I've been looking at expenditure systems and have been really interested in the behaviour of the demand system that underlies bliss points:

Consider the bliss point utility function of the following form:


for two dimensions the corresponding hicksian demands are:

$$x_1^c=\delta_1-\left[\frac{\bar{U}}{1+\frac{p_2}{p_1}}\right]^\frac{1}{2}$$ $$x_2^c=\delta_2-\left[\frac{\bar{U}}{1+\frac{p_1}{p_2}}\right]^\frac{1}{2}$$

It follows that the expenditure function is: $$e(p_1,p_2,\bar{U})=p_1\delta_1-p_1\left[\frac{\bar{U}}{1+\frac{p_2}{p_1}}\right]^\frac{1}{2}+p_2\delta_2-p_2\left[\frac{\bar{U}}{1+\frac{p_1}{p_2}}\right]^\frac{1}{2}$$

Obviously expenditure functions can be much larger. however I'm having a hard time for generating a expenditure function for a number of $n$ goods.

tldr What would the hicksian demands look like for the utility function: $$U(\mathbf{x})=-\sum_{i=1}^n(x_i-\delta_i)^2$$


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