Part I
These notes are for sections through Exam One.
1 Introduction
An introduction to mathematical techniques used in ECON 401 and applications to budget constraints: Plotting a line, budget set, budget line, derivatives, chain rule, optimization
2 Preferences and utility functions
Axioms of consumer preferences, monotonicity, convexity, diminishing MRS, perfect substitutes, quasilinear, Cobb–Douglas, Leontieff or perfect complements
- WB 3.11, example 2.4
- WB 4.3, example 2.11
3 Preferences and choices
Deriving the marginal rate of substitution, monotone transformations, convexity, strict convexity, demand functions
- WB 4.7, example 3.4
- WB 4.10, example 3.1
- WB 4.11, example 3.6
- WB 5.1, example 3.8
- Demands for quasilinear preferences
4 Choice, demand, and revealed preference
Definitions, demand function for \(\max\) preferences, income offer curve and Engel curve for Cobb–Douglas, price offer curve for Cobb–Douglas, quasilinear preferences, WARP
5 WARP/SARP, Slutsky, income effect, substitution effect
The sign of the substitution effect, endowments, labor endowments
- WB 7.4, example 5.4
- WB 8.4, example 5.8
- WB 9.2, example 5.11
- WB 9.4, proposition 5.2
- WB 9.3, example 5.14
- WB 9.7, example 9.7
- WB 9.11, example 5.16
6 Intertemporal choice, consumer surplus/EV/CV, market demand
The notes are separated into three parts
- Intertemporal choice
- Consumer surplus, EV, CV
- Market demand
- Practice Exam 41, example 6.7
- WB 14.5, example 6.11
- WB 14.3, example 6.12
- WB 14.6, example 6.13
- WB 14.7, example 6.14
- WB 15.1, exmaple 6.15
- Dog breeders’ demand for electric polishers (?), example 6.17
- WB 15.5, example 6.18
- WB 15.9, example 6.19
- WB 15.10, example 6.20
Worked problems for Exam One
Part II
7 Equilibrium, taxes
Equilibrium supply and demand, quantity tax, tax incidence, deadweight loss, it doesn’t matter who pays the tax
- WB 16.1, example 7.4
- WB 16.2, example 7.5
- WB 16.3, example 7.6
- WB 16.4, example 7.7
- WB 16.6, example 7.8
- WB 16.11, example 7.9
8 Technology, profit maximization, cost minimization, and cost curves
The notes for section 8 are broken into four parts:
- Technology
- Profit maximization
- Cost minimization
- Cost curves
- WB 18.3, example 8.5
- WB 19.9, example 8.6
- WB 20.1, example 8.7
- WB 20.9, example 8.10
- WB 21.1, example 8.12
- WB 21.2, example 8.13
- WB 21.3, example 8.14
- WB 21.4, example 8.15
9 Supply
Firm supply: \(P = MC\) and \(p \geq \min \{ AC(y)\} \). Industry supply: \(S(p) = \sum_{i=1}^{n} S_i(p) \) and zero-profit condition.
The notes for section 9 are broken into two parts:
- Firm supply
- Industry supply
- WB 23.1, example 9.2
- WB 23.3, example 9.3
- WB 23.5, example 9.4
- WB 23.7, example 9.5
- WB 23.8, example 9.6
- WB 23.9, example 9.7
- WB 24.4, example 9.8
10 Monopoly
\(MR=MC\), firms will not produce where the demand curve is inelastic, the markup.
- WB 25.3, example 10.2
- WB 25.5, example 10.3
Review for Exam 2
A very brief review for Exam 2
Part III
11 Monopoly behavior and game theory
First-, second-, and third-degree price discrimination; the monopolist that sells to two markets charges a higher price in the market where demand is less elastic with respect to price (not because students are poorer!), definition of a game, Nash equilibrium, contributions to a public good
The notes for section 11 have two parts:
- Monopoly behavior—getting to know the consumer
- A too brief introduction to game theory
- WB 29.1, example 11.9
- WB 29.2, example 11.10
- WB 29.3, example 11.11
- WB 29.4, example 11.12
- WB 29.8, example 11.13
12 Oligopoly and auctions
The notes for this section primarily present proposed solutions for the workbook questions.
The notes for section 12 have two parts:
- Oligopoly
- Auctions
Oligopoly:
- WB 28.4, example 12.2
- WB 28.5, example 12.3
- WB 28.7, example R3.16
- WB 28.8 (or 27.8 from the edition on the Canvas website), this handout
- WB 28.9, example R3.18
Auctions:
- WB 18.1, example 12.5
- WB 18.2, example 12.6
- WB 18.3, example 12.7
- WB 18.4, example 12.8
- WB 18.5, example 12.9
- WB 18.6, example 12.10
- WB 18.7, example 12.11
- WB 18.8, example 12.12
- WB 18.9, example 12.13
- WB 18.10, example 12.14
13 General equilibrium
These notes go through economic analysis of the Edgeworth box. Allocation, feasible allocation, Pareto dominates, Pareto optimal, contract curve.
IMPORTANT: In Varian, the set of Pareto optima is the contract curve. What I call in the notes the contract curve is called the core in Varian. The bright part of this is that you can now identify the set of Pareto optimal, the contract curve, and the core—one more item to cross off the list of things to know.
The notes for section 13 have two parts. There are also two handouts. You may want to start with the two handouts, which go through finding the set of Pareto optima and the set of Walrasian equilibria more slowly.
- General equilibrium: The Edgeworth box
- More on the Edgeworth box including Walras’s law
- Handout for perfect complements
- Handout for quasilinear
- WB 32.1, example 13.1
- WB 32.2, example 13.3. In this example, \(A\)’s preferences are Cobb–Douglas; in particular, \(A\) utility is equivalent to the utility function \(U_{A}=x_{A1}^{1/2} x_{A2}^{1/2}\). When \(p_1 = 1\) and \(p_2 = p\), the budget constraint faced by \(A\) is \(x_{A1} + p x_{A2} = \omega_{A1} + p \omega_{A2}\). Which makes the demand functions \[ x_{A1}^{\star} = ``\frac{\alpha m}{p_1}" = \frac{\frac{1}{2} (\omega_{A1} + p \; \omega_{A2} )}{1}\] and \[ x_{A2}^{\star} = \frac{\frac{1}{2} (\omega_{A1} + p \; \omega_{A2})}{p}.\] So you already know a tremendous amount about demand functions.
- WB 32.4, example 13.4
- WB 32.5, example 13.6
- WB 32.7, example 13.7
- WB 32.8, example 13.8
14 Externalities,
Externalities
- WB 35.1, example 14.1
- WB 35.3, example 14.2
- WB 35.5, example 14.5
- WB 35.8, example 14.3
- WB 35.9, example 14.4
15 Public goods
Here are the notes on public goods
- WB 37.1, example 15.1
- WB 37.3, example 15.2
- WB 37.4, example 15.3
- WB 37.5, example 15.4
- WB 37.6, example 15.5
- WB 37.7, example 15.6
R3: Review for Exam 3
Review notes
- Old Exam 3.15, this handout. My answer agrees with the book: What is called the “contract curve” in the handout is called the “core” in Varian. Varian calls the set of Pareto optima the contract cure and what I’ve been calling the contract curve the “core.” Got that?
- Suggested solutions for selected problems from Exam 2. Unfortunately there we do not have a common version of the exam to work of off (not everyone has adopted the Ryan labeling method). Anyway, here are a few suggested solutions. And a few problems from Old Exam 3.
- Extra Practice Problems, #75. The demand curve is \(y\) is given by \(y=256/p^2\). Only two firms produce \(y\). Each firm faces costs \(c(y)=y^2\). If they agree to collude and maximize their joint profits, how much output will each firm produce? Let \(y_1\) and \(y_2\) be the output of the two firms. Right away, from proposition 8.2 we know that both firms will produce that same amount of output. The profit function is \[ \pi (y_1,y_2) = 16(y_1+y_2)^{-1/2}(y_1 + y_2) - y_1^2 - y_2^2.\] Which can be simplified to \[16 (y_1+y_2)^{1/2} - y_1^2 - y_2^2.\] The first-order condition for \(y_1\) is \[8(y_1+y_2)^{-1/2}-2y_1 =0.\] The first-order condition for \(y_2\) is similar. Now we want to use proposition 8.2. How do we do that? What if we add the two first-order conditions? Doing so yields \[16(y_1+y_2)^{-1/2}=2(y_1+y_2).\] Let \(y=y_1=y_2\). Then our developing equation becomes \[8 = (2y)^{3/2} = 8^{1/2}y^{3/2}.\] Then \(8^{1/2}=y^{3/2}\) or \(8 = y^3\) or \(y=2\). That is, both firms produce \(2\). Or you can use proposition 8.2 immediately—the marginal costs must be equal at the two firms, which means \(y_1=y_2=y\) and the profit function can be writte \(\pi (y) = 16(2y)^{1/2}-2y^2\). Now profit-max with respect to \(y\). See remark 8.18.