The Game

Suppose that there are K firms in a market, and that each of the k ∈ K firms produces n_k products, some of which may be produced domestically and some of which may be produced abroad.³ Further, suppose that the government imposes an ad valorem tariff t_i on a subset of products produced abroad. Note that we assume that the tariff is calculated as a proportion of consumer price, so that the price received by the firm is 1 − t_i the price faced by consumers.

Let $n=\sum\limits_{k\in K}n_k$ denote the number of products sold by all K firms. The Bertrand model assumes that firms simultaneously set their products’ prices in order to maximize their profits. This model also assumes that all firms can perfectly observe each others’ prices, quantities, costs, and product characteristics.

This version of the Bertrand model also assumes that each product is produced using its own distinct constant marginal cost technology c_i, for all i ∈ n. As we will see, this assumption is necessary when information is limited.

Firm k ∈ K chooses the prices {p_i}_i = 1^n_k of its products so as to maximize profits. Mathematically, firm k solves:

where t_i equals 0 if the product is not subject to a tariff. q_i, the quantity sold of product i, is assumed to be a twice differentiable function of all product prices.

Differentiating profits with respect to each p_i yields the following first order conditions (FOCs):

which may be rewritten as

where $r_i\equiv\frac{p_iq_i}{\sum\limits_{j=1}^np_jq_j}$ is product i’s revenue share, $\hat{c}_i\equiv \frac{c_i}{(1-t_i)}$ is product i’s effective marginal cost, $m_i\equiv\frac{p_i-\hat{c}_i}{p_i}$ is product i’s gross margin, and $\epsilon_{ij}\equiv\frac{\partial q_i}{\partial p_j}\frac{p_j}{q_i}$ is the elasticity of product i with respect to the price of product j.

The FOCs for all products may be stacked and then represented using the following matrix notation:

where r and m are n-length vectors of revenue shares and margins, $E = \left(\begin{smallmatrix} \epsilon_{11}&\ldots&\epsilon_{1n}\\\vdots &\ddots&\vdots\\\epsilon_{n1}&\ldots&\epsilon_{nn} \end{smallmatrix}\right)$ is a n × n matrix of own- and cross-price elasticities, and $\Omega=\left(\begin{smallmatrix} \omega_{11}(1-t_1)&\ldots&\omega_{1n}(1-t_1)\\\vdots &\ddots&\vdots\\\omega_{n1}(1-t_n)&\ldots&\omega_{nn}(1-t_n)\end{smallmatrix}\right)$ is an n × n matrix whose i, jth element equals ω_ij, the share of product j’s profits owned by the firm setting product i’s price, multiplied by 1 − t_i, product i’s tariff rate. In many cases, product i and j are wholly owned by a single firm, in which cases ω_ij equals 1 if i and j are owned by the same firm and 0 otherwise. ‘diag’ returns the diagonal of a square matrix and ‘∘’ is the Hadamard (entry-wise) product operator.

The solution to system @ref(eq:FOC) yields equilibrium prices conditional on the ownership structure Ω and tariffs t_i, ∀i ∈ n. A change in tariff policy is modeled as the solution to system @ref(eq:FOC) where Ω and ĉ_i, ∀i ∈ n are updated to reflect the change in tariffs.

Calibrating Model Demand and Cost Parameters

For the Logit, CES, and AIDS demand specifications allowed under this implementation of the Bertrand model, the calibration strategy is the same. First, we assume that quantities/shares and (with the exception of LA-AIDS) prices are observed for all products in the market, and that margins for some products are observed. Our decision to treat quantities, prices, and margins as primitives comes directly from equation @ref(eq:FOC).

In addition to quantities, prices, some margins and capacities, we assume that users observe diversion ratios. Diversion ratios come in two forms: quantity diversion and revenue diversion. The quantity diversion from product i to product j, d_ij^q, is defined as the percentage of all of i’s lost unit sales that switch to j due to a price increase in product i, while the revenue diversion from product i to product j, d_ij^r, is defined as the percentage of all of i’s lost revenue that switches to j due to a price increase in product i. Mathematically, quantity and revenue diversion may be represented as

Note that d_ij^q, d_ij^r are restricted to be between -1 and 1, and are positive if products i and j are substitutes and negative if they are complements. Additionally, conditional on customers switching from product i, they must switch to another product (i.e. ∑_jd_ij ≤ 0). For all the models included in trade, we assume that i and j are not complements (d_ij ≥ 0) and for some demand models (i.e. AIDS) we will assume that ∑_jd_ij = 0.

Although diversion ratios are not present in equation @ref(eq:FOC), these definitions indicate that diversion ratios may be helpful in recovering the matrix of own- and cross-price elasticities E. Indeed, for a number of the demand systems described below, diversions will be used for just this purpose.

We further assume that all of this information represents the outcome of the unique current equilibrium for firms in the market playing the static Bertrand pricing game described above. We then substitute observed margins, shares and prices into equation @ref(eq:FOC), which is now solely a function of demand parameters, and then solve for the coefficient(s) on prices. Once the price coefficients have been estimated, we use observed prices and the demand equations to estimate the intercepts.

Often, there are more FOCs than unknown price coefficients. For instance, under Logit demand, only the price parameter α and the share of the outside good s₀ needs to be estimated and up to n FOCs with which to estimate it. This means that at a minimum, users need only supply enough margin information to complete a single product’s FOC. If that product happens to be owned by a single-product firm, then only one margin is necessary. On the other hand, if the product happens to be owned by a multi-product firm, then at a minimum, all the margins for products owned by that firm must be supplied.

The (Marshallian) demand specifications used in trade can be grouped into two categories: demand systems that are derived from a representative consumer’s expenditure function and demand systems that are derived from a representative consumer’s indirect utility function. The LA-AIDS demand system fall into the former category, while the Logit and CES fall into the latter category. Below, we briefly discuss these demand systems as well as the assumptions and/or data needed to recover estimates of the demand parameters.

We conclude this section with a discussion of how calibrated demand parameters and the FOCs can be used to calibrate product-specific constant marginal costs.

The Game

Suppose that there are K firms in a market, each producing a subset J_k of J products. Further, suppose that that each of the k ∈ K firms manufactures its J_k products at n_k plants. Let n = ∑_k ∈ Kn_k denote the total number of plants producing any of the J products. The Cournot model assumes that firms simultaneously set the amount of each product produced at each plant in order to maximize their profits. This model also assumes that all firms can perfectly observe each others’ quantities, and costs, as well as the demand for each product. Further, suppose that the government imposes an ad valorem tariff t_i on a subset of products produced at plants located abroad. Note that we assume that the tariff is calculated as a proportion of consumer price, so that the price received by the firm is 1 − t_i the price faced by consumers.

Functions in trade’s Cournot model also adopt the additional assumption that each firm’s plant has its own distinct marginal cost technology.

Firm k ∈ K chooses product output at each plant $\{q_j^r\}_{\substack{j\in j_k,\\ r\in n_k}}$ so as to maximize profits. Mathematically, firm k solves:

subject to

where p_j, the price sold of product j, is assumed to be a twice differentiable function of all firm quantities with $\frac{\partial p_j}{\partial q_j^r}< 0$ for all plants r and products j. Likewise, additively separable plant variable costs c^r assumed to be twice differentiable with $\frac{\partial c^r}{\partial q_j^r}> 0$ . Finally, t_j^r is the tariff imposed on plant r producing product j.

Differentiating profits with respect to each q_j^r yields the following first order conditions (FOCs):

Calibrating Model Demand and Cost Parameters

The Cournot model can yield different equilibrium quantity and price predictions depending on 1) the curvature of plant variable costs and 2) the curvature of demand. trade allows users to explore the consequences of different cost and demand assumptions.

Currently, cournot contains two different ways to specify plant costs. First, users can set the ‘cost’ option equal to a n-length character vector whose values are either equal to “linear” for linear marginal costs ($\frac{\partial c^r}{\partial q_j^r}= 0.5\gamma_r\sum_{j\in n_r}q_j^r$) or “constant” for constant marginal costs ($\frac{\partial c^r}{\partial q_j^r}= \gamma_r$) . When the ‘cost’ option is employed, cournot’s calcSlopes method uses observed costs (implied by predicted margins and prices) and prices to calibrate plant-level cost parameters γ_r.

Alternatively, users can specify both plant-level marginal and variable costs. Marginal costs may be specified by setting the ‘mcfunPre’ argument equal to a length n list of functions that each take as an input the vector of product quantities produced at a plant and then return the marginal cost associated with producing that level of output at that plant. Likewise, the ‘vcfunPre’ argument can be similarly used to specify plant-level variable costs. Note that when marginal costs and variable costs are specified in this manner, cournot makes no attempt to calibrate any parameters on the cost side.

Currently, cournot allows users to specify that product demand is either linear (p_j = b_j + a_j∑_k ∈ nq_j^k) or log-linear (ln (p_j) = b_j + a_jln (∑_k ∈ nq_j^k)). Users can specify product demand by setting cournot‘s ’demand’ argument equal to a j-length character vector equal to either “linear” for linear demand or “log” for log-linear demand.