You wish to retire after 20 years; at which time you want to have accumulated enough money to receive an annuity of $60,000 a year for 25 years of retirement. During the period before retirement you can earn 4 percent annually, while after retirement you can earn 3 percent on your money. What annual contribution to the retirement fund will allow you to receive the 60,000 annually?

David consumes two things: gasoline (g) and bread (b). david's utility function is u(g, b) = 10g^0.25 b^0.75. use the lagrange technique to solve for david's optimal choices of gasoline and bread as a function of the price of gasoline, p_g, the price of bread, p_b, and his income m. with recent decrease in the price of gasoline (maybe due to external shock such as shale gas production) does david increase his consumption of gasoline? for david, how does partial differential g/partial differential p_g depend on his income m? that is, how does david's change in gasoline consumption due to an increase in the price of gasoline depend on his income level? to answer these questions, find the cross-partial derivative, |partial differential^2 g/partial differential m partial differential p_g.