Monopoly profit is the top curve. Each extra firm shifts the curve down; the market carries N firms once the N-firm curve clears that firm's own dotted entry cost. Raising the subsidy walks the market up the ladder.
Free entry
Blocked entry (N = 1 throughout)
At a free-entry threshold the entrant earns exactly its entry cost, so its own net gain is zero. The social gain is the consumer-surplus gain plus the externality on the extra output, less what incumbents lose to business stealing, less the entrant's sunk cost, less the extra subsidy paid on the larger volume. e* is the external benefit per installation at which a step breaks even.
A heat pump displaces a gas boiler, so the abatement is the gas emissions avoided less the electricity emissions added. With clean power the whole gas footprint is abated.
Inverse demand in the gross price the installer invoices, with the household paying \(p-s\):
\[ p = a + s - b Q. \]Firms differ in two ways
Firms are ordered best to worst. Firm \(k\) has unit cost and entry cost
\[ c_k = c + (k-1)\zeta, \qquad \Phi_k = (k-1)\Phi \;\text{(rising)} \;\text{ or }\; \Phi \;\text{(flat)}, \]with firm 1 the incumbent, which has the lowest cost and pays nothing to be there. Both gradients say the same thing in different places: the marginal entrant is progressively worse placed — further from a training provider, a less suitable existing trade business.
Equilibrium
Write \(M=a+s-c\) and the marginal firm's residual gap \(M_N = M - \zeta (N+2)(N-1)/2\). Then
\[ p = c + \mu, \quad \mu=\frac{M+\zeta N(N-1)/2}{N+1}, \qquad Q = \frac{N\,[\,M-\zeta (N-1)/2\,]}{b (N+1)}, \] \[ p - c_k = \mu-(k-1)\zeta, \qquad \Pi_k = \frac{(p-c_k)^2}{b}, \qquad \Pi_N = \frac{M_N^2}{b (N+1)^2}. \]At \(\zeta=0\) these are the symmetric formulas. Firm \(N\) can produce at all only if \(M_N>0\).
Entry
Firm \(N\) is viable iff \(\Pi_N \ge \Phi_N\), that is
\[ M \;\ge\; \frac{\zeta (N+2)(N-1)}{2} + (N+1)\sqrt{b \Phi_N}, \]and the \(k\)-th firm enters where that binds.
The three facts that matter
Inside a regime the gross price rises with the subsidy, \(\partial p/\partial s = 1/(N+1)\), so pass-through to the household is \(N/(N+1) < 1\).
At an entry threshold the gross price jumps down by \(M_N/(N(N+1))\), which at the threshold equals
\[ \Delta p = -\frac{\sqrt{b \Phi_N}}{N}. \]Note what is absent: \(\zeta\). A unit-cost gradient pushes the entry threshold to the right but leaves the size of the price drop unchanged. Entry becomes harder to trigger, not less potent once triggered. This is the only route to pass-through above one in the model.
The envelope through the thresholds is
\[ p = c + \sqrt{b \Phi_N} + (N-1)\zeta . \]With flat entry costs and \(\zeta=0\) this is flat in \(s\): cheap entry means prices near marginal cost and near-complete pass-through. Either gradient tilts it up, so deep markets stop converging to marginal cost — because the marginal installer really is worse.
Why this is the paper's mechanism
Anything that raises the cost of acquiring the installation skill — distance to the nearest training provider, a scarce and expensive local plumber workforce — raises \(\Phi\) or \(\zeta\). Either raises the price envelope, raises the subsidy needed to trigger entry, and leaves more of any given grant as margin rather than installations.
Caveats
Marginal costs are constant within a firm, so there are no capacity constraints and no waiting-time margin; entry is instantaneous; demand is linear. The price fall is also discrete rather than smooth, and the jumps shrink in \(N\), so the model rationalises a price fall in thin markets more naturally than a larger fall in thick ones.
On welfare: entry at a free-entry threshold is excessive in the Mankiw–Whinston sense — the entrant's private gain is zero, incumbents lose to business stealing, and the entry cost is burned. Total surplus therefore falls at an entry step unless the externality is very large, while consumer surplus rises sharply. The two criteria disagree, and which one you use is a choice the model cannot make for you.