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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.

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  • $\begingroup$ Yes, also for R^2. I just edited my question to specify this. Thank you $\endgroup$ Jan 19 at 16:18
  • $\begingroup$ 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? $\endgroup$
    – Giskard
    Jan 19 at 16:51

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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.

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  • $\begingroup$ Thank you for your answer. I just included an additional paragraph to further explain myself. $\endgroup$ Jan 19 at 16:26
  • $\begingroup$ @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? $\endgroup$
    – Giskard
    Jan 19 at 16:29
  • $\begingroup$ @MistahWhite Perhaps you are looking for something like a demand function or an income offer curve? Those are related to, but not visualizations of, the utility maximization problem. $\endgroup$
    – Giskard
    Jan 19 at 16:32
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    $\begingroup$ Just a small add-on to @Giskard's comment. This website has some beautiful illustrations/ tools to help visualize various concepts. It would likely (I think) be useful to anyone studying economics! :) Link: econgraphs.org $\endgroup$ Jan 19 at 18:00

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