(good x) and clothing (good y): 1) u₁(x, y) = 3x²y 2) u₂(x, y) = 2√x + y 3) uz(x, y) = x0.6y0.4 4) u₁(x, y) = x² + y² 5) u5(x, y) = x + 3y For each of these people: a) Compute their marginal utilities of good x, MUx= Ju(x,y) ду du(x,y) ax = MUY b) Check whether the property of "more is better" is satisfied for both goods? Explain. [Hint: Check whether marginal utilities are positive assuming positive amounts of good x and good y] and marginal utility of good y, c) Does the marginal utility of good x diminish, remain constant, or increase as each of the individuals buys more x? Explain. [Hint: There are 2 ways to do it: 1) visually check what happens to the expression of MU, when x increases (does it decrease, keep constant or decrease?); or 2) take the partial derivative of this marginal utility with respect to x, that is аMUX If ƏMUX ƏMUX < 0, the marginal utility of x is diminishing/decreasing in x; = 0, the ?х marginal utility of x is constant in x; and if Ux>0, the marginal utility of x is increasing in x] əx əx d) Does the marginal utility of good y diminish, remain constant, or increase as each of the individuals buys more y? Explain. e) Find the marginal rate of substitution of good x for good y, MRSxy? [Hint: Take and simplify as much as you can the ratio of marginal utilities MRSxy= MUX MUY f) Is the MRSxy diminishing, constant, or increasing as more x is consumed? OMRSxy. If [Hint: There are 2 ways to do it: 1) visually check what happens to the expression of MRSxy when x increases (does it decrease, keep constant or decrease?); or 2) take the partial derivative of the marginal rate of substitution with respect to x, that is ax ƏMRSx.y = 0, əx ƏMRSx.y əx > 0, the marginal rate of the marginal rate of substitution is constant in x; and if substitution of x is increasing in x] < 0, the marginal rate of substitution is diminishing/decreasing in x; ƏMRSxy ax
(good x) and clothing (good y): 1) u₁(x, y) = 3x²y 2) u₂(x, y) = 2√x + y 3) uz(x, y) = x0.6y0.4 4) u₁(x, y) = x² + y² 5) u5(x, y) = x + 3y For each of these people: a) Compute their marginal utilities of good x, MUx= Ju(x,y) ду du(x,y) ax = MUY b) Check whether the property of "more is better" is satisfied for both goods? Explain. [Hint: Check whether marginal utilities are positive assuming positive amounts of good x and good y] and marginal utility of good y, c) Does the marginal utility of good x diminish, remain constant, or increase as each of the individuals buys more x? Explain. [Hint: There are 2 ways to do it: 1) visually check what happens to the expression of MU, when x increases (does it decrease, keep constant or decrease?); or 2) take the partial derivative of this marginal utility with respect to x, that is аMUX If ƏMUX ƏMUX < 0, the marginal utility of x is diminishing/decreasing in x; = 0, the ?х marginal utility of x is constant in x; and if Ux>0, the marginal utility of x is increasing in x] əx əx d) Does the marginal utility of good y diminish, remain constant, or increase as each of the individuals buys more y? Explain. e) Find the marginal rate of substitution of good x for good y, MRSxy? [Hint: Take and simplify as much as you can the ratio of marginal utilities MRSxy= MUX MUY f) Is the MRSxy diminishing, constant, or increasing as more x is consumed? OMRSxy. If [Hint: There are 2 ways to do it: 1) visually check what happens to the expression of MRSxy when x increases (does it decrease, keep constant or decrease?); or 2) take the partial derivative of the marginal rate of substitution with respect to x, that is ax ƏMRSx.y = 0, əx ƏMRSx.y əx > 0, the marginal rate of the marginal rate of substitution is constant in x; and if substitution of x is increasing in x] < 0, the marginal rate of substitution is diminishing/decreasing in x; ƏMRSxy ax
Chapter10: Consumer Choice Theory
Section: Chapter Questions
Problem 9P
Related questions
Question
![The following five individuals have different utility functions over food
(good x) and clothing (good y):
1) u₁(x, y) = 3x²y
2) u₂(x, y) = 2√x + y
3) Uz(x, y) = x0.6y0.4
4) u₁(x, y) = x² + y²
1
5) us(x, y) = x + 3y
For each of these people:
a) Compute their marginal utilities of good x, MUx =
Ju(x,y)
MU, = ду
du(x,y)
ax
b) Check whether the property of "more is better" is satisfied for both goods? Explain. [Hint:
Check whether marginal utilities are positive assuming positive amounts of good x and
good y]
ƏMUX
əx
and marginal utility of good y,
c) Does the marginal utility of good x diminish, remain constant, or increase as each of the
individuals buys more x? Explain. [Hint: There are 2 ways to do it: 1) visually check what
happens to the expression of MUx when x increases (does it decrease, keep constant or
decrease?); or 2) take the partial derivative of this marginal utility with respect to x, that is
ƏMUx .aMUX
ƏMUX
If < 0, the marginal utility of x is diminishing/decreasing in x; = 0, the
ax
əx
əx
marginal utility of x is constant in x; and if > 0, the marginal utility of x is increasing
in x]
d) Does the marginal utility of good y diminish, remain constant, or increase as each of the
individuals buys more y? Explain.
MUX
MUy
e) Find the marginal rate of substitution of good x for good y, MRSxy? [Hint: Take and
simplify as much as you can the ratio of marginal utilities MRSx,y =
f) Is the MRSxy diminishing, constant, or increasing as more x is consumed?
ƏMRSxy If
[Hint: There are 2 ways to do it: 1) visually check what happens to the expression of
MRSxy when x increases (does it decrease, keep constant or decrease?); or 2) take the
partial derivative of the marginal rate of substitution with respect to x, that is
əx
ƏMRSxy= 0,
ax
ƏMRSxy
əx
> 0, the marginal rate of
< 0, the marginal rate of substitution is diminishing/decreasing in x;
ƏMRSx.y
əx
the marginal rate of substitution is constant in x; and if
substitution of x is increasing in x]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3f0128e8-6553-498c-afa0-dd14e88e258d%2Fc11602c4-98ab-46ac-87a2-77d1e2dde538%2Fat3zxz_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The following five individuals have different utility functions over food
(good x) and clothing (good y):
1) u₁(x, y) = 3x²y
2) u₂(x, y) = 2√x + y
3) Uz(x, y) = x0.6y0.4
4) u₁(x, y) = x² + y²
1
5) us(x, y) = x + 3y
For each of these people:
a) Compute their marginal utilities of good x, MUx =
Ju(x,y)
MU, = ду
du(x,y)
ax
b) Check whether the property of "more is better" is satisfied for both goods? Explain. [Hint:
Check whether marginal utilities are positive assuming positive amounts of good x and
good y]
ƏMUX
əx
and marginal utility of good y,
c) Does the marginal utility of good x diminish, remain constant, or increase as each of the
individuals buys more x? Explain. [Hint: There are 2 ways to do it: 1) visually check what
happens to the expression of MUx when x increases (does it decrease, keep constant or
decrease?); or 2) take the partial derivative of this marginal utility with respect to x, that is
ƏMUx .aMUX
ƏMUX
If < 0, the marginal utility of x is diminishing/decreasing in x; = 0, the
ax
əx
əx
marginal utility of x is constant in x; and if > 0, the marginal utility of x is increasing
in x]
d) Does the marginal utility of good y diminish, remain constant, or increase as each of the
individuals buys more y? Explain.
MUX
MUy
e) Find the marginal rate of substitution of good x for good y, MRSxy? [Hint: Take and
simplify as much as you can the ratio of marginal utilities MRSx,y =
f) Is the MRSxy diminishing, constant, or increasing as more x is consumed?
ƏMRSxy If
[Hint: There are 2 ways to do it: 1) visually check what happens to the expression of
MRSxy when x increases (does it decrease, keep constant or decrease?); or 2) take the
partial derivative of the marginal rate of substitution with respect to x, that is
əx
ƏMRSxy= 0,
ax
ƏMRSxy
əx
> 0, the marginal rate of
< 0, the marginal rate of substitution is diminishing/decreasing in x;
ƏMRSx.y
əx
the marginal rate of substitution is constant in x; and if
substitution of x is increasing in x]
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