Consider the following allocation problem. One unit of a perfectly divisible good is allocated among three agents. Each agent has quasi linear utility, with valuations $v_{1}\left(x_{1}\right)=\sqrt{x_{1}}, v_{2}\left(x_{2}\right)=\sqrt{x_{2}}$, and $v_{3}\left(x_{3}\right)=x_{3}$ when the agents receive amounts $x_{1}, x_{2}$, and $x_{3}$ of the good respectively. (a) Show that the Pareto optimal allocation is $\left(x_{1}, x_{2}, x_{3}\right)=\left(\frac{1}{4}, \frac{1}{4}, \frac{1}{2}\right)$, by maximizing the total valuation function $$ \mathrm{TU}\left(x_{1}, x_{2}, x_{3}\right)=\sqrt{x_{1}}+\sqrt{x_{2}}+x_{3} $$ subject to the constraints $x_{1}+x_{2}+x_{3}=1$ and $x_{i} \geq 0$ for $i=1,2,3$. (You may assume that the maximum location is an interior point.) $$ $$ (b) Suppose that we sell the unit of the good via the VCG mechanism. Further suppose that all three agents bid according to dominant strategy equilibrium. Compute the payments given by the VCG mechanism. $$ $$ (c) Suppose that we sell the good via the uniform price mechanism, where the auctioneer computes price $p \geq 0$ in such a way that total demand equals total supply (which is 1 in this case). Suppose that all three agents report demand truthfully. Show that the equilibrium price is $p=1$, and the the allocation of the good is $\left(x_{1}, x_{2}, x_{3}\right)=\left(\frac{1}{4}, \frac{1}{4}, \frac{1}{2}\right)$
My try
So my understanding of the VCG mechanism is alright, however, I got more confused when the question was structured like this, and forgive the sloppiness. So here we go:
a) First we write down the maximization problem as a Lagrangian:
$$ \mathcal{L} = (\sqrt{x_1} + \sqrt{x_{2}} + x_{3} ) - \lambda (x_{1} + x_{2} + x_{3} - 1) $$ Then by taking partial derivatives w.r.t all x_i $$ 1. \frac{d \mathcal{L}}{d x_{1}} = \frac{1}{2\sqrt(x_{1})} - \lambda = 0\\ 2.\frac{d \mathcal{L}}{d x_{2}} = \frac{1}{2\sqrt(x_{2})} - \lambda = 0\\ 3. \frac{d \mathcal{L}}{d x_{3}} = 1 - \lambda = 0\\ 4. \frac{d \mathcal{L}}{d \lambda} = x_1 + x_2 + x_3 = 1 , \text{we do not have to check for C.S. as we have an equality} $$ Thus $\lambda $ is strictly positive as $\lambda = 1$. By equations 1 and 2, we have that $ \frac{1}{2 \sqrt x_1} = \frac{1}{2 \sqrt x_2} = 1$ and so you will get that $x_1 = x_2 = \frac{1}{4}$ and $x_3 = \frac{1}{2} $.
Then the Total Utility is 1.5.
b) Now we introduce the VCG mechanism. We have a product that is perfectly divisible, that means we can split it up and allocate a part such that everyone is happy with what they have, we assume they play efficiently by bidding their true valuations. The price they pay? Their externality, which I understand as the what welfare loss when a particular player has joined the bidding/auction. Because players 1 and 2 have the same function, that is also different from player 3. We have two sub-maximization problems. My understanding is that we want to find the price they all pay if they were to win. However, we have now a change of assumption, and that is, we only sell a unit and not parts. So this means, however has the highest value, wins.
Suppose that $v_1(x_1)$ won and we want to figure out his externality.
$$ 1. \ max_{x_2, x_3} \ \ p^{\mathcal{L}}_{1} = \bigl( (\sqrt x_{2} + x_3) - (\sqrt x_{2}) \bigr) - \lambda(x_2 + x_3 -1) $$
x_2 Becomes absolete and so we take FOC w.r.t $x_3$ and $\lambda$ which leads to a price of 1 and $x_3 = 1$. Which implies that $x_1 = x_2 = 0 $
So I repeated this for $x_3$ which a slight different formulation, same price but here $x_2 = 1$.
The reason for this formulation, is that we are interested in the price. Which is defined as (in my interpretation): maximum allocation then counting him out of it - the maximum allocation if he never played. However, I dont think this is correct and that I got p=1 by luck. Can someone please help?