# Finding individual utility

There are N agents living in an economy with two goods, $$X$$ and $$Y$$. Their preferences are described by the following utility function $$u(X,Y) = 2 \sqrt{XY}$$. Each agent is endowed with 1 unit of $$X$$ and $$y_{i}$$ units of $$Y$$. Each unit of $$Y$$ is sells for $$p$$ units of $$X$$.

The question is to show that each agent gets a utility of $$\frac{1 + py_{i}}{\sqrt p}$$.

Here is what I have tried:

The agent chooses $$X,Y$$ in order to maximise $$2\sqrt{XY}$$ subject to $$X + pY = 1 + p y_{i}$$ So I have expressed the resource constraint in units of good $$X$$. The LHS is what the agent can buy and the RHS is the endowment, also expressed in units of $$X$$.

I then re-arrange the constraint in terms of $$Y$$, i.e, $$Y = \frac{1}{p} + y_{i} - \frac{X}{p}$$ and substitute this in to the utility function, with the FOC with respect to $$X$$ and then substitute the optimal value of $$X$$ back in to the utility function to get an expression in terms of $$y_{i}$$. However, the expression is not what I am supposed to get.

If anyone wants to check my algebra, here it is:

The utility function becomes $$2\sqrt {(X/p) + Xy_{i} - X^2/p}$$ Differentiating this w.r.t $$X$$ I find that $$X^* = (1/2)(1 + y_{i}p)$$. I substitute this in the utility function, $$2\sqrt{X^*y_{i}}$$ but this does not equal $$\frac{1 + py_{i}}{\sqrt p}$$.

Any ideas on where I am going wrong?

Your mistake probably occurred at the final substitution stage. Optimal utility should be $$2\sqrt{x^*y^*}$$, which is not the same as $$2\sqrt{x^*y_i}$$.
In general, maximizing a two-good Cobb-Douglas utility function $$u(x,y)=Ax^\alpha y^\beta$$ subject to budget constraint $$p_xx+p_yy=m$$ has general solution $$$$x^*=\frac{\alpha}{\alpha+\beta}\frac{m}{p_1} \quad\text{and}\quad y^*=\frac{\beta}{\alpha+\beta}\frac{m}{p_2}.$$$$
In your case, the parameters $$A,\alpha,\beta,p_x,p_y,m$$ take the following values: $$$$A=2,\quad\alpha=\beta=\frac12,\quad p_x=1,\quad p_y=p,\quad m=1+py_i.$$$$ Plugging in the values, you get optimal consumption: $$$$x^*=\frac{1+py_i}{2},\quad y^*=\frac{1+py_i}{2p}.$$$$ Hence optimal utility is $$$$u(x^*,y^*)=2\sqrt{\frac{1+py_i}{2}\cdot \frac{1+py_i}{2p}}=\frac{1+py_i}{\sqrt p}$$$$