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Consider the mean-variance utility used in CAPM. The budget line when allocating a risk-free and a risky asset is the line connecting the $r_f$ and the risky asset.

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Suppose that I have fixed amount of wealth and I want to allocate between two risky asset.

In the Arrow Debreu framework, the budget line when allocation between two assets is also the line connecting the two assets.

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How about the mean-variance framework?

My guess is that, if we are allocating two risky assets, then the budget "line" is $\bf not$ the line connecting the two assets. For example, considering the allocation between two identical assets with identical mean and variance but independent correlation. Then, the allocated portfolio reduces the variance but keeps the mean the same: the "budget line" is not the line connecting two assets.

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    $\begingroup$ What should the axes be? $\endgroup$
    – Michael Greinecker
    Commented Apr 18, 2022 at 7:55
  • $\begingroup$ @MichaelGreinecker Maybe take the most convenient axes? If the axes are mean and variance, then the budget line will be actually a curve, which is not very convenient. $\endgroup$
    – High GPA
    Commented Apr 18, 2022 at 17:07
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    $\begingroup$ The most convenient axes will make this as simple as the problem of dividing one's money between burgers and bus tickets. There's no real difference. $\endgroup$
    – Michael Greinecker
    Commented Apr 18, 2022 at 17:40

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For example, considering the allocation between two identical assets with identical mean and variance but independent correlation. Then, the allocated portfolio reduces the variance but keeps the mean the same: the "budget line" is not the line connecting two assets.

In CAPM the set of risky investment opportunities is agiven. This will also include any combinations like the one you propose.

What exactly is a budget line? The budget set is the set of possible solutions in the consumer's utility maximization problem. The budget line is a face of this. Assuming that the consumer's preferences are monotonic, the optimal solution will be somewhere on this line.

The question in CAPM is how to allocate your wealth. In the coordinate system that you gave each investment opportunity is characterized by an expected risk and expected return. You can invest as little or as much as you want to into any of these, thus the question is basically what portion of your wealth to allocate where. The "budget set" is the set of all possible investment opportunities. If we assume that the investor likes higher returns and dislikes higher risk, then we can also say that the optimal choice will be somewhere on the efficient frontier. I think this is as close as you will get to the budget line in this model. (You get further insights if you assume the existence of a risk-free option.)

As Michael Greineker points out, axes matter. A scooter is shorter and lighter then a truck, while a Rolls-Royce falls inbetween in both dimensions. Possibly there is a scooter and a truck for which $$ 70\% \cdot \text{length}_{\text{scooter}} + 30\% \cdot \text{length}_{\text{truck}} = \text{length}_{\text{Rolls-Royce}} $$ and $$ 70\% \cdot \text{weight}_{\text{scooter}} + 30\% \cdot \text{weight}_{\text{truck}} = \text{weight}_{\text{Rolls-Royce}}, $$ thus in this coordinate system the Rolls-Royce will be on a line between the scooter and the truck. Yet this is not a budge line: for approx. \$50,000 I could buy a truck and a scooter, but it would not be enough for a Rolls-Royce.

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