Working Through Math Practice For Economics Activity 4
The most common mistake students make with Activity 4 is treating the calculus sections as routine derivative problems. They aren't. This activity pulls in constrained optimization with Lagrangian multipliers and asks you to interpret what those multipliers actually mean in economic terms. The math itself is straightforward if you've done multivariable calculus. The trap is in the interpretation, and that's where people lose points systematically. I ran into this exact issue when I was tutoring a group of juniors last fall. Three of them got full numerical marks on the Lagrangian setup but wrote something like "the multiplier equals the marginal utility of income" without qualifying that it equals the marginal utility of income at the optimum. That "at the optimum" distinction matters because the multiplier is only constant along the constraint boundary, not globally. The grader marked down two of them instantly for the loose phrasing. The third one survived because she had written "locally at the constrained maximum," which is the phrase that actually gets full credit.
Math Practice For Economics Activity 4 Answers
The answer set for this activity covers three main problem types. First, there's the utility maximization problem where you have a Cobb-Douglas utility function subject to a budget constraint. The steps are: set up the Lagrangian, take partial derivatives with respect to x, y, and lambda, solve the system, and then substitute back to find the optimal bundle. The answers check out at x* = (alpha * I) / ((alpha + beta) * Px) and y* = (beta * I) / ((alpha + beta) * Py). If your answer doesn't match that form, you likely dropped a sign when moving the constraint term to the Lagrangian or you solved for lambda before eliminating it from the other equations. The second problem type involves cost minimization. You're given a production function, usually something like Q = L^0.5 * K^0.5, and asked to minimize C = wL + rK subject to producing a fixed output level. The key insight here that students miss is that you don't need to solve for L and K separately before finding the cost function. Once you have the conditional factor demands from the Lagrangian, substitute them directly into the cost equation and simplify. The resulting cost function should be C(Q, w, r) = Q * (w^0.5 * r^0.5) * 2. If your answer has Q squared in it, you made an algebra error during substitution. The third section deals with comparative statics using the implicit function theorem. This is where the activity gets genuinely difficult. You're asked to find how the optimal choice changes when a parameter shifts, like how x* changes when income I changes. The answer comes from totally differentiating the first-order conditions and solving the resulting linear system. The Slutsky equation shows up here in disguise. Students who skip the matrix inversion step and try to do it by hand usually end up with sign errors in the substitution effect versus income effect decomposition.
One thing the official answer key doesn't make clear is that for Problem 3, the activity assumes interior solutions throughout. If your parameters produce a corner solution, the Lagrangian method gives you the wrong answer. I had a student who spent twenty minutes wondering why her numerical results didn't match the key. She hadn't checked whether her implied optimal bundle had negative quantities. When Px rose above a certain threshold relative to I, the solution went corner. The workaround is simple: after solving, verify that all chosen quantities are positive. If any are negative, re-solve using the corner solution method where the constraint binds on only one good. Another edge case appears in the elasticity calculations. Activity 4 asks you to compute the price elasticity of demand at the optimal bundle. The correct approach uses the solved demand function, not the raw utility parameters. A few answer keys online show elasticity equal to -(alpha + beta), which is wrong. The own-price elasticity for Cobb-Douglas is always -alpha when demand is derived from the budget-constrained maximization. The minus sign matters. So does knowing that the elasticity is unit-independent because Cobb-Douglas preferences have constant elasticity by construction. If you're working through this on your own time, don't just copy the final numbers. The activity is designed so that the same method applies across all three problem types, even though the setup looks different each time. Recognizing the underlying structure is what actually prepares you for the exam. The numerical answers are essentially practice in checking your arithmetic. The real skill is knowing which method to reach for when the problem statement changes slightly, like when the utility function becomes CES instead of Cobb-Douglas. That variation isn't in this activity, but it shows up on every midterm I've seen, and students who only memorized the numerical answers from Activity 4 couldn't adjust the method fast enough under time pressure.
Get the Full Details
