Question

Suppose the utility function is given by U(x_{1},
x_{2}) = 14 min{2x, 3y}. Calculate the optimal consumption
bundle if income is m, and prices are p_{1}, and
p_{2}.

Answer #1

U(x,y) = 14 min{2x, 3y}

Since the utility fucntion remains unchanged under monotonic increasing transformation, so we can write the utility function as U(x,y) = min{2x, 3y}

The price of x is p1 and y is p2. The consumer has a income m. Thus the budget constraint is p1*x+p2*y=m

Since the utility function is the min function, for the consumer to be in equilibrium he/she should derive the same amount of utility from x and y. If does not get the same utility from x and y then the consumer will get lower amount that is 2x or 3y which ever is less.

Thus for equilibrium 2x=3y which implied x=1.5*y.

Substituting the above x in the budget constraint of the consumer p1*x+p2*y=m, we get p1*(1.5*y.)+p2*y=m. Solving for y, we get the optimal consumption of y as y=m/(1.5*p1+p2). Thus the optimal consumption of x is x=1.5*y=1.5*m/(1.5*p1+p2)

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*)?

Consider the following utility function: U(x1,x2)
X11/3 X2
Suppose a consumer with the above utility function faces prices
p1 = 2 and
p2 = 3 and he has an income m = 12. What’s his optimal
bundle to consume?

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...

. Suppose utility is given by the following function:
u(x, y) = min(2x, 3y) Suppose Px = 4, Py =
6, and m = 24.
Use this information to answer the following questions:
(a) What is the no-waste condition for this individual?
(b) Draw a map of indifference curves for these preferences. Be
sure to label your axes, include the no-waste line, and draw at
least three indifference curves.
(c) Given prices and income, what is the utility-maximizing
bundle of...

Consider utility function u(x1,x2)
=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?)

Change the Humphrey and Lauren example such that Lauren’s
utility function is uL(x1,x2) = min{x1, x2} and Humphrey’s utility
function is uH (x1, x2) = 2√x1 + √x2. Their endowments are eL =
(4,16) and eH = (2,24).
1)Suppose Humphrey and Lauren are to simply just consume their
given endowments. State the definition of Pareto efficiency. Is
this a Pareto efficient allocation? As part of answering this
question, can you find an alternative allocation of the goods that
Pareto dominates...

Determine the optimal quantities of both x1 and
x2 for each utility function. The price of good 1
(p1) is $2. The price of good 2 (p2) is $1.
Income (m) is $10.
a.) U(x1,x2) =
min{2x1, 7x2}
b.) U(x1,x2) =
9x1+4x2
c.) U(x1,x2) =
2x11/2 x21/3
Please show all your work.

The utility function is given by u (x1,x2) = x1^0.5 + x2^0.5
1) Find the marginal rate of substitution (MRSx1,x2 )
2) Derive the demand functions x1(p1,p2,m) and x2(p1, p2,m) by
using the method of Lagrange.

Qin has the utility function U(x1, x2) = x1 + x1x2, where x1 is
her consumption of good 1 and x2 is her consumption of good 2. The
price of good 1 is p1, the price of good 2 is p2, and her income is
M.
Setting the marginal rate of substitution equal to the price
ratio yields this equation: p1/p2 = (1+x2)/(A+x1) where A is a
number. What is A?
Suppose p1 = 11, p2 = 3 and M...

The utility function is given by u (x1, x2) = x1^0.5+x2^0.5
1) Find the marginal rate of substitution (MRSx1,x2 )
2) Derive the demand functions x1(p1, p2, m) and x2(p1,p2, m) by
using the method of Lagrange.

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