Active Learning Exercise #1 Assume the utility function is 𝑈 𝐶, 𝐿 = 𝐶 !/! + 2𝐿!/! Optimization problem is 𝑚𝑎𝑥!,! 𝐶 !/! + 2𝐿!/! 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝐶 ≤ 𝑊 100 − 𝐿 𝑜𝑟 𝐶 + 𝑊𝐿 ≤ 100𝑊, 0 ≤ 𝐿 ≤ 100 Find the optimal values for C and L. SOLUTION Derive MRS 𝜕𝑈
𝐿!!/!
𝐶
𝑀𝑅𝑆 = − 𝜕𝐿 = −
=2
1
𝜕𝑈
𝐿
(2)𝐶 !!/!
𝜕𝐶
!/!
Substitute 𝑊 100 − 𝐿 for C so that all income is used (budget constraint is satisfied) 𝐶
𝑀𝑅𝑆 = −2
𝐿
!
!
𝑊 100 − 𝐿
= −2
𝐿
!/!
Equate with slope of budget line: 𝑊 100 − 𝐿
𝑀𝑅𝑆 = −2
𝐿
𝑊 100 − 𝐿
2
𝐿
4𝑊 100 − 𝐿
!/!
=𝑊→4
!/!
= −𝑊 = −
𝑊 100 − 𝐿
𝐿
= 𝑊 ! 𝐿 → 4 100 − 𝐿
𝑃!
𝑃!
= 𝑊 ! = 𝑊𝐿 → 𝐿∗ =
400
𝑊+4
Confirm that the optimum respects the condition: 0 ≤ 𝐿 ≤ 100. It does as long as W is non-­‐negative. Alternative Solution Use the FOC by taking the derivative of the objective function and set it equal to zero. The optimization problem is 𝑚𝑎𝑥!,! 𝐶 !/! + 2𝐿!/! 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝐶 ≤ 𝑊 100 − 𝐿 , 0 ≤ 𝐿 ≤ 100 Substitute: 𝑚𝑎𝑥! (𝑊 100 − 𝐿 )!/! + 2𝐿!/! 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 0 ≤ 𝐿 ≤ 100 Derive the FOC: 𝜕∙
1
=−
𝜕𝐿
2
𝑊 100 − 𝐿
!
!
!𝑊
!
+ 𝐿!! = 0 → 𝐿∗ =
400
𝑊+4
Active Learning Exercise #2 Assume the utility function is 𝑈 𝐶, 𝐿 = 𝐶 + 2𝐿 Optimization problem is 𝑚𝑎𝑥!,! 𝐶 + 2𝐿 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝐶 ≤ 𝑊 100 − 𝐿 𝑜𝑟 𝐶 + 𝑊𝐿 ≤ 100𝑊, 0 ≤ 𝐿 ≤ 100 Find the optimal values for C and L. SOLUTION 𝜕𝑈
2
𝑀𝑅𝑆 = − 𝜕𝐿 = − = −2 𝜕𝑈
1
𝜕𝐶
Given MRS = -­‐2, consumers are left as well off when they trade-­‐off one more unit of leisure in exchange for two units of consumption and that is true regardless of the values for C and L. Indifference curves are then parallel lines with a constant slope of -­‐2. The slope of the budget line is −
𝑃!
𝑊
= − = −𝑊 𝑃!
1
There is a corner solution except when W = 2. The problem can also be solved as follows. After substituting 𝑊 100 − 𝐿 for C, the problem is 𝑚𝑎𝑥! 𝑊 100 − 𝐿 + 2𝐿 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 0 ≤ 𝐿 ≤ 100 𝑚𝑎𝑥! 𝑊100 + (2 − 𝑊)𝐿 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 0 ≤ 𝐿 ≤ 100 Optimal solution: If W < 2 then L =100. If W > 2 then L = 0. If W = 2 then any value of L is optimal. Intuition: MU of leisure and MU of consumption are constant. The price of leisure is W and the price of consumption is 1. By increasing leisure by one unit, utility is raised by 2 due to more leisure but is reduced by W due to less consumption. If 2 > W then utility goes up so that the consumer should take on more leisure. Given that the net gain in utility is 2-­‐W regardless of how much leisure is consumed, a consumer will want to take as much leisure as she can. Thus, the optimal bundle is a corner solution in which L = 100, C = 0. The reason is that MU is constant. If instead MU of leisure is decreasing in the amount of leisure then, perhaps at some point, the net effect on taking more leisure will be negative. In that case, there will be some amount of leisure between 0 and 100 at which MRS = slope of the budget line which will be the optimal level of leisure.