18 Econ Models
According to (Varian 2016, 1997), basics of economic model include:
People making the choices
Possible constraints they face
Interaction among agents
What adjusts if the choices aren’t mutually consistent?
Always start with the simplest model
one period
2 goods
2 people
linear utility
The goal is to have the simplest model, not to have the most complex one, because model is just a simplified version of reality. Then, you can generalize your model when you have the simplest one.
This sections is a summary of professor Matilde Machado’s course on industrial organization
18.1 Concentration Measures
Competition lies between two extremes:
Percent competition (minimum concentration)
Monopoly (maximum concentration)
Concentration measures approximates the level of competition in a market between the two extremes.
We can use concentration measures to
Compare different markets
Regulate the market
Concentration indices should be independent of the market size and in the interval of [0,1] where 1 means monopoly and 0 means perfect competition
Concentration curve: x-axis = number of firms, y-axis = % of total production
Hannah and Kay (1977) argue for characteristics of concentration index:
- Classification can be shown on the concentration curve between markets
- Principles of Transfer of Sales (e.g., a small firm sells to a large on will increase the concentration index)
- Entry or exit condition (e.g., entries decreases concentration, while exits decrease concentration, typically applies to small firms only)
- Merger condition (should increase concentration)
Examples of concentration indexes
-
The inverse of the number of firms (\(\frac{1}{n}\))
Good when firms are of the same size
Does not account for the “transfer of sales” (i.e., number of firms does not change with transfer of sales)
-
The concentration ratio (\(C_r\))
sum of the market shares of the largest \(r\) firms
but \(r\) is rather arbitrary
hard to compare between industries
-
Hirschman-Herfindahl index
account for all points of the concentration curve
satisfies Hannah and Kay (1977) criteria
gives more weights to larger firms
Issues with concentration measures
Can’t account for cross-ownership
-
Depend heavily on boundary conditions of the industry/market
- Definitions usually contain geography and products.
Can’t reflect market evolution (i.e., static measures). Hence, one can also look at market stability
Market power can be measured by
Surplus over marginal cost (but MC is hard calculate)
Price-elasticity of the residual-demand (high elasticity means lower power)
Market shares or concentration measures
18.2 Monopoly
Def: only player in a market
Barriers to entry:
- Economies of scale or sunk costs
- Patents/Licenses
- Cost advantages (based on technology)
- Customer loyalty (or high switching costs)
18.2.1 The standard model
\[ p = P(Q) \]
price is a function of quantity produced
Assumptions:
Single goods
Full Info (i.e., consumers know product’s characteristics)
Negatively sloped demand curve: \(\frac{d D(p)}{d p}<0\)
Marginal costs are non-negative: \(\frac{d C(q)}{dq} \ge 0\)
Uniform pricing (i.e., same price for all consumers and all goods)
Rational monopolists maximize their profits:
\[ \displaystyle \max_{q} \Pi = p(q) q - C(q) = TR - TC \]
FOC: \(p(q) + p'(q) q = c'(q)\). Hence, marginal revenue = marginal cost
Intuitively,
If \(MR < MC\), it means each extra unit we produce will be lower than its cost. Hence, we should produce more until it reaches \(MR = MC\)
If \(MR > MC\), it means each extra unit we produce will be higher than its cost. Hence, we should reduce production until it reaches \(MR=MC\)
FOC:
\[ \begin{aligned} p(q) + p'(q) q &= c'(q) \\ p(q) - c'(q) &= - p'(q)q \\ \frac{p(q) - c'(q)}{p(q)} &= - \frac{\partial p}{\partial q} \frac{q}{p} = \frac{1}{\epsilon (q)} \\ p(q) [1 - \frac{1}{\epsilon (q)}] &= c'(q) \\ p(q) &= \frac{c'(q)}{1 - \frac{1}{\epsilon (q)}} \end{aligned} \]
where
\(\frac{p(q) - c'(q)}{p(q)}\) is the Lerner index (a measure of market power that allows cross-market comparisons)
\(\frac{1}{\epsilon (q)}\) is the inverse of the demand elasticity
Hence, monopolists always choose to produce at \(\epsilon (q) >1\) (from the last derivation)
Alternatively, we can also write the maximization problem in terms of quantity (instead of price)
\[ \displaystyle \max_{p} \Pi = p D(p) - C(D(p)) \]
FOC
\[ \begin{aligned} D(p) + pD'(p) &= C'(D(p))D'(p) \\ D'(p) [p- C'(D(p))] &= -D(p) \\ \frac{p - C'(D(p))}{p} &= \frac{1}{D'(p)} \frac{D(p)}{p}= \frac{1}{\epsilon (q)} \end{aligned} \]
If costs are linear \(c(q) = c \times q\) and linear demand \(p(q) = a - bq\)
\[ \displaystyle \max_{q} \Pi = p(q) q - C(q) = (a-bq) q - cq \]
FOC:
\[ a - 2bq = c \\ q^M = \frac{a-c}{2b} \]
\(q^M >0\) iff \(a>c\) where \(a\) represents the willingness to pay for the first unit
Thus, \(p^M = a -b \frac{a-c}{2b} = \frac{a+c}{2} >c\) since \(a > c\)
Thus, \(\Pi^M = \frac{1}{b}(\frac{a-c}{2})^2\)
18.2.1.1 Perfect Competition
Assumptions
A large number of firms leads to price-taking
Free entry and exit
Homogeneous products across firms
At equilibrium,
Price = Marginal cost (\(p^c = MC\))
Zero profits (\(\pi^c = 0\))
Efficiency (maximizes total welfare = consumer surplus + producer surplus)
Comparison:
- \(p^M > (p^c = c) \to CS^M < CS^c\) where \(CS\) means consumer surplus
- \(\Pi^M > \Pi^c = 0 \to PS^M > PS^c = 0\) where \(PS\) means producer surplus
- \(DWL = TS^c - TS^M > 0\) where DWL = Deadweight Loss (i.e., loss of total surplus)
- Consumers with valuation higher than MC (but lower than \(p^M\)) can’t buy
18.3 Pricing
18.3.1 Price Discrimination
Examples:
Discounts (e.g., students, seniors, coupons)
Tariff
Def: When a seller price the good at 2 different prices (either to the same or different consumers), we say there is price discrimination (exception: different prices at different locations that reflect differences in costs).
Firms can only do price discrimination when arbitrage is not possible, which are
-
Product arbitrage (linked to the transferability of the commodity) happens when transaction costs are low, which allow those with lower price to resell to other consumers. And prevention include
Services
Product warranties
Product specificity
Transaction costs
Contractual clauses
Vertical integration
Government Intervention
Demand arbitrage (linked to transferability of the demand)
Types of price discrimination
-
First-degree (perfect price discrimination):
customer willingness to pay (\(v_i\)) = price (\(p_i\))
Two-part tariff: \(T(q) = A + pq\) where A (access fee) is fixed and paid independently of the quantity consumed and p (unit fee) is the variable part.
Individual tariffs - continuous demand: \(T_i(q) = A_i + p^c q\) where \(p^c = MC\) and \(A_i = S_i^c\) (i.e., consumer i’s surplus when \(p = p^c\))
Second-degree (incomplete info forces the use of self-selection devices)
Third-degree (leverage signal to separate markets)
18.3.2 Bundling
selling more unit of the same good together
similar to quantity discounts (under Price Discrimination)
18.4 Cournot Model
Assumptions:
Homogeneous product across firms
Market price is a function of the total supply
Simultaneous game (decision on production)
Quantity is the variable of interest, which has Cournot-Nash equilibrium
18.4.1 Symmetric Duopoly
Consider a duopoly (\(n=2\)) and \(MC = c\), then residual demand of firm 1 is
\[ RD_1(p, q_2) = D(p) - q_2 \]
Then this is a similar set up as The standard model. And optimal value for firm 1 production is a function of function 2’s production \(q^*1 (q_2) = R_1 (q_2)\)
Case 1: When \(q_2 = 0\), then \(RD_1(p,0) = D(p)\) then firm 1 becomes a monopolist \(q^*_1(0) = q^M\)
Case 2: \(q_2 = q_c\) then \(RD_1(p, q^c) = D(p) - q^c\), since both demand and cost functions are linear, the reaction function will also be linear.
Market price will be set as
\[ P = a - bQ = a-b(q_1 + q_2) \\ MC_1 = MC_2 = c \]
Consider firm 1:
\[ \displaystyle \max_{q_1} \Pi_1(q_1,q_2) = (p-c)q_1 = (a-b(q_1 + q_2)-c)q_1 \]
FOC:
\[ \begin{aligned} \frac{\partial \Pi_1}{\partial q_1} &= 0 \\ a -bq_1 - b_2 - c- b_1 &= 0 \\ q_1 &= \frac{a-c}{2b} - \frac{q_2}{2} \end{aligned} \]
Since the two firms are symmetrical, FOC for firm 2 should be identical
Now we solve for the optimal quantity, which results in
\[ q^N = \frac{a-c}{3b} = q_1^N = q_2^N \]
where \(N\) superscript means Cournot solution
Total quantity is
\[ Q^N = q_1^N + q_2^N = \frac{2}{3}(\frac{a-c}{b}) \]
and market price is
\[ p^N = a - bQ^N = a - \frac{2}{3}(a-c) = \frac{a+2c}{3} \]
18.4.2 Symmetric Oligopoly
In equilibrium
| Perfect Competition (\(C\)) | Cournot (\(N\)) | Monopoly (\(M\)) | ||
|---|---|---|---|---|
| \(p^c = c\) | < | \(p^N = \frac{a+2c}{3}\) | < | \(p^M = \frac{a+c}{2}\) |
| \(\frac{\partial p^c}{\partial c} =1\) | > | \(\frac{\partial p^N}{\partial c} = \frac{2}{3}\) | < | \(\frac{\partial p^M}{\partial c} = \frac{1}{2}\) |
In general (\(n \ge 2\)) firms
\[ \displaystyle \max_{q_1} \Pi_1(q_1, \dots, q_N) = (a - b(q_1 + \dots + q_N) - c) q_1 \]
FOC
\[ a-b(q_1 + \dots + q_n) - c- bq_1 = 0 \\ q_1 = \frac{a-b(q_2 + \dots + q_N) - c}{2b} \]
If all firms are symmetric (i.e., \(q_1 = \dots = q_N = q\))
\[ q_1 = \frac{a - b(n-1)q - c}{2b} \\ q^N = \frac{a-c}{(n+1)/b} \]
Equilibrium quantity
\[ Q^N = nq^N = \frac{n}{n+1} \frac{a-c}{b} \underset{n \to \infty}{\operatorname{\to}} \frac{a-c}{b} = q^c \]
Equilibrium price
\[ p^N = a- bQ^N = a - b\frac{n}{n+1}\frac{a-c}{b} = \frac{a}{n+1} \frac{n}{n+1}c \underset{n \to \infty}{\operatorname{\to}} c \]
Hence, as n goes to infinity (i.e., oligopoly converges to a perfect competition), which is the same equilibrium quantity and price as in the perfect competition case.
Deadweight loss in the Cournot model is the area where the willingness to pay is higher than MC.
\[ \begin{aligned} DWL &= \frac{1}{2}(p^N - p^c) (Q^c - Q^*) \\ &= \frac{1}{2b}(\frac{a-c}{n+1})^2 \underset{n \to \infty}{\operatorname{\to}} 0 \\ \end{aligned} \]
Asymmetric duopoly
with constant marginal costs
Assume we have linear demand \(P(q_1 + q_2) = a-b(q_1 + q_2)\)and
\(c_1 = MC_1\) and \(c_2 = MC_2\)