Note that it does not matter whether you pull in the constant of the right-hand side of the respective constraint.
The marginal utility curve is $U'(d)=70-6d$ [\euro/MWh]. At
$\pi=5$, the demand would be determined by $5=70-6d$, i.e. $d =
$\pi=5$ and if the demand were unconstrained, the demand would be determined by $5=70-6d$, i.e. $d =
65/6=10.8333$, which is above the consumption limit
$d_{max}=10$. Therefore the optimal demand is $d^*=10$, the upper limit is binding $\mu_{max}
\geq0$ and the lower limit is non-binding $\mu_{min}=0$.
$d_{max}=10$. Therefore the optimal demand is $d^*=10$, the upper limit is binding such that $\mu_{max}
\geq0$ and the lower limit is non-binding such that $\mu_{min}=0$.
To determine the value of $\mu_{max}$ we use \eqref{eq:2stat} to get
$\m_{max}= U'(d^*)-\pi= U'(10)-5=5$.
$\m_{max}= U'(d^*)-\pi+\mu_{min}= U'(10)-5+0=5$.
\begin{shaded}
\item Suppose now the electricity price is $\pi=60$~\euro/MWh. What are
the optimal demand $d^*$, $\mu_{max}$ and $\mu_{min}$ now?
\end{shaded}
At $\pi=60$, the demand would be determined by $60=70-6d$, i.e. $d =10/6=1.667$, which is below the consumption limit $d_{min}=2$. Therefore the optimal demand is $d^*=2$, the upper limit is non-binding $\mu_{max}
=0$ and the lower limit is binding $\mu_{min}\geq0$.
At $\pi=60$, the demand would be determined by $60=70-6d$, i.e. $d =10/6=1.667$, which is below the consumption limit $d_{min}=2$. Therefore the optimal demand is $d^*=2$, the upper limit is non-binding such that $\mu_{max}
=0$ and the lower limit is binding such that $\mu_{min}\geq0$.
To determine the value of $\mu_{min}$ we use \eqref{eq:2stat} to get
Consider an electricity market with two generator types, one with the cost function $C_1(G_1)=c_1G_1$ with variable cost $c_1=20\emwh$, capacity $K_1=300\mw$ and a dispatch rate of $G_1$~[MW], and another with the cost function $C_2(G_2)=c_2G_2$ with variable cost $c_2=50\emwh$, capacity $K_2=400\mw$ and a dispatch rate of $G_2$~[MW]. The demand has utility function $U(D)=8000D -5D^2$~[\euro/h] for a consumption rate of $D$~[MW].
Consider an electricity market with two generator types, one with the cost function $C_1(g_1)=c_1g_1$ with variable cost $c_1=20\emwh$, capacity $G_1=300\mw$ and a dispatch rate of $g_1$~[MW], and another with the cost function $C_2(g_2)=c_2g_2$ with variable cost $c_2=50\emwh$, capacity $G_2=400\mw$ and a dispatch rate of $g_2$~[MW]. The demand has utility function $U(d)=8000d -5d^2$~[\euro/h] for a consumption rate of $d$~[MW].
\begin{enumerate}[(a)]
\begin{shaded}\item What are the objective function and constraints required for an optimisation problem to maximise short-run social welfare in this market?\end{shaded}
The optimisation problem has objective function:
The optimisation problem has the objective function:
Primal feasibility is just the constraints above. Dual feasibility is $\bar{\m}_i,\ubar{\m}_i \geq0$ and complementary slackness is $\bar{\m}_i(G_i-K)=0$ and $\ubar{\m}_i G_i =0$ for $i=1,2$.
Primal feasibility is just the generator limits above in (a). Dual feasibility is $\bar{\m}_i,\ubar{\m}_i \geq0$ and complementary slackness is $\bar{\m}_i(G_i-K)=0$ and $\ubar{\m}_i G_i =0$ for $i=1,2$.
\begin{shaded}\item Determine the optimal rate of production of the generators and the value of all KKT multipliers. What is the interpretation of the respective KKT multipliers?\end{shaded}
The marginal utility at the full output of the generators, $K_1
+K_2=$ 700~MW is $U'(700)=8000-10\cdot700=1000$\euro/MWh,
The marginal utility at the full output of the generators, $G_1
+G_2=$ 700~MW is $U'(700)=8000-10\cdot700=1000$\euro/MWh,
which is higher than the costs $c_i$, so we'll find optimal rates
$G_1^*= K_1$ and $G_2^*= K_2$ and $D^*= K_1+K_2$. This means $\l
= U'(K_1+K_2)=1000$\euro/MWh, which is the market price. Because
$g_1^*= G_1$ and $g_2^*= G_2$ and $d^*= G_1+G_2$. This means $\l
= U'(G_1+G_2)=1000$\euro/MWh, which is the market price. Because
the lower constraints on the generator output are not binding, from
complementary slackness we have $\ubar{\m}_i =0$. The upper
constraints are binding, so $\bar{\m}_i \geq0$.
From stationarity $\bar{\m}_i =
\l- c_i$, which is the increase in social welfare if Generator $i$
\l- c_i+\ubar{\m}_i$, which is the increase in social welfare if Generator $i$