Question from Past Microeconomics Qualifying Exam

Fall 2000 - Section II, Question four, George Mason University

Consider the plight of a husband. His utility funciton is ${\displaystyle U(c,a)}$ where ${\displaystyle c}$ is his annual consumption and ${\displaystyle a}$ is the amount of "female attention" he receives in a given year. Assume ${\displaystyle Uc>0}$ and ${\displaystyle Ua>0}$. Female attention, ${\displaystyle a}$, is bestowed by the husband's wife, with ${\displaystyle a=f(g,y)}$, where ${\displaystyle g}$ is the annual amount of gifts (of consumption goods) the husband gives to his wife, and y is how long they have been married. The husband earns a fixed income ${\displaystyle W}$ (of consumption goods) each year.

• a. Write out the optimization problem faced by the husband each year. Clearly label his choice variables, his objective, and any constraints he faces. Use calculus to solve for the husband's optimal choices, and verbally interpret any first-order conditions.
• b. Suppose ${\displaystyle f'g>0}$ and ${\displaystyle f'y<0}$. Interpret these conditions verbally. How would the evolution of the marriage depend on the sign and magnitude of ${\displaystyle f'g}$? Use the logic of income and substitution effects to explain what happensto the husband's gift giving as marriage matures.
• c. Mathematically show how the duration of the marriage, ${\displaystyle y}$, affects gift giving by the husband. Explain your results.