Question

Consider the following utility function: U(x1,x2)
X_{1}^{1/3} X_{2}

Suppose a consumer with the above utility function faces prices
p_{1 =} 2 and

p_{2 =} 3 and he has an income m = 12. What’s his optimal
bundle to consume?

Answer #1

U(x1,x2) = X_{1}^{1/3} X_{2}

P1 = 2 and P2 = 3. Income m =12

Consumers optimise where their Marginal Rate of Substitution between x1 and x2 equals the price ratio of x1 and x2

MRS = MUx1/MUx2

Differentiating the utility function with respect to x1 and x2 we get,

MUx1 =

MUx2 =

MRS = (x2)/(3x1)

Price ratio of x1 and x2 equals = 2/3

Hence at equilibrium,

(x2)/(3x1) = 2/3

x2 = 2*x1

The consumer's budget constraint can be written as:

P(x1)*x1 + P(x2)*x2 = m

2*x1+3*x2 = 12

Put x2 = 2*x1, we get

2*x1+6*x1 = 12

**x1 = 1.5**

**x2 = 2*x1 = 1.5*2 = 3**

Hence optimal consumption bundle, x1=1.5 and x2 = 3

Suppose x1 and x2 are perfect substitutes
with the utility function U(x1, x2) =
2x1 + 6x2. If p1 = 1,
p2 = 2, and income m = 10, what it the optimal bundle
(x1*, x2*)?

Suppose the utility function is given by U(x1,
x2) = 14 min{2x, 3y}. Calculate the optimal consumption
bundle if income is m, and prices are p1, and
p2.

3. Suppose that a consumer has a utility function
u(x1, x2) =
x1 + x2. Initially the
consumer faces prices (1, 2) and has income 10. If the prices
change to (4, 2), calculate the compensating and equivalent
variations. [Hint: find their initial optimal consumption
of the two goods, and then after the price increase. Then show this
graphically.]
please do step by step and show the graph

7.
Suppose you have the following utility function for two
goods:
u(x1, x2) = x
1/3
1 x
2/3
2
. Suppose your initial income is I, and prices are p1 and
p2.
(a) Suppose I = 400, p1 = 2.5, and p2 = 5. Solve for the
optimal bundle. Graph the budget
constraint with x1 on the horizontal axis, and the
indifference curve for that bundle.
Label all relevant points
(b) Suppose I = 600, p1 = 2.5, and...

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=1/4x12
+1/9x22. Suppose the prices of good
1 and
good 2 are p1 andp2, and income is
m.
Do bundles (2, 9) and (4, radical54) lie on the same
indifference curve?
Evaluate the marginal rate of substitution at
(x1,x2) = (8, 9).
Does this utility function represent
convexpreferences?
Would bundle (x1,x2) satisfying (1)
MU1/MU2 =p1/p2 and (2)
p1x1 + p2x2 =m be an
optimal choice? (hint: what does an indifference curve look
like?)

Consider a consumer with a utility function U =
x2/3y1/3, where x and y are the quantities of
each of the two goods consumed. A consumer faces prices for x of $2
and y of $1, and is currently consuming 10 units of good X and 30
units of good Y with all available income. What can we say about
this consumption bundle?
Group of answer choices
a.The consumption bundle is not optimal; the consumer could
increase their utility by...

2. A consumer has the utility function U ( X1,
X2 ) = X1 + X2 +
X1X2 and the budget constraint
P1X1 + P2X2 = M ,
where M is income, and P1 and P2 are the
prices of the two goods. .
a. Find the consumer’s marginal rate of substitution (MRS)
between the two goods.
b. Use the condition (MRS = price ratio) and the budget
constraint to find the demand functions for the two goods.
c. Are...

Suppose Alex only consumes 3 units of x1 with 8 units of x2.
That is, if he is consuming more x1 or x2 in a different ratio, it
does not increase his utility
a) Write down Alex’s utility function. What kind of utility
function does he have?
b) Suppose Alex wants to have a utility 48. If he desires to
make the best use of his money, based on your utility function in
a) how many units of x1 and...

Consider the utility function:
u( x1 , x2 ) = 2√ x1 +
2√x2
a) Find the Marshallian demand function. Use ( p1 ,
p2 ) to denote the exogenous prices of x1 and
x2 respectively. Use y to denote the consumer's
disposable income.
b) Find the indirect utility function and verify Roy's
identity
c) Find the expenditure function
d) Find the Hicksian demand function

Imran consumes two goods, X1 and X2. his
utility function takes the form: u(X1, X2)=
4(X1)^3+3(X2)^5. The price of X1
is Rs. 2 and the price of X2 is Rs. 4. Imran has
allocated Rs. 1000 for the consumption of these two goods.
(a) Fine the optimal bundle of these two goods that Imran would
consume if he wants to maximize his utility. Note: write bundles in
integers instead of decimals.
(b) What is Imran's expenditure on X1? On
X2?...

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