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The Consumer Budget Assignment Paper
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
[(1−t)x1+tx2,(1−t)y1+ty2,(1−t)z1+tz2]
must also be in B.
Let's assume that (x1, y1, z1) and (x2, y2, z2) are in B, which means that
px1+qy1+rz1≤ m and px2+qy2+rz2≤ m,and x1,y1,z1,x2,y2,and z2 are all non−negative⋅Then, we can write
px=p(1−t)x1+pt x2
qy=q(1−t)y1+qt y2
rz=r(1−t)z1+rt z2
Adding these inequalities and using the fact that p, q, and r are all positive, we get:
px+qy+rz≤ m
Therefore, the point
[(1−t)x1+tx2,(1−t)y1+ty2,(1−t)z1+tz2]
is also in B.
Thus, the budget set B is convex.

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
B= (x, y, z) : x + y + 2z ≤ 3, x ≥ 0, y ≥ 0, z ∈ {0, 1, 2} ⋅
Let's consider the points (0, 0, 0) and (1, 1, 0) in B. These points satisfy the constraints
0 + 0 + 2(0) = 0 ≤ 3
1 + 1 + 2(0) = 2 ≤ 3 The Consumer Budget Assignment Paper

However, the point in the middle of the line segment connecting these two points, namely (0.5, 0.5, 0), is not in B, since

0.5 + 0.5 + 2(0) = 1 < 3

Final answer
Therefore, the budget set is not convex.

The Consumer Budget Assignment Paper

Expert Answer

The Consumer Budget Assignment Paper

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

[(1−t)x1+tx2,(1−t)y1+ty2,(1−t)z1+tz2]

must also be in B.

Let's assume that (x1, y1, z1) and (x2, y2, z2) are in B, which means that

px1+qy1+rz1≤ m and px2+qy2+rz2≤ m,and x1,y1,z1,x2,y2,and z2 are all non−negative⋅Then, we can write

px=p(1−t)x1+pt x2

qy=q(1−t)y1+qt y2

rz=r(1−t)z1+rt z2

Adding these inequalities and using the fact that p, q, and r are all positive, we get:

px+qy+rz≤ m

Therefore, the point

[(1−t)x1+tx2,(1−t)y1+ty2,(1−t)z1+tz2]

is also in B.

Thus, the budget set B is convex.

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

B= (x, y, z) : x + y + 2z ≤ 3, x ≥ 0, y ≥ 0, z ∈ {0, 1, 2} ⋅

Let's consider the points (0, 0, 0) and (1, 1, 0) in B. These points satisfy the constraints

0 + 0 + 2(0) = 0 ≤ 3

1 + 1 + 2(0) = 2 ≤ 3 The Consumer Budget Assignment Paper

However, the point in the middle of the line segment connecting these two points, namely (0.5, 0.5, 0), is not in B, since

0.5 + 0.5 + 2(0) = 1 < 3

Final answer

Therefore, the budget set is not convex.

The Consumer Budget Assignment Paper

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