Lagrangian
LEARNING GOALS:
- II.3: Solve multivariable constrained optimization problems with Lagrangians; e.g., utility maximization, expenditure minimization, profit maximization with inputs, cost minimization.
- II.4: Interpret the economic meanings of the first order conditions, tangency conditions, and Lagrange multipliers in the various problems.
- III.A.1: Given prices and income, determine a consumer’s budget set. Determine how the budget set changes with changes in those variables.
Consider the Cobb-Douglas utility function $u(x_1, x_2) = x_1^\frac{1}{4}x_2^\frac{3}{4}$. Let the budget constraint be $p_1x_1 + p_2x_2 = I$, where $p_1$, $p_2$ are the prices and $I$ denotes the income.
- Write the Lagrangian for this utility maximization problem.
- Solve the first-order conditions to find the demand functions for both good 1 and good 2. [Hint: Your results should only depend on the parameters $p_1$, $p_2$, $I$.]
- Using the adjustable slider in the graph above, increase the price of the first good, $p_1$, while holding $p_2$ constant. What do you notice about demand for each good? What happens to the consumer’s utility at the optimal choice? Again, adjust the slider above, this time holding $p_1$ constant and increasing $p_2$. Based on this intuition and the demand functions you derived in part (b), explain the behavior you observed.
- Lastly, solve for $\lambda$, the Lagrange multiplier using the values $p_1 = 4.0$, $p_2 = 3.0$, $I = 120$. Calculate the utility of the consumer given these values. Recalculate the consumer’s utility if their income is instead $I = 121$. How much has their utility increased? How does this value compare to $\lambda$, the Lagrange multiplier?