Question
In: Economics
The utility of an agent who consumes x amounts of good X and y amounts of...
The utility of an agent who consumes x amounts of good X and y amounts of good Y is given by the following utility function: u = U(x, y) = 3 + 2x 2 + xy + y 2 Assume that the agent has an income equal to £10, that the price of good X is px = £2 and X and the price of good Y is py = £1. a) [3 marks] Find the budget constraint of the agent. b) [3 marks] Write down the constrained optimisation problem for this agent. c) [12 marks] Assuming that the agent spends all the income, solve for the optimal consumption bundle using the Lagrangian method. d) [2 marks] How can you interpret the Lagrange multiplier, λ? e) [5 marks] Illustrate on a graph the agent’s optimal consumption bundle. Be sure to label all axes, curves and relevant points in your graph.
Solutions
Rahul Sunny answered 2 years agoRelated Solutions
A utility-maximizing consumer who consumes quantity x and y of two goods has a quadratic utility...
A utility-maximizing consumer who consumes quantity x and y of
two goods has a quadratic utility function given by
Ux,y=x+y-0.05x2-0.05y2
subject to
2x+5y=128
Where $128 is the consumer’s budget and the prices of the two
goods are, respectively, 2 and 5. Assuming marginal utilities Ux,
Uy > 0,
a. Find the quantity x and y that maximize the utility function.
b. Using bordered Hessian, check utility for a maximum.
c. What is the maximum utility of the consumer?
A utility-maximizing consumer who consumes quantity x and y of two goods has a quadratic utility...
A utility-maximizing consumer who consumes quantity x and y of
two goods has a quadratic utility function given by
Ux,y=x+y-0.05x2-0.05y2
subject to
2x+5y=128
Where $128 is the consumer’s budget and the prices of the two
goods are, respectively, 2 and 5. Assuming marginal utilities Ux,
Uy > 0,
a. Find the quantity x and y that maximize the utility function.
b. Using bordered Hessian, check utility for a maximum.
c. What is the maximum utility of the consumer?
A utility-maximizing consumer who consumes quantity x and y of two goods has a quadratic utility...
A utility-maximizing consumer who consumes quantity x and y of
two goods has a quadratic utility function given by
Ux,y=x+y-0.05x2-0.05y2
subject to
2x+5y=128
Where $128 is the consumer’s budget and the prices of the two
goods are, respectively, 2 and 5. Assuming marginal utilities Ux,
Uy > 0,
a. Find the quantity x and y that maximize the utility function.
b. Using bordered Hessian, check utility for a maximum.
c. What is the maximum utility of the consumer?
A utility-maximizing consumer who consumes quantity x and y of two goods has a quadratic utility...
A utility-maximizing consumer who consumes quantity x and y of
two goods has a quadratic utility function given by
Ux,y=x+y-0.05x2-0.05y2
subject to
2x+5y=128
Where $128 is the consumer’s budget and the prices of the two
goods are, respectively, 2 and 5. Assuming marginal utilities Ux,
Uy > 0,
a. Find the quantity x and y that maximize the utility function.
b. Using bordered Hessian, check utility for a maximum.
c. What is the maximum utility of the consumer?
Esther consumes goods X and Y, and her utility function is U(X,Y)=XY+Y For this utility function,...
Esther consumes goods X and Y, and her utility
function is
U(X,Y)=XY+Y
For this utility function,
MUX=Y
MUY=X+1
a. What is Esther's MRSXY?
Y/(X + 1)
X/Y
(X + 1)/Y
X/(Y + 1)
b. Suppose her daily income is $20, the price of X is $4
per unit, and the price of Y is $1 per unit. What is her
best choice?
Instructions: Enter
your answers as whole numbers.
X =
Y =
What is Esther's utility when her...
A consumes two goods, x and y. A ’s utility function is given by u(x, y)...
A consumes two goods, x and y. A ’s utility function is given
by u(x, y) = x 1/2y 1/2
The price of x is p and the price of y is 1. A has an income
of M.
(a) Derive A ’s demand functions for x and y.
(b) Suppose M = 72 and p falls from 9 to 4. Calculate the
income and substitution effects of the price change.
(c) Calculate the compensating variation of the price
change....
The utility that Jane receives by consuming good X and good Y is given by u(X,Y)...
The utility that Jane receives by consuming good X and good Y is
given by u(X,Y) = XY.
5.1) Draw the indifference curve associated with a utility level
of 12 and the indifference curve associated with the utility level
of 36.
5.2) Suppose that X costs $1 per unit and Y $3 per unit. Jane
has $12 to spend on X and Y. Graph the budget line that she
faces.
5.3) Derive Jane’s demand functions. What is the
utility-maximizing choice...
The utility that James receives by consuming good X and good Y is given by u(X,Y)...
The utility that James receives by consuming good X and good Y
is given by u(X,Y) = XY.
5.1) Draw the indifference curve associated with a utility level
of 12 and the indifference curve associated with the utility level
of 36.
5.2) Suppose that X costs $1 per unit and Y $3 per unit. James
has $12 to spend on X and Y. Graph the budget line that he
faces.
5.3) Derive James' demand functions. What is the
utility-maximizing choice...
13. Pip consumes two goods, x and y. Pip’s utility function is given by u(x, y)...
13. Pip consumes two goods, x and y. Pip’s utility function is
given by u(x, y) = x 1/2y 1/2 The price of x is p and the price of
y is 1. Pip has an income of M.
(a) Derive Pip’s demand functions for x and y. [5 marks]
(b) Suppose M = 72 and p falls from 9 to 4. Calculate the income
and substitution effects of the price change. [5 marks]
(c) Calculate the compensating variation of...
Leon consumes two goods, X and Y. His utility function is given by U(X,Y) = 56XY....
Leon consumes two goods, X and Y. His utility function is given
by U(X,Y) = 56XY. The price of X is $14 a unit; the price of Y is
$8 a unit; and Leon has $672 to spend on X and Y.
a. Provide the equation for Leon’s budget line. (Your answer for
the budget line should be in the form Y = a – bX, with specific
numerical values given for a and b.)
b. Provide the numerical value...
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