Advanced Microeconomics Exam Solutions

Working Through Advanced Micro Problems Without Losing Your Mind

Most students hit a wall in the second semester of graduate micro. The first semester taught you to optimize. The second semester expects you to optimize within systems where every other agent is also optimizing, and nothing is linear, and your professor will ask you to prove existence before you've even had coffee. I spent about eight years TA-ing this material and grading exams across several programs. The solutions you find online tend to fall into two categories: hand-wavy sketch answers that pass for partial credit but would get you a C, or overly formal proofs that ignore what the question was actually testing. There's a middle ground, but you won't find it in the standard test banks.

Advanced Microeconomics Exam Solutions: What Actually Helps

The best solutions I've seen do three things consistently. They state the theorem or result being applied in full generality before using it. They separate the economic intuition from the mathematical machinery so you can spot which part your professor actually cares about grading. And they flag where a standard assumption fails and how to adjust. Take a typical general equilibrium question involving existence. The standard approach invokes the Brouwer fixed-point theorem. That gets you partway. But the trick is recognizing when the exercise is really about continuous dependence on parameters, not just existence. I once graded an exam where a student wrote a perfectly valid Brouwer proof and then lost half the points because they never addressed the question's actual ask about comparative statics implications. The solution should have noted that existence alone doesn't give you the derivative sign you need, and you'd have to invoke differentiability of the excess demand function and the Sonnenschein-Mantel-Debreu result to proceed further. That distinction shows up constantly. Professors dress up questions in different clothing — sometimes it's welfare theory, sometimes it's mechanism design, sometimes it's an asymmetric information problem — but the core algebraic move is often identical. If you're memorizing answers to specific past exams, you'll miss that pattern. If you're learning to identify the underlying structure, you'll solve things faster on the real exam.

Here's a concrete example that comes up more often than it should. A question asks you to characterize the set of incentive-compatible mechanisms in a two-agent setting with quasi-linear utilities and private values. The naive approach sets up the full Bayesian program and tries to solve the differential equation system directly. That works, but it's tedious and prone to sign errors under exam conditions. The faster approach uses the envelope theorem to reduce the problem to the allocation rule's monotonicity condition and a single integral constraint on payments. You verify monotonicity first, which takes about two minutes, then compute the transfer via the integral formula. This cuts the solving time from roughly fifteen minutes down to five or six, and it's the method I ended up teaching my own students because the full program rewards people who don't make arithmetic mistakes under pressure. A few structural things to keep in mind. When dealing with convex analysis in duality problems, the subdifferential characterization is almost always cleaner than working with the primal directly. I found this out the hard way during my own qualifying exam when I tried to derive the conjugate of a non-smooth cost function by brute force and spent twenty minutes on a path that led nowhere. Switching to the subdifferential gave me the answer in three lines. Another counter-intuitive point that doesn't get enough attention: the revelation principle is often more limiting than students realize. It guarantees you can restrict attention to direct truthful mechanisms, but it says nothing about whether those mechanisms are implementable in repeated play or robust to strategic manipulation outside the model's assumptions. I've seen exams reward students who stated the revelation principle as a silver bullet. The better answers flagged the gap between one-shot dominant-strategy implementation and dynamic settings where the principle's power fades fast.

Now, about where the available solution sets fall short. A lot of the freely circulated materials skip over boundary cases and corner solutions. They assume interior optima without checking first-order conditions against the constraint set. If you're practicing exclusively with these, you will lose points on exams that deliberately place you at a kink or a corner. The production function with an essential boundary, the utility function with bliss points, the exchange economy where one agent's endowment is zero in a good — these show up. The solutions usually hand-wave through them. You need to work through at least one example manually per topic to internalize the pattern. There's also a real bottleneck with solution quality for problem sets drawn from Mas-Colell, Whinston, and Green or the Osborne-Rubinstein game theory texts. The MWG exercises are notoriously open-ended. An exam question might ask for a construction that admits multiple valid approaches, and the posted solutions typically present only one. That's fine if you're just checking your answer, but it leaves you unprepared for an examiner who expects you to justify why your particular construction is preferable or complete. I started keeping my own notes on alternate solution paths for the harder problems, and I'd recommend the same habit rather than relying on a single source. If you want something to work from, I maintain a running collection of detailed solutions covering general equilibrium, games with incomplete information, and mechanism design. The files go straight to the standard notation, show the skipped steps, and flag where assumptions matter. You can find it at advancedmicroexamsolutions.org. It's not exhaustive, and it doesn't cover everything, but the sections on incentive compatibility and fixed-point applications are where most students struggle.

The one limitation of the collection I should be honest about: the proofs are written to be correct, not to match any particular professor's preferred level of formality. Some instructors want every epsilon-delta spelled out. Others want the high-level argument with just enough rigor to show you understand the logic. You'll need to calibrate based on your course's track record. I've included remarks at the start of each chapter noting which style the solution leans toward, but it's not perfect. What tends to work best is pairing the solutions with your own attempts. Spend the full allotted time on the problem before looking at anything. Write down the theorem you're going to invoke before you start computing. Afterward, compare your work and note where your path diverged and whether the divergence was due to a missing step or a genuinely different valid approach. That second layer of comparison is where the actual learning happens, not in reading someone else's answer passively. I've watched students go from struggling to pass the midterms to doing solid work on the final by following this routine, and I've also watched capable students stall because they treated the solutions as a shortcut instead of a feedback tool. The material doesn't care how you study it. It only cares whether you can reproduce the arguments under time pressure and extend them when the question changes shape slightly. The solutions help with the first part. The practice does the rest.