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## Expert Answer

## Question

a. Imagine a consumer who can buy any non-negative amounts*x, y*and

*z*of three commodities, subject only to the constraint that her total expenditure is at most

*m*when the prices of the two commodities are

*p > 0, q > 0*and

*r > 0*, respectively. The consumer’s budget set is

*B = { (x, y, z) : px + qy + rz ≤ m, x ≥ 0, y ≥ 0, z ≥ 0 }*. Argue that this set is convex. b. In the context of the part a, suppose that the third commodity in indivisible: it can only be consumed in integer amounts,

*z ∈ {0, 1, 2, . . .}.*Determine whether the budget set in this setting is convex. The Consumer Budget Assignment Paper

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### Step-by-step

**a.**To show that the budget set B is convex, we need to show that for any two points (x1, y1, z1) and (x2, y2, z2) in B, the line segment connecting them is also contained in B. That is, for any t between 0 and 1, the point

**b.**When the third commodity is indivisible, the budget set is no longer convex. To see why, let's consider an example. Suppose that p = q = 1 and r = 2, and the total expenditure is m = 3. Then, the budget set B is given by

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