Question

1. Al Einstein has a utility function that we can describe by
u(x_{1}, x_{2}) = x^{21} +
2x_{1}x_{2} + x^{22}

. Al’s wife, El Einstein, has a utility function v(x_{1},
x_{2}) = x_{2} + x_{1}.

(a) Calculate Al’s marginal rate of substitution between
x_{1} and x_{2}.

(b) What is El’s marginal rate of substitution between
x_{1} and x_{2}?

(c) Do Al’s and El’s utility functions u(x_{1},
x_{2}) and v(x_{1}, x_{2}) represent the
same preferences?

(d) Is El’s utility function a monotonic transformation of
Al’s?

2. Marie Curie has a utility function U(x, y) = max{x, 2y}

(a) Draw a graph, and plot the lines given by the equations x = 10,
and 2y = 10.

(b) What’s the value of U(x, y) when x = 10 and 2y < 10?

(c) What’s the value of U(x, y) when x < 10 and 2y = 10?

(d) On the same chart, plot the indifference curve along which U(x,
y) = 10.

(e) Are Marie’s preferences convex?

3. Convert these marketing pitches into mathematical utility
functions:

(a) Everything goes with Crackers

(b) Bank Brewery asks that you drink responsibly.

(c) Sunflower margarine tastes just like butter!

4. Suppose each individual in a society gets utility from varying
degrees of a specific policy P, which can

be between 0 and 1 so that 0 < P < 1. Individual 1’s utility
function is given by u_{1}(P) = P^{0.5},

while individual 2’s utility function is given by u_{2}(P)
= 0.3P, individual 3’s utility function is given by

u_{3}(P) = P^{-0.5}, and individual 4’s utility
function is given by u_{4} = u_{1}(P) +
u_{2}(P) + u_{3}(P). Assuming

these are cardinal utility functions:

(a) In words, describe how the policy affects each
individual.

(b) Calculate the utility of each individual for each value of P
from 0.05 to 0.95 in increments of 0.5.

(c) There is no correct answer to this question: how would you
choose a value of P if you were the

government? Why? What value of P would you choose?

(d) Why is there no correct answer to the question above?

(e) Which one of these individuals would you elect as your leader
and why?

5. Here are some utility functions. Find a) the marginal utility
with respect to

x_{1}, b) the marginal utility with respect to
x_{2}, c) the marginal rate of substitution between
x_{1} and x_{2}.

(a) u(x_{1}, x_{2}) = 2x_{1} +
3x_{2}

(b) u(x_{1}, x_{2}) = 4x_{1} +
6x_{2}

(c) u(x_{1}, x_{2}) = ax_{1} +
bx_{2}

(d) u(x_{1}, x_{2}) = 2√x_{1} +
x_{2}

(e) u(x_{1}, x_{2}) = ln(x_{1}) +
x_{2}

(f) u(x_{1}, x_{2}) = v(x_{1}) +
x_{2}

(g) u(x_{1}, x_{2}) =
x_{1}x_{2}

(h) u(x_{1}, x_{2}) =
x^{a1}x^{b2}

(i) u(x_{1}, x_{2}) = (x_{1} +
2)(x_{2} + b)

(j) u(x_{1}, x_{2}) = (x_{1} +
a)(x_{2} + b)

(k) u(x_{1}, x_{2}) = x^{a1} +
x^{a2}

6. Tom Edison is eating 5 avocados and 20 sugar apples. The
marginal rate

of substitution is the rate at which Tom is willing to trade sugar
apples for an extra avocado. At his

current consumption bundle of 5 avocados and 20 sugar apples, his
marginal rate of substitution is 4.

Compared to his current

(a) Will Tom be better off if he instead consumes 6 avocados and 20
sugar apples?

(b) Will Tom be better off if he instead consumes 6 avocados and 15
sugar apples?

(c) Will Tom be better off if he instead consumes 7 avocados and 16
sugar apples?

(d) Will Tom be better off if he instead consumes 6 avocados and 16
sugar apples?

7. (To be done on your own) Give three examples of utility
functions that give exactly the same preferences

as u(x, y) = xy. Derive the marginal rate of substitution in each
case and check whether they are equal.

Answer #1

1

Sol

Given , Al's utility function u(x_{1} , x_{2} )
= x_{1}^{2} + 2x_{1}x_{2} +
x_{2}^{2}

Al's wife's utility function u(x_{1} , x_{2}) =
x_{1} + x_{2}

Al Einstein has a utility function that we can describe by u(x1,
x2) = x 2 1 + 2x1x2 + x 2 2 . Al’s wife, El Einstein, has a utility
function v(x1, x2) = x2 + x1. (a) Calculate Al’s marginal rate of
substitution between x1 and x2. (b) What is El’s marginal rate of
substitution between x1 and x2? (c) Do Al’s and El’s utility
functions u(x1, x2) and v(x1, x2) represent the same preferences?
(d) Is El’s...

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.

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.

Consider a consumer with preferences represented by the utility
function
u(x,y)=3x+6 sqrt(y)
(a) Are these preferences strictly convex?
(b) Derive the marginal rate of substitution.
(c) Suppose instead, the utility function is:
u(x,y)=x+2 sqrt(y)
Are these preferences strictly convex? Derive the marginal rate
of substitution.
(d) Are there any similarities or differences between the two
utility functions?

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...

Show that utility u(x1,x2)=2√x1+√x2 is strictly quasi
concave(Hint: You can prove it by showing the utility function has
diminishing marginal rate of substitution).

2. Consider a consumer with preferences represented by the
utility function:
u(x,y)=3x+6sqrt(y)
(a) Are these preferences strictly convex?
(b) Derive the marginal rate of substitution.
(c) Suppose instead, the utility
function is:
u(x,y)=x+2sqrt(y)
Are these preferences strictly convex?
Derive the marginal rate of sbustitution.
(d) Are there any similarities or diﬀerences between the two
utility functions?

Consider a consumer with preferences represented by the utility
function:
U(x,y) = 3x + 6 √ y
Are these preferences strictly convex?
Derive the marginal rate of substitution
Suppose, the utility function is:
U(x,y) = -x +2 √
y
Are there any similarities or differences between the two
utility functions?

Consider the Cobb-Douglas utility function
u(x1,x2)=x1^(a)x2^(1-a).
a. Find the Hicksian demand correspondence h(p, u) and the
expenditure function e(p,u) using the optimality conditions for the
EMP.
b. Derive the indirect utility function from the expenditure
function using the relationship e(p,v(p,w)) =w.
c. Derive the Walrasian demand correspondence from the Hicksian
demand correspondence and the indirect utility function using the
relationship
x(p,w)=h(p,v(p,w)).
d. vertify roy's identity.
e. find the substitution matrix and the slutsky matrix, and
vertify the slutsky equation.
f....

Suppose an individual consumers two goods, with utility function
U (x1; x2) = x1 + 6(x1x2)^1/2 + 9x2. Formulate the utility
maximization problem when she faces a budget line p1x1 + p2x2 = I.
Find the demand functions for goods 1 and 2.
(b) Now consider an individual consumers with utility function U
(x1; x2) = x1^1/2 + 3x2^1/2. Formulate the utility maximization
problem when she faces a budget line p1x1 + p2x2 = I. Find the
demand functions for...

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