Working Through the Edgeworth Box and Contract Curve
The answer key for this section covers the exchange economy problem set from my intermediate micro course. You are looking at Edgeworth box diagrams, contract curves, competitive equilibrium, and whether allocations are Pareto efficient or not. I have seen students stall on this one for weeks because the geometry looks deceptively simple until you try to derive actual numbers. The problem asks you to take two consumers with Cobb-Douglas utilities over two goods and find every allocation that cannot be improved upon without making someone worse off. The utility functions look like uA = xA^0.6 * yA^0.4 and uB = xB^0.3 * yB^0.7 with total endowments of X = 10 and Y = 8. The answer key walks through the algebra of setting MRS_A equal to MRS_B, which gives you the contract curve equation. If you skip that step you will guess wrong on part c, which asks you to verify whether a specific allocation lies on the curve. I spent an entire grading cycle watching students write the contract curve as MRS_A = MRS_B and stop there. That is not an answer, it is a restatement of the question. The actual work is expanding the ratio of marginal utilities and solving for y in terms of x. For these parameters it simplifies to yA = 0.75 * xA / (10 - xA) when you account for the box constraints. The answer key shows this derivation across three sub-steps so you can catch arithmetic mistakes before they compound into the general equilibrium price ratio.
Competitive Equilibrium and the First Welfare Theorem
Part d shifts from the contract curve to competitive equilibrium. You pick a price vector, calculate demand for each consumer using their budget constraint, then clear each market. The trick here is that the endowment point matters for individual budgets even though total resources are fixed. I used to tell students to normalize prices to p_x + p_y = 1 but that just hides the ratio p_x/p_y that actually determines the equilibrium allocation. The answer key keeps the relative price throughout so the final allocation checks out against the contract curve independently. There is a boundary case that the textbook glosses over. If one consumer has an endowment of zero in both goods, their demand is undefined at any positive price. In the answer key this shows up as a corner solution where the competitive equilibrium allocates all of good Y to consumer B and all of good X to consumer A, but only if the utility weights support it. When I encountered a version with identical preferences but asymmetric endowments, the contract curve collapsed to a single point and the competitive equilibrium became indeterminate without an extra normalization condition. The standard answer key does not cover this, so I added a footnote explaining that you need to check interiority conditions before applying the Walrasian approach.
Pareto Efficiency Checks and Common Mistakes
The answer key includes a verification section where you test three candidate allocations. Two lie on the contract curve and one does not. The incorrect one fails because the marginal rates of substitution diverge. Students usually miss this by rounding too early. I recommend keeping four decimal places through the MRS calculation and only rounding the final allocation. That habit cut my grading time from about forty minutes per problem set to roughly twelve. Another thing the answer key does not emphasize enough is that the contract curve only exists inside the box. If you extend the MRS equality beyond the feasible region you get mathematically correct but economically irrelevant solutions. I once graded a midterm where a student derived a contract curve segment that required negative quantities. The algebra was flawless but the diagram was wrong. The answer key's graph includes the box boundary explicitly so you can see where the curve enters and exits.
Using the Answer Key Without Losing Learning Value
The most practical use is to attempt the problem set first, then check only the final allocation numbers and the MRS equality verification. If your contract curve equation differs, trace back through the MRS derivation rather than copying the answer. The competitive equilibrium price ratio should match the slope of the supporting budget line at the equilibrium allocation. If it does not, you likely normalized prices incorrectly or mixed up the endowment budget constraint. I keep a separate spreadsheet where I enter different parameter combinations and watch how the contract curve bends. Changing the exponent on good X from 0.6 to 0.8 shifts the curve toward consumer B's origin. This visual check takes about five minutes and prevents the common error of assuming the contract curve is always a straight line from one corner to the other. It is only straight when preferences are homothetic and symmetric, which these Cobb-Douglas specifications satisfy only under special parameter restrictions.
Microeconomics Lesson 4 Activity 34 Answer Key download notes
The file is a PDF with seven pages, including the problem statement, full derivations, and a separate answer sheet for self-checking. There is a typo on page four where the endowment total for good Y is listed as 9 instead of 8. The rest of the algebra is correct. I flagged this with the instructor and they released a corrected version the following week. If you are using the original, substitute Y-bar = 8 in the market-clearing condition for good Y and the rest follows. The answer key does not include a section on the second welfare theorem, but it is worth noting that any Pareto efficient allocation on the contract curve can be supported as a competitive equilibrium with appropriate lump-sum transfers. The theorem assumes convex preferences and no externalities, which holds here but breaks down if you switch to Leontief or perfect substitutes utilities. I added a practice problem testing Giffen behavior in a related activity and found that the competitive equilibrium computation requires a different approach when demand curves slope upward. The standard answer key method assumes downward-sloping demand, so students should not apply it blindly outside the Cobb-Douglas case.
