Deterministic Welfare and Capital Investment
Consider a firm facing a downward sloping demand function
\[ q = D(p), \qquad D'(p) < 0, \]
with inverse demand
\[ p = P(q). \]
The firm has a fixed short-run capacity constraint \(K\). Output cannot exceed this capacity:
\[ q \leq K. \]
Short-run marginal cost is constant and equal to \(c\). Hence short-run cost is
\[ C_{SR}(q) = cq \quad \text{for } q \leq K. \]
Installing capacity is costly. If the cost of capacity is \(r\) per unit, long-run cost is
\[ C_{LR}(q,K) = cq + rK, \]
with \(q \leq K\).
In the short run the capacity \(K\) is fixed, so the cost \(rK\) is sunk for the regulator’s pricing problem.
Firm Behavior
Suppose the regulator sets a price \(p\).
Demand at that price is \(D(p)\), but output cannot exceed capacity. Therefore actual output is
\[ q(p) = \min\{D(p),K\}. \]
The firm’s profit is
\[ \pi(p) = (p-c)\min\{D(p),K\} – rK. \]
Welfare
Total welfare is the sum of consumer surplus and firm profit. Since price payments are transfers, welfare depends only on the quantity served.
Let \(P(q)\) denote inverse demand. Then welfare is
\[ W(q) = \int_0^q P(x)\,dx – cq – rK. \]
Since \(rK\) is fixed in the short run, the regulator maximizes
\[ \int_0^q P(x)\,dx – cq \]
subject to
\[ q \leq K. \]
Ignoring the capacity constraint, the first-best condition is
\[ P(q) = c. \]
Optimal Regulated Price
Low-Demand Case
Suppose
\[ 0 < D(c) < K. \]
At price \(p=c\), demand is below capacity, so the constraint does not bind. The efficient quantity is therefore
\[ q^* = D(c), \]
which can be implemented by setting
\[ p^* = c. \]
High-Demand Case
Suppose
\[ D(c) \geq K. \]
Under marginal cost pricing, demand would exceed capacity. Therefore the efficient feasible quantity is
\[ q^* = K. \]
To implement this quantity, price must satisfy
\[ D(p) \geq K. \]
Using inverse demand, this is equivalent to
\[ p \leq P(K). \]
To ensure the firm is willing to supply, price must also satisfy
\[ p \geq c. \]
Therefore any price in the interval
\[ p \in [c, P(K)] \]
supports output \(K\).
Proposition
Proposition. The welfare-maximizing regulated price is
\[ p^* = \begin{cases} c & \text{if } D(c) < K, \\ p \in [c,P(K)] & \text{if } D(c) \geq K. \end{cases} \]
Proof.
Total welfare is
\[ W(q) = \int_0^q P(x)\,dx – cq – rK. \]
Since \(rK\) is fixed in the short run, the regulator maximizes
\[ \int_0^q P(x)\,dx – cq \]
subject to \(q \leq K\).
The first-order condition for an interior optimum is
\[ P(q) = c. \]
If \(D(c) < K\), the capacity constraint does not bind and the optimal quantity is \(q^* = D(c)\), implemented by \(p=c\).
If \(D(c) \geq K\), the constraint binds and the optimal feasible quantity is \(q^* = K\). Any price satisfying \(D(p) \geq K\) implements this outcome. Using inverse demand, this is equivalent to \(p \leq P(K)\). Combining this with the supply condition \(p \geq c\) gives \(p \in [c,P(K)]\).
Investment Incentives
So far the analysis considered the short run where capacity \(K\) is fixed and the cost \(rK\) is sunk. In the long run, however, the firm must decide whether to invest in capacity. The regulated price must therefore allow the firm to recover the cost of capacity.
Suppose the firm installs capacity \(K\) at cost \(r\) per unit. Total cost is
\[ C(q,K) = cq + rK, \]
with \(q \leq K\).
If the capacity constraint binds, the firm produces \(q=K\). Profit is then
\[ \pi = pK – cK – rK = (p-c)K – rK. \]
For the firm to be willing to invest in capacity \(K\), profit must be nonnegative:
\[ (p-c)K – rK \geq 0. \]
This implies
\[ p \geq c + r. \]
Thus the price must cover both the marginal production cost and the marginal cost of capacity.
Proposition. A regulated price that ensures investment in capacity must satisfy
\[ p \geq c+r. \]
Proof. If the firm installs capacity \(K\) and the constraint binds, profit is
\[ \pi = (p-c)K – rK. \]
Nonnegative profit requires \((p-c)K \ge rK\), which implies \(p \ge c+r\).
The price \(p=c+r\) corresponds to pricing at long-run marginal cost. Therefore, while marginal-cost pricing \(p=c\) is efficient in the short run when capacity is fixed, a higher price may be required in the long run to ensure efficient investment in capacity.
If demand is sufficiently high so that \(D(c) \ge K\), a price that both sustains production at capacity and ensures investment must satisfy
\[ p \in [c+r,\,P(K)]. \]
Such a price exists only if
\[ P(K) \ge c+r, \]
meaning that demand at quantity \(K\) is high enough to cover both operating and capital costs.
Investment When the Regulator Sets \(p=P(K)\)
Suppose firms know in advance that the regulator will set the price equal to the inverse demand evaluated at installed capacity:
\[ p=P(K). \]
If capacity binds, output is \(q=K\), so profit is
\[ \pi(K)=P(K)K-cK-rK. \]
Equivalently,
\[ \pi(K)=\bigl(P(K)-c-r\bigr)K. \]
The firm therefore chooses capacity \(K\) to solve
\[ \max_K \; \pi(K)=\max_K \; \bigl(P(K)-c-r\bigr)K. \]
Assuming an interior solution, the first-order condition is
\[ \frac{d\pi}{dK}=P'(K)K+P(K)-c-r=0, \]
or
\[ P(K)+K P'(K)=c+r. \]
This is the firm’s private investment condition under the regulatory rule \(p=P(K)\).
By contrast, the socially efficient level of capacity maximizes welfare
\[ W(K)=\int_0^K P(q)\,dq-cK-rK, \]
so the first-order condition for the efficient capacity choice is
\[ P(K)=c+r. \]
Since inverse demand is downward sloping, \(P'(K)<0\), and therefore
\[ P(K)+K P'(K)<P(K). \]
It follows that the firm’s investment condition is satisfied at a lower level of capacity than the socially efficient one. Hence the rule \(p=P(K)\) induces underinvestment in capacity.
The intuition is that when the firm expands capacity, the regulator lowers the allowed price from \(P(K)\) to \(P(K+dK)\). The firm therefore takes into account not only the revenue from the additional unit of capacity, but also the reduction in price on all inframarginal units. This lowers the private return to investment and leads to inefficiently low capacity.
Comparison with Overinvestment Concerns for DSOs
The previous analysis shows that if the regulator commits to the rule
\[ p = P(K), \]
the firm’s investment incentives coincide with those of an unregulated monopolist. The firm chooses capacity to maximize
\[ \pi(K) = P(K)K – cK – rK, \]
leading to the first-order condition
\[ P(K) + K P'(K) = c+r. \]
Since inverse demand is downward sloping (\(P'(K)<0\)), the resulting capacity level is smaller than the socially efficient level defined by
\[ P(K) = c+r. \]
Hence this regulatory rule leads to underinvestment in capacity.
This prediction contrasts with a common concern in the regulation of electricity Distribution System Operators (DSOs), where regulators often fear overinvestment. This concern is typically associated with rate-of-return regulation. Under such regulation, the firm is allowed to earn a regulated return \(s\) on its capital stock \(K\), so that allowed revenue satisfies
\[ R \ge cq + sK. \]
If the allowed return exceeds the true cost of capital (\(s>r\)), capital becomes artificially profitable for the regulated firm. Increasing the capital stock raises the firm’s allowed revenue, creating incentives to expand the regulated asset base. This phenomenon is known as the Averch–Johnson effect and can lead firms to overinvest in capital, sometimes referred to as “gold-plating”.
The key difference between the two frameworks lies in how investment affects the firm’s profits. Under the pricing rule \(p=P(K)\), increasing capacity lowers the market price, which reduces the profitability of investment. This discourages capacity expansion and leads to underinvestment. In contrast, under rate-of-return regulation, expanding the capital base increases the firm’s allowed revenue, which encourages investment and may lead to excessive capital accumulation.
Thus, while the rule \(p=P(K)\) replicates monopoly investment incentives and results in too little capacity relative to the social optimum, rate-of-return regulation may produce the opposite distortion by inducing firms to invest too much in capital.