# EC 101A Final

### Introduction

Notes are from the 12th edition of Microeconomic Theory by Nicholson and Snyder, which you can find here. Check out the notes on Midterm 1 and Midterm 2. Good luck on finals!

#### Oligopoly

Under an oligopoly, only $$p_1 = c = p_2$$ is a unique Nash Equilibrium. Neither firm has an incentive to deviate, and for any other case, at least one firm can profitably deviate.

Under Cournot oligopoly, firms choose quantities. Under Bertrand oligopoly, firms choose prices and then produce the quantity demanded by market. Bertrand Competition thus concludes that just two firms guarantee perfect competition, whereas Cournot oligopolies produce above the marginal cost. In a Stackelberg oligopoly, one firm makes the quantity decision first.

Suppose we have cost $$c(y) = cy$$ with demand $$p(Y) = a-bY$$. Under monopoly, we have $$y_M = \frac{a-c}{2b}$$ and $$p_M = \frac{a+c}{2}$$. Under Cournot, we have $$Y_D = 2\frac{a-c}{3b}$$ and $$p_D = \frac{1}[3} + \frac{2}{3} c$$. Under Stackelberg, we have $$Y_D = 3\frac{a-c}{4b}$$ and $$p_D = \frac{1}{4}a + \frac{3}{4}c$$.

#### Auctions

In second-price auctions, each individuals has a private value $$v_i$$ and a bid $$b_i$$. Through casework, we can show that $$u_i(v_i, b_{-i}) \geq u_i(b_i, b_{-i})$$. In other words, betting one’s own value is weakly dominant. However, in the real world, users of eBay may bid multiple times or overbid.

#### Dynamic Games

 In dynamic games, one player plays after the other. The second player must always express their strategy in the form of a condition (Choice 2 Choice 1). We can consider the subgame-perfect equilibriums, where the player chooses the highest-payoff strategy given the other players’ strategy. Dynamic games can enable some cooperation.

#### Edgeworth Box

An edgeworth box involves two consumers $$i = a, b$$ with two goods $$x_1, x_2$$ where each consumer $$i$$ has an endowment $$\omega^i_j$$ of good $$j$$. We denote total endowment $$(\omega_1, \omega_2) = (\omega_1^a + \omega_1^b, \omega_2^a + \omega_2^b)$$. The individual rationality condition is satisfied where $$u_i(x_1^{i*}, x_2^{i*}) \geq u_i (w_1^i, w_2^i)$$ while Pareto efficiency denotes the optimal utility for all agents. The barter lens is the area that satisfies the individual rationality condition. The Pareto set is the set of points where indifferent curves are tangent; the subset within the individually rational area is called the contract curve.

Suppose both consumers face budget constraints such that $$p_1x_1 + p_2x_2 \leq p_1\omega_1 + p_2\omega_2$$ A Walrasian Equilibrium denotes a point where each consumer maximizes utility subject to their budget constraint. The offer curve for each consumer is the set of points that maximize utility as a function of $$p_1, p_2$$. At the intersection of these points, both individuals maximize utility given prices and the total quantity demanded is equal to the total endowment.

An increase in the endowment $$\omega$$ increases final consumption. An increase in $$p_2/p_1$$ increases $$x_1$$ and decreases $$x_2$$.

#### Asymmetric Information

For workers, drivers, and CEOs, the agent’s effort is unobserved, so principal, whether the employer, insurer, or shareholder, has to use a proxy. Agents may get bad luck and be punished, or get lucky and be rewarded for low effort. This is a hidden action or moral hazard.

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