A find the optimum consumption bundle for both goods as a

Question

Question #1:

Suppose you spend your daily income ($100) into two goods: foods and other goods (composite goods), and suppose your preference over these two goods are represented by the following utility function: U(x1,x2) = αlnx1 + x2, where α is a constant number.

Let's denote foods as x₁ (horizontal axis) and other composite goods as x₂ (vertical axis).

If foods price is $5 per unit,

• a) Find the optimum consumption bundle (for both goods as a function of α)
• b) Find the range for α that could result in interior optimum solution for this problem.
• c) Give an example number for α that could result in corner optimum solutions and write down the utility function.

Question #2

In this question you are asked to create a utility function and to find the optimum consumption bundle.

• a) Find an example of two perfect complement goods (please do not use the examples we used in the lectures, quizzes, and exams). Try to find your own unique example.
• b) Assign a number to the ratio of the two complement goods (for example, when ratio is 1 for tea and sugar,onepack of sugar is used together withonecup of tea), and also assign numbers to the prices of the two goods.
• c) Suppose the total budget to be spent for these two goods is $I. Find the optimum consumption basket.
• d) Change price of any of the two goods by 10 percent. To consume the same amount of the two goods as before how the other price must change (assuming no change in the income)?

Question #3

Choose one utility function from the following Cobb-Douglas utility functions:

a) U(x,y) = xy

b) U(x,y) = (xy)^1/2

c) U(x,y) = x(y)^1/2

d) U(x,y) = y(x)^1/2

• a) Assign numbers for P_xand P_yand total income. (For example, P_x=$5, P_y=$1, , etc.)
• b) Change (increase or decrease) P_xby any amount,  and find the substitution and income effects of the price change on the consumption of good X.