<p>The basic peak load pricing problem is formulated in</p>
<p>Boiteux, M. (1949). “De la tarification des pointes de demande.” In: Revue générale
de l’électricité, pp. 321–340.</p>
<p>and</p>
<p>Steiner, P. O. (1957). “Peak loads and efficient pricing.” In: The Quarterly Journal of Economics, pp. 585–610.</p>
<p>The most simple set up considers <span class="math-container">$T=2$</span> periods. The important assumption is that storage is impossible hence demand must balance supply in each period. Furthermore, the same capital <span class="math-container">$K$</span> is used in both periods and production <span class="math-container">$q_1$</span> and <span class="math-container">$q_2$</span> must both be weekly smaller than the capacity/capital <span class="math-container">$K$</span>.</p>
<p>With inverse demand given as <span class="math-container">$P_1(q_1)$</span> and <span class="math-container">$P_2(q_2)$</span> and the cost as
<span class="math-container">\begin{align}
C(q_1,q_2,K) = b(q_1+q_2) + rK,
\end{align}</span>
the welfare optimization problem can be formulated as</p>
<p><span class="math-container">\begin{align}
\max_{q_1,q_2,K} &amp;\ \int_{0}^{q_1} P_1(z) dz + \int_{0}^{q_2} P_2(z) dz - C(q_1,q_2,K)\\[8pt]
s.t. &amp;\ \ \ q_1 \leq K \ \wedge \ q_2 \leq K
\end{align}</span>
For which the first order conditions of the Lagrangian is given as
<span class="math-container">\begin{align}
P_1(q_1) - b - \lambda_1 = 0\\[8pt]
P_2(q_2) - b - \lambda_2 = 0\\[8pt]
\lambda_1 + \lambda_2 - r = 0.
\end{align}</span>
When the cost of capital <span class="math-container">$r&gt;0$</span> it follows that <span class="math-container">$\lambda_1 + \lambda_2&gt;0$</span> so at least one of the shadow prices is strictly above 0 implying that
<span class="math-container">\begin{align}
q_1 = K \ \ \vee  \ \ q_2 = K,
\end{align}</span>
such that there will never be invested in capacity that is not used.</p>
<p>Furthermore, in this case
<span class="math-container">\begin{align}
P_{1}(K) - b + P_2(K) - b = r,
\end{align}</span>
states that the sum of prices net variable cost <span class="math-container">$b$</span> equals the unit cost of capital. So if demand is peaking in both periods then prices in both periods are above short-run marginal cost <span class="math-container">$b$</span>.</p>
<p>In case one shadow price is <span class="math-container">$0$</span> - assume <span class="math-container">$\lambda_2 = 0$</span> - because the demand is so low that production is less than capacity <span class="math-container">$q_2&lt;K$</span> then
<span class="math-container">\begin{align}
P_{1}(K) &amp;= b+r \\[8pt]
P_2(q_2) &amp;= b,
\end{align}</span>
such that only the peak period <span class="math-container">$t=1$</span> where production is at capacity <span class="math-container">$q_1=K$</span> is priced for the capital cost <span class="math-container">$r$</span>.</p>
<p>Peak pricing is commonly used in transportation, pricing for hotel rooms, plane tickets as well as network service. The essential component is time-varying demand combined with the impossibility of storage.</p>

CC BY-SA 4.0
2024-07-24T20:47:28.723Z
2024-07-24T20:53:47.837Z
2024-07-24T20:53:47.837Z
Jesper Hybel
<p>Electricity is distributed in a network with a max capacity <span class="math-container">$K$</span>. Let <span class="math-container">$Y_i(t)$</span> be demand of person <span class="math-container">$i$</span> at time <span class="math-container">$t$</span>. Then at no time <span class="math-container">$t \in [0,T]$</span> can <span class="math-container">$\sum_i Y_i(t) &gt; K$</span> or networks breaks down.</p>
<p>Obviously, investing in capacity higher than <span class="math-container">$Y^*:=\max_t \{\sum_i Y_i(t)\}$</span> would be uneconomical so <span class="math-container">$K$</span> is set equal to the <span class="math-container">$Y^*$</span>. Then obviously any unit of electricity used at any time where <span class="math-container">$\sum_i Y_i(t) &lt; K$</span> would require no extra net capacity costs. So marginal costs are 0 any time <span class="math-container">$t$</span> where <span class="math-container">$\sum_i Y_i(t) &lt; K$</span> and else equal to whatever the marginal unit of capacity costs. According to the marginal cost principle only the users of electricity at times <span class="math-container">$t$</span> where <span class="math-container">$\sum_i Y_i(t) = K$</span> should pay. This is a basic principle of peak pricing.</p>
<p>This reasoning is fairly straightforward. However, I suppose that there is economic literature investigating and analyzing this problem of peak pricing in more depth, maybe some lecture slides or a good textbook.</p>
<p>So, in short, my question is:</p>
<p>Can anyone refer me to good resources on the question: Why should electricity network capacity costs according to long-run marginal cost pricing be paid by peak consumers only?</p>

CC BY-SA 4.0
2024-04-19T12:42:25.857Z
Your notation seems to imply that a person's demand for electricity at time $t$ depends only on the time and not on the price they are charged.  Is that your intention?
Adam Bailey
CC BY-SA 4.0
2024-04-21T14:16:48.787Z
@AdamBailey it is unintentional. I guess you could read $Y_i(t):= H_i(t,p(t))$ so demand at time $t$ depends on price but price itself being a function of time - many electricity consumers today pay prices that are different across the hours of the day - of course, some don't.
Jesper Hybel
<cost><marginal><price-discrimination>
Why should electricity network capacity costs according to long run marginal cost pricing be paid by peak consumers only?
Jesper Hybel
# Why should electricity network capacity costs according to long run marginal cost pricing be paid by peak consumers only?

(Asked by: username_0 on 2024-04-18)

Electricity is distributed in a network with a max capacity $K$. Let $Y\_i(t)$ be demand of person $i$ at time $t$. Then at no time $t \in [0,T]$ can $\sum\_i Y\_i(t) > K$ or networks breaks down.

Obviously, investing in capacity higher than $Y^\*:=\max\_t \{\sum\_i Y\_i(t)\}$ would be uneconomical so $K$ is set equal to the $Y^\*$. Then obviously any unit of electricity used at any time where $\sum\_i Y\_i(t) < K$ would require no extra net capacity costs. So marginal costs are 0 any time $t$ where $\sum\_i Y\_i(t) < K$ and else equal to whatever the marginal unit of capacity costs. According to the marginal cost principle only the users of electricity at times $t$ where $\sum\_i Y\_i(t) = K$ should pay. This is a basic principle of peak pricing.

This reasoning is fairly straightforward. However, I suppose that there is economic literature investigating and analyzing this problem of peak pricing in more depth, maybe some lecture slides or a good textbook.

So, in short, my question is:

Can anyone refer me to good resources on the question: Why should electricity network capacity costs according to long-run marginal cost pricing be paid by peak consumers only?

**Tags:** cost, marginal, price-discrimination

> **Comments:**
> **username_1**: Your notation seems to imply that a person's demand for electricity at time $t$ depends only on the time and not on the price they are charged.  Is that your intention?  
> **username_0**: @AdamBailey it is unintentional. I guess you could read $Y_i(t):= H_i(t,p(t))$ so demand at time $t$ depends on price but price itself being a function of time - many electricity consumers today pay prices that are different across the hours of the day - of course, some don't.  

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## 1 answers

### Answer by username_0 on 2024-07-24. Score: 2 | Accepted answer

The basic peak load pricing problem is formulated in

Boiteux, M. (1949). “De la tarification des pointes de demande.” In: Revue générale
de l’électricité, pp. 321–340.

and

Steiner, P. O. (1957). “Peak loads and efficient pricing.” In: The Quarterly Journal of Economics, pp. 585–610.

The most simple set up considers $T=2$ periods. The important assumption is that storage is impossible hence demand must balance supply in each period. Furthermore, the same capital $K$ is used in both periods and production $q\_1$ and $q\_2$ must both be weekly smaller than the capacity/capital $K$.

With inverse demand given as $P\_1(q\_1)$ and $P\_2(q\_2)$ and the cost as
\begin{align}
C(q\_1,q\_2,K) = b(q\_1+q\_2) + rK,
\end{align}
the welfare optimization problem can be formulated as

\begin{align}
\max\_{q\_1,q\_2,K} &\ \int\_{0}^{q\_1} P\_1(z) dz + \int\_{0}^{q\_2} P\_2(z) dz - C(q\_1,q\_2,K)\\[8pt]
s.t. &\ \ \ q\_1 \leq K \ \wedge \ q\_2 \leq K
\end{align}
For which the first order conditions of the Lagrangian is given as
\begin{align}
P\_1(q\_1) - b - \lambda\_1 = 0\\[8pt]
P\_2(q\_2) - b - \lambda\_2 = 0\\[8pt]
\lambda\_1 + \lambda\_2 - r = 0.
\end{align}
When the cost of capital $r>0$ it follows that $\lambda\_1 + \lambda\_2>0$ so at least one of the shadow prices is strictly above 0 implying that
\begin{align}
q\_1 = K \ \ \vee \ \ q\_2 = K,
\end{align}
such that there will never be invested in capacity that is not used.

Furthermore, in this case
\begin{align}
P\_{1}(K) - b + P\_2(K) - b = r,
\end{align}
states that the sum of prices net variable cost $b$ equals the unit cost of capital. So if demand is peaking in both periods then prices in both periods are above short-run marginal cost $b$.

In case one shadow price is $0$ - assume $\lambda\_2 = 0$ - because the demand is so low that production is less than capacity $q\_2 then
\begin{align}
P\_{1}(K) &= b+r \\[8pt]
P\_2(q\_2) &= b,
\end{align}
such that only the peak period $t=1$ where production is at capacity $q\_1=K$ is priced for the capital cost $r$.

Peak pricing is commonly used in transportation, pricing for hotel rooms, plane tickets as well as network service. The essential component is time-varying demand combined with the impossibility of storage.


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