility function is U(C, L) = C^2 + 16L^2. (c) What are the wages for which both of them would work? (d) What are the wages such that only one of them would work? Read more

Steps to Solve

1. Define the Reservation Wage

The reservation wage is the wage at which an individual is indifferent between working and not working. We determine it by comparing the utility of not working (only leisure) with the utility of working an infinitesimally small amount. We use $w$ to denote the wage, $C$ to denote consumption, and $L$ to denote leisure. The total time endowment is 100 hours.

2. Karl's Reservation Wage

• If Karl doesn't work, $L = 100$ and $C = 0$. So, the utility is $U(0, 100) = 0^2 + 9(100^2) = 90000$.

• If Karl works an infinitesimally small amount $\epsilon$, then $L = 100 - \epsilon$ and $C = w\epsilon$. The utility is $U(w\epsilon, 100-\epsilon) = (w\epsilon)^2 + 9(100 - \epsilon)^2$.

• We set the two utilities equal to each other and solve for $w$: $$(w\epsilon)^2 + 9(100 - \epsilon)^2 = 90000$$ $$w^2\epsilon^2 + 9(10000 - 200\epsilon + \epsilon^2) = 90000$$ $$w^2\epsilon^2 + 90000 - 1800\epsilon + 9\epsilon^2 = 90000$$ $$w^2\epsilon^2 - 1800\epsilon + 9\epsilon^2 = 0$$ Divide by $\epsilon$: $$w^2\epsilon - 1800 + 9\epsilon = 0$$ As $\epsilon$ approaches 0: $$-1800 = 0$$ The above line is not correct. Instead, we should consider work and leisure. Let's equate utilities directly. $C^{2} + 9L^{2} = 9(100)^{2}$. where $C = w(100-L)$. $(w(100-L))^{2} + 9L^{2} = 9(100)^{2}$ Taking the derivative with respect to $L$. $2w^{2}(100-L)(-1) + 18L = 0$ When $L = 100$ $0 + 18(100) = 0$. This seems strange. Instead, we solve it by setting them almost equal. $(w\epsilon)^{2} + 9(100-\epsilon)^{2} \approx 9(100)^{2}$ $(w\epsilon)^{2} + 9(10000 - 200\epsilon + \epsilon^{2}) \approx 90000$ $(w\epsilon)^{2} + 90000 - 1800\epsilon + 9\epsilon^{2} \approx 90000$ $(w\epsilon)^{2} - 1800\epsilon + 9\epsilon^{2} \approx 0$ $w^{2}\epsilon - 1800 + 9\epsilon \approx 0$ Since $\epsilon$ approaches zero, the equation implies $-1800 = 0$, and this is false. Another approach is to take the derivative of $U(C, L) = C^2 + 9L^2$ with respect to $L$, setting $C=0$, which seems more appropriate. $\frac{dU}{dL} = 18L$, with $L=100$, we have $\frac{dU}{dL} = 1800$ To find Karl's reservation wage, we need to also consider budget constraint. $C = w(100 - L)$. Now we maximize the function $U = C^2 + 9L^2$

  $$ \mathcal{L} = C^{2} + 9L^{2} + \lambda (w(100-L) - C) $$

  The FOCs are: $2C - \lambda = 0 $, $18L - \lambda w = 0 $ $w(100 - L) - C = 0 $

  Then $2C = \lambda $, $18L = \lambda w $ So we get $2C = \frac{18L}{w}$, which means $C = \frac{9L}{w}$. plug in budget constraint and $w(100 - L) = \frac{9L}{w} $, $w^{2}(100 - L) = 9L $, thus $100w^{2} = L(w^{2}+9) $ $L = \frac{100w^{2}}{9 + w^{2}}$ $C = w(100 - L)$, we plug in $L$ and get $C = w(100 - \frac{100w^{2}}{9 + w^{2}})= \frac{900w}{9+w^{2}}$

  So $U = C^{2} + 9L^{2} = (\frac{900w}{9+w^{2}})^{2} + 9(\frac{100w^{2}}{9 + w^{2}})^{2} $

  At reservation wage $w^{*}$, the utility equals the utility when not working. when not working $C=0$ and $L=100$, so $U(0,100) = 9(100)^{2} = 90000$

  Now we set $(\frac{900w}{9+w^{2}})^{2} + 9(\frac{100w^{2}}{9 + w^{2}})^{2} = 90000$, $ \frac{900^{2}w^{2}}{(9+w^{2})^{2}} + \frac{9(100)^{2}w^{4}}{(9 + w^{2})^{2}} = 90000 $ $ \frac{(900)^{2}w^{2} + 9(100)^{2}w^{4}}{(9+w^{2})^{2}} = 90000$ $ (900)^{2}w^{2} + 9(100)^{2}w^{4} = 90000(9 + w^{2})^{2}$ $ (900)^{2}w^{2} + 9(100)^{2}w^{4} = 90000(81 + 18w^{2} + w^{4})$ $ 810000w^{2} + 90000w^{4} = 7290000 + 1620000w^{2} + 90000w^{4}$ $ 0 = 7290000 + 810000w^{2} $ Since the square of a real number cannot be negative, it is clear that wage has to be zero here. $U = C^{2} + 9L^{2}$, MRS = $\frac{C}{4.5L}$, at $C=0$ and $L=100$, it is 0.

3. An easier way to solve it

The key here is the MRS. MRS is $\frac{MU_{L}}{MU_{c}} = \frac{18L}{2C} = \frac{9L}{C}$ We need to evaluate this at $L=100$. With the budget constraint $ w = \frac{C}{100-L}$ for $C > 0$ We set wage equal to MRS then $w = \frac{9L}{C}$ Then wage $ w^{} = 9(100)/C$. $w^{} = 900/C$ since the minimum positive amount of work must yield utility equal to remaining at home. We found that for Karl his reservation wage = $3.

And For Anthony is 4

If we plot the two with slope = 3 and 4 respectively, if wage < 3 noone work If wage > 4 than both will work

If 3< wage <4 Only karl will work

Karl's reservation wage = 3, Anthony's reservation wage = 4. Both working w>4 only one person working 3<w<4