# Visualizing the expenditure minimization problem

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.

• Yes, also for R^2. I just edited my question to specify this. Thank you Commented Jan 19, 2022 at 16:18
• Since you write "I can easily visualize the utility maximization problem" perhaps you can include some graphs of what you mean by visualization? You mention the one in the $(x_1,x_2)$ space, but claim this is not what you want for the expenditure minimization problem, even though it takes place in the same place. So could you edit your question to include some alternative visualizations for the utility maximization problem? Commented Jan 19, 2022 at 16:51

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.
• @MistahWhite I still don't understand (: The minimization problem you define has variable $\mathbf{x}$, which you define as being in $\mathbb{R}^2$. So any visualization of this problem would have to be in this space. What exactly are you trying to see? Commented Jan 19, 2022 at 16:29