Visualizing the expenditure minimization problem

Question

I can easily visualize the utility maximization problem ie. $$v(\mathbf{p},m^{*})= \max_{\mathbf{x}} \ u(\mathbf{x}) \ \ s.t \ \ \mathbf{px}\leq m$$ Since it is pretty easy to graph the indifference curves and the budget constraint (in $\mathbb{R}^{2}$), in the $(x_{1},x_{2})$ space. But I am having trouble finding references that will show me an illustration of the expenditure minimization problem (which of course is the dual problem to utility max). That is, how do I visualize

$$e(\mathbf{p},u^{*})= \min_{\mathbf{x}} \ \mathbf{px} \ \ s.t \ \ u(\mathbf{x}) \geq u^{*}$$

My purpose for wanting to understand this, is that I want to visually see how under a different price vector (say $\mathbf{p}'$), I can visualize how we can still achieve utility $u^{*}$ (also in some parameter space under $\mathbb{R}^{2}$); similar to how we can adjust price vectors to find another budget that will give us identical utility as the budget $\mathbf{px}=m$ would give.

Another parameter space that I could think of is the $(m,u)$ space. That is, (income, utility) space. We could write an inverse expenditure as some positively sloped function of utility, and the indirect utility function, non-decreasing in income (maybe concave), and find some tangency there.

Answer 1

I am not sure what you mean - the visualization is essentially the same, only the roles of the goal function and the constraint are switched. Given the appropriate utility and income levels the optimal solutions $\mathbf{x^*}$ coincide, thus so will both the budgets and utility levels.
This is also true for any graph showing the relationship of the optimal solution of utility maximization problem and a parameter such as price.

I made a nice illustrative Desmos graph, but I am not sure this is what you want.