Question

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?

Answer #1

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?

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 whose preferences over the goods are
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this function the marginal utilities are given by MUx(x, y) = y^2
and MUy(x, y) = 2xy.
(a) What are the formulas for the indifference curves
corresponding to utility levels of u ̄ = 1, u ̄ = 4, and u ̄ = 9?
Draw these three indifference curves in one graph.
(b) What is the marginal rate of substitution...

Consider a consumer with the following utility function:
U(X, Y ) = XY.
(a) Derive this consumer’s marginal rate of substitution,
MUX/MUY
(b) Derive this consumer’s demand functions X∗ and Y∗.
(c) Suppose that the market for good X is composed of 3000
identical consumers, each with income of $100. Derive the market
demand function for good X. Denote the market quantity demanded as
QX.
(d) Use calculus to show that the market demand function
satisfies the law-of-demand.

A consumer has preferences represented by the utility function
u(x, y) = x^(1/2)*y^(1/2). (This means that
MUx=(1/2)x^(−1/2)*y^(1/2) and MUy =1/2x^(1/2)*y^(−1/2)
a. What is the marginal rate of substitution?
b. Suppose that the price of good x is 2, and the price of good
y is 1. The consumer’s income is 20. What is the optimal quantity
of x and y the consumer will choose?
c. Suppose the price of good x decreases to 1. The price of good
y and...

3. Suppose that a consumer has a utility function given by
U(X,Y) = X^.5Y^.5 . Consider the following bundles of goods: A =
(9, 4), B = (16, 16), C = (1, 36).
a. Calculate the consumer’s utility level for each bundle of
goods.
b. Specify the preference ordering for the bundles using the
“strictly preferred to” symbol and the “indifferent to” symbol.
c. Now, take the natural log of the utility function. Calculate
the new utility level provided by...

Suppose a consumer has the utility function u(x, y) = x + y.
a) In a well-labeled diagram, illustrate the indifference curve
which yields a utility level of 1.
(b) If the consumer has income M and faces the prices px and py
for x and y, respectively, derive the demand functions for the two
goods.
(c) What types of preferences are associated with such a utility
function?

Consider a consumer with the following utility function: U(X,
Y ) = X1/2Y 1/2
(a) Derive the consumer’s marginal rate of substitution
(b) Calculate the derivative of the MRS with respect to
X.
(c) Is the utility function homogenous in X?
(d) Re-write the regular budget constraint as a function of PX
, X, PY , &I. In other words, solve the equation for Y .
(e) State the optimality condition that relates the marginal
rate of substi- tution to...

Suppose a consumer has the utility function U (x, y) = xy + x +
y. Recall that for this function the marginal utilities are given
by MUx(x,y) = y+1 and MUy(x,y) = x+1.
(a) What is the marginal rate of substitution MRSxy?
(b)If the prices for the goods are px =$2 and py =$4,and if the
income of the consumer is M = $18, then what is the consumer’s
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(c) What if instead the prices are...

George has preferences of goods 1 (denoted by x) and 2 (denoted
by y) represented by the utility function u(x,y)= (x^2)+y:
a. Write an expression for marginal utility for good 1. Does he
like good 1 and why?
b. Write an expression for George’s marginal rate of
substitution at any point. Do his preferences exhibit a diminishing
marginal rate of substitution?
c. Suppose George was at the point (10,10) and Pete offered to
give him 2 units of good 2...

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