Lesson test: The consumer's problem

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The consumer's problem

1.[2p]
A consumer has $u = x^{0.4} y^{0.6}$ , income £120, $p_{x} = 4$ and $p_{y} = 3$ . How many units of $x$ do they buy?

CorrectNot quite: 12

2.[2p]
What does the tangency condition $MRS = p_{x} / p_{y}$ say?

The rate at which the consumer will trade the goods equals the rate at which the market will

Correct

The consumer gets equal utility from both goods

The answer is: The rate at which the consumer will trade the goods equals the rate at which the market will

The consumer spends equal amounts on both goods

The answer is: The rate at which the consumer will trade the goods equals the rate at which the market will

Marginal utility is the same for both goods

The answer is: The rate at which the consumer will trade the goods equals the rate at which the market will

3.[2p]
With $u = x^{a} y^{b}$ , what fraction of income goes to $x$ when $a = 0.25$ and $b = 0.75$ ?

CorrectNot quite: 0.25

4.[1p]
Doubling both prices and income leaves the consumer's chosen bundle unchanged.

True

Correct

False

The answer is: True

5.[3p]
A consumer with $u = 3 x + 2 y$ faces $p_{x} = 2$ and $p_{y} = 1$ with income £60. What do they buy?

60 units of $y$ and none of $x$

Correct

30 units of $x$ and none of $y$

The answer is: 60 units of $y$ and none of $x$

15 units of $x$ and 30 of $y$

The answer is: 60 units of $y$ and none of $x$

20 units of $x$ and 20 of $y$

The answer is: 60 units of $y$ and none of $x$

6.[3p]
A consumer has $u = 10 \sqrt{x} + y$ with $p_{x} = 2$ , $p_{y} = 1$ and ample income. How many units of $x$ do they buy?

CorrectNot quite: 6.25

7.[3p]
Which situations make the tangency condition the wrong way to find the optimum?

Select all that apply

Preferences are perfect substitutes and the MRS never equals the price ratio

Correct

The tangency bundle costs more than the consumer's income

Correct

Indifference curves are bowed away from the origin, so the tangency is a minimum along the budget line

Correct

Income rises while prices stay fixed

The answer is: Preferences are perfect substitutes and the MRS never equals the price ratio, The tangency bundle costs more than the consumer's income, Indifference curves are bowed away from the origin, so the tangency is a minimum along the budget line

8.[2p]
Why does a rise in $p_{x}$ pivot the budget line about the $y$ intercept rather than shifting it?

A consumer buying no $x$ is unaffected by the price of $x$ , so that intercept cannot move

Correct

Income has not changed, so the line must keep its position

The answer is: A consumer buying no $x$ is unaffected by the price of $x$ , so that intercept cannot move

The $y$ intercept is fixed by convention when drawing budget lines

The answer is: A consumer buying no $x$ is unaffected by the price of $x$ , so that intercept cannot move

Only income changes can shift a budget line

The answer is: A consumer buying no $x$ is unaffected by the price of $x$ , so that intercept cannot move

9.[3p]
Match each Lagrange result to what it means.

${MU}_{x} / p_{x} = {MU}_{y} / p_{y}$
$λ$
$MRS = p_{x} / p_{y}$
$p_{x} x + p_{y} y = m$

indifference curve tangent to budget line
equal marginal utility per pound
the whole income is spent
the marginal utility of income

Show the answer

${MU}_{x} / p_{x} = {MU}_{y} / p_{y}$ : equal marginal utility per pound $λ$ : the marginal utility of income $MRS = p_{x} / p_{y}$ : indifference curve tangent to budget line $p_{x} x + p_{y} y = m$ : the whole income is spent

1.[2p] A consumer has u=x0.4y0.6, income £120, px=4 and py=3. How many units of x do they buy?

2.[2p] What does the tangency condition MRS=px/py say?

3.[2p] With u=xayb, what fraction of income goes to x when a=0.25 and b=0.75?

4.[1p] Doubling both prices and income leaves the consumer's chosen bundle unchanged.

5.[3p] A consumer with u=3x+2y faces px=2 and py=1 with income £60. What do they buy?

6.[3p] A consumer has u=10x+y with px=2, py=1 and ample income. How many units of x do they buy?

7.[3p] Which situations make the tangency condition the wrong way to find the optimum?

8.[2p] Why does a rise in px pivot the budget line about the y intercept rather than shifting it?

9.[3p] Match each Lagrange result to what it means.

1.[2p]
A consumer has $u = x^{0.4} y^{0.6}$ , income £120, $p_{x} = 4$ and $p_{y} = 3$ . How many units of $x$ do they buy?

2.[2p]
What does the tangency condition $MRS = p_{x} / p_{y}$ say?

3.[2p]
With $u = x^{a} y^{b}$ , what fraction of income goes to $x$ when $a = 0.25$ and $b = 0.75$ ?

4.[1p]
Doubling both prices and income leaves the consumer's chosen bundle unchanged.

5.[3p]
A consumer with $u = 3 x + 2 y$ faces $p_{x} = 2$ and $p_{y} = 1$ with income £60. What do they buy?

6.[3p]
A consumer has $u = 10 \sqrt{x} + y$ with $p_{x} = 2$ , $p_{y} = 1$ and ample income. How many units of $x$ do they buy?

7.[3p]
Which situations make the tangency condition the wrong way to find the optimum?

8.[2p]
Why does a rise in $p_{x}$ pivot the budget line about the $y$ intercept rather than shifting it?

9.[3p]
Match each Lagrange result to what it means.