Model | RDP 2025-05: How Costly are Mark-ups in Australia? The Effect of Declining Competition on Misallocation and Productivity

RDP 2025-05: How Costly are Mark-ups in Australia? The Effect of Declining Competition on Misallocation and Productivity 3. Model

Jonathan Hambur and Owen Freestone

August 2025

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Below we outline a very high-level sketch of the EMX model and its key mechanisms. A more detailed description of the model follows. We focus on the components needed to study the misallocation effects, and leave the reader to refer to EMX for a fuller account of the model.

In the model there are heterogeneous firms that differ in terms of their productivity. In the baseline model, firms compete in a monopolistically competitive market, giving rise to mark-ups. Unlike some models, where all firms have equal mark-ups, in this model firms' mark-ups endogenously differ based on their size. In turn, their size relates to their productivity. How mark-ups vary with firm size is dictated by the ‘superelasticity’. This is an important parameter in the model, and captures how demand elasticities (and so mark-ups) change as firm size changes. If the superelasticity is positive, larger firms tend to have higher mark-ups, which is a standard finding in the literature. If it is zero, there is no variation in firm mark-ups, and we are back in the standard model with equal mark-ups and no misallocation across firms.

In the model, mark-ups create costs in the economy via three mechanisms:

When average mark-ups are higher, decisions around consumption, investment and output are distorted, so the economy is smaller than it would be in the first best case, for a given level of aggregate productivity (the traditional deadweight loss channel).
Mark-ups lead to inefficient rates of entry, and so an inefficient number of varieties in the economy (entry channel).
Dispersion in mark-ups creates misallocation. So the level of aggregate productivity is lower than it could be, as resources could be moved between firms in ways that could raise aggregate productivity (misallocation channel).

As noted, while the model contains several important channels, it abstracts away from others. For example, it does not consider the dynamic effects of competition on firms' incentives to innovate, improve, and adopt technologies.^[6] Moreover, our main focus is on the final channel though we return to the other channels briefly at the end of the paper.

3.1 Consumer

The model has a representative consumer. Their utility function has the following form:

(1)

\sum_{t = 0}^{\infty} β^{t} (\log C_{t} - φ \frac{L_{t}^{1 + v}}{1 + v})

subject to the budget constraints

(2)

C_{t} + I_{t} = W_{t} L_{t} + R_{t} K_{t} + Π_{t}

where C_t, I_t, W_t, L_t, R_t, K_t, $Π_{t}$ are consumption, investment, wages, labour hours, the return on capital, the capital stock and aggregate profits, respectively. $β$ is the discount factor, and v is the elasticity of labour supply. The price is taken to be the numeraire P(t) = 1.

Capital follows the law of motion (with $δ$ being the depreciation rate):

(3)

K_{t + 1} = (1 - δ) K_{t} + I_{t}

As is standard, households have the static labour supply decision:

(4)

φ C_{t} L_{t}^{v} = W_{t}

3.2 Intermediate goods producers

In the baseline model, there is a continuum of monopolistically competitive firms $i \in [0, n (s)]$ working within each industry s producing differentiated goods.

Firms pay a fixed cost $κ$ to enter and then get a random TFP draw $z_{i t} (s) \sim G (z)$ , where in practice this distribution is taken to be a Pareto distribution. Each firm produces y_it(s) using a gross output production function:

(5)

y_{i t} (s) = z_{i} (s) {[ϕ^{\frac{1}{θ}} v_{i t} {(s)}^{\frac{θ - 1}{θ}} + {(1 - ϕ)}^{\frac{1}{θ}} x_{i t} {(s)}^{\frac{(θ - 1)}{θ}}]}^{\frac{θ}{(θ - 1)}}

where z_i(s) is the firm's TFP, x_it(s) is intermediate materials, and v_it(s) is the value added of firm i, $θ$ is the elasticity of substitution between value added and materials. Value-added is constructed using capital and labour via a Cobb-Douglas production function:

(6)

v_{i t} (s) = k_{i t} {(s)}^{α} l_{i t} {(s)}^{1 - α}

where $α$ is the capital share of income.

Within each sector, output is aggregated using a Kimball aggregator of the form:

(7)

\int_{0}^{n (s)} ϒ (\frac{y_{i t} (s)}{y_{t} (s)}) d i

where the function $ϒ : ℝ_{+} \to ℝ_{+}$ is strictly increasing and strictly concave.

The assumption about the aggregator implies that firms face an inverse demand curve f (q), demand elasticity $σ (q)$ , and mark-up $μ (q)$ of the following form

(8)

\begin{matrix} f (q) = ϒ^{'} (q_{i t}), & σ (q) : = - \frac{ϒ^{'} (q)}{ϒ^{″} (q) q}, & μ (q) : = \frac{σ (q)}{σ (q) - 1} \end{matrix}

where q is the firm's share of industry output.

Unlike under the standard constant elasticity of substitution (CES) aggregator, this means that a firm's mark-up over marginal cost depends on its size (or market share q). Dispersion in mark-ups, and therefore marginal revenue products, leads to lower productivity within the sector, as discussed extensively elsewhere, such as Hsieh and Klenow (2009). This is because aggregate productivity could be raised by shifting resources and output from one firm to another. So the endogenous dispersion in mark-ups leads to misallocation and lower productivity in the industry, compared to a version of the model with CES aggregators.

How the elasticity of demand, and so mark-ups, change with firm size is referred to as the superelasticity of demand. The superelasticity (along with the firm size distribution) pins down the dispersion in mark-ups in the model, and so the degree of misallocation. If the superelasticity is zero, we return to the CES case with constant mark-ups. But there may still be aggregate mark-ups in the economy, and so costs through other channels (e.g. the deadweight loss channel).

EMX also consider other competitive structures in their fairly general set-up. This includes an oligopoly model, where non-atomistic firms compete within industries. We return to this in Section 6.

3.3 Final goods producers

The products of the industries are aggregated into a final good Y. This final good is used for consumption, investment and as an intermediate good. So there is a fairly simple input-output structure, sometimes referred to as a ‘roundabout’ structure.

The final good is produced by perfectly competitive firms who aggregate industry inputs, leading to the following form:

(9)

Y_{t} = {(\int_{0}^{1} y_{t} {(s)}^{(η - 1) / η} d s)}^{η / (η - 1)}

where $η > 1$ is elasticity of substitution across sectors.

3.4 Aggregate mark-ups and productivity

As shown in EMX, sector and aggregate productivity can be written as sales-weighted harmonic averages of firm and industry productivity, respectively:^[7]

(10)

z_{t} (s) = {(\int_{0}^{n (s)} \frac{q_{i t} (s)}{z_{i} (s)} d i)}^{- 1}, Z_{t} = {(\int_{0}^{1} \frac{q_{t} (s)}{z_{t} (s)} d s)}^{- 1}

Similarly, sector and aggregate mark-ups are sales-weighted harmonic averages:

(11)

μ_{t} (s) = {(\int_{0}^{n (s)} \frac{1}{μ_{i t} (s)} \frac{p_{i t} (s) y_{i t} (s)}{p_{t} (s) y_{t} (s)} d i)}^{- 1}

(12)

M_{t} = {(\int_{0}^{1} \frac{1}{μ_{t} (s)} \frac{p_{t} (s) y_{t} (s)}{Y_{t}} d s)}^{- 1}

Each industry's share of aggregate sales is a function of its mark-up and productivity:

(13)

q_{t} (s) = {(\frac{μ_{t} (s)}{M_{t}} \frac{Z_{t}}{z_{t} (s)})}^{- η}

Inputting this in into Equation (10), aggregate productivity takes the form:

(14)

Z_{t} = {(\int_{0}^{1} {(\frac{μ_{t} (s)}{M_{t}})}^{- η} z_{t} {(s)}^{η - 1} d s)}^{1 / (η - 1)}

As discussed, dispersion in mark-ups leads to lower industry-level productivity, and therefore lower aggregate productivity. We can see this from Equation (14) (at an industry level): if $μ_{t} (s) = M_{t}$ the first term in the integral will be 1, so there will be no distortion down in aggregate productivity coming through via Jensen's inequality.

Note that we can also consider the case of value-added productivity, which relates more directly to GDP, as GDP is measured on a value-added basis. The value-added productivity in the economy (or industry) can be written as:

(15)

Z_{v a} = ϕ^{1 / (θ - 1)} \frac{1 - (1 - ϕ) Z^{θ - 1} M^{- θ}}{{[1 - (1 - ϕ) Z^{θ - 1)} M^{1 - θ)}]}^{θ / (θ - 1)}} Z

While only the dispersion of mark-ups influences gross output productivity, the level of mark-ups in aggregate does affect value-added productivity. This is because it distorts the choice of materials relative to value added, leading to too many intermediate inputs being used relative to what would be efficient.

Footnotes

That said, it does capture some dimensions of innovation, particularity through firm entry and related gains from new varieties. [6]

Alternatively they can be expressed as labour- or input-weighted arithmetic averages. [7]