Skip has the following utility function: U(x, y) = x(y + 1), where x and y are quantities of two consumption goods whose prices are px and py respectively. Skip has a budget of B. Therefore the Skip’s maximization problem is x(y + 1) + λ(B − pxx − pyy) a) From the first order conditions find expressions for the demand functions x∗ = x(px, py, B) y∗ = y(px, py, B) Carefully graph x∗ and y∗. Graph Skip’s indifference curves. What kind of good is y? b) Verify that skip is at a maximum by checking the second order conditions. c) By substituting x∗ and y∗ into the utility function find an expressions for the indirect utility function, U = U(px, py, B)
Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.
Skip has the following utility function: U(x, y) = x(y + 1), where x and y are quantities of
two consumption goods whose prices are px and py respectively. Skip has a budget of B.
Therefore the Skip’s maximization problem is
x(y + 1) + λ(B − pxx − pyy)
a) From the first order conditions find expressions for the demand functions
x∗ = x(px, py, B) y∗ = y(px, py, B)
Carefully graph x∗ and y∗. Graph Skip’s indifference curves. What kind of good
is y?
b) Verify that skip is at a maximum by checking the second order conditions.
c) By substituting x∗ and y∗ into the utility function find an expressions for the
indirect utility function,
U = U(px, py, B)
d) By rearranging the indirect utility function, derive an expression for the
expenditure function,
B∗ = B(px, py, U0)
Interpret this expression. Find ∂B/∂px and ∂B/∂py.
Skip’s maximization problem could be recast as the following minimization problem:
pxx + pyy s.t. U0 = x(y + 1)
e) Write down the lagrangian for this problem.
f) Find the values of x and y that solve this minimization problem and show
that the values of x and y are equal to the partial derivatives of the expenditure function, ∂B/∂px and ∂B/∂py respectively. (Hint: use the indirect utility
function)
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