Utility function used to indicate bliss point

Question

How does one create a utility function to indicate existence of a bliss point? what do the goods marshillian demands look like in such a situation?

Answer 1

An answer that meets all the current demands of the question:

Let $(x_b,y_b)$ be the blisspoint. Let $$ U(x,y) = \left\{ \begin{array}{cc} 1 & \mbox{ if } (x,y) = (x_b,y_b) \\ 0 & \mbox{ if } (x,y) \neq (x_b,y_b). \end{array}\right. $$ The demand function for $x$ in this case is $$ D_x(I,p_x,p_y) = \left\{ \begin{array}{cc} x_b & \mbox{ if } p_x \cdot x_b + p_y \cdot y_b \leq I \\ \left[0,\frac{I}{p_x}\right] & \mbox{ if } p_x \cdot x_b + p_y \cdot y_b > I. \end{array}\right. $$

Answer 2

For a continuous version of a bliss point we can have the function: $$U(x_1,x_2)=-(x_1-\delta)^2-(x_2-\delta_2)^2$$

where $\delta_1$ and $\delta_2$ are bliss requirements.

The corresponding Marshallian: demands for $x_1$ and $x_2$ are:

$$x_1(p_1,p_2,I)=\frac{p_1m+p_2^2\delta_1+p_1\delta_2}{p_1^2+p_2^2}$$

$$x_2(p_1,p_2,I)=\frac{p_2m+p_2^2\delta_2+p_2\delta_1}{p_1^2+p_2^2}$$

The corresponding Hicksian Demands for this function is:

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

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