Question

Consider a consumer with preferences over two goods (good x and good y) given by

u ( x , y ) = x ⋅ y.

Given income of I and price of good yas $ P yper pound and price of good xgiven as $ P xper pound, the consumer chooses the optimal consumption bundle given as

x ∗ = I 2 P x

and

y ∗ = I 2 P y .

Given P x = $ 1per pound and P y = $ 2per pound, income I = $ 100as in table below and

(a) find the optimal consumption bundle, i.e.,

( x ∗ , y ∗ )and u ( x ∗ , y ∗ ) .

(b) Let the income change as given in the table below (in increments of one hundred dollar).

I x ∗ y ∗ $ 100 $ 200 $ 300 $ 400 $ 500 $ 600 $ 700 $ 800 $ 900

The two prices do not change. Find the new optimal consumption bundle, i.e.,

( x ∗ , y ∗ )and complete the table.

(c) Draw the income consumption curve with optimal quantity of good xon the horizontal axis and with optimal quantity of good yon the vertical axis.

Answer 1

Given:

Utility function (x,y) = x*y

The consumer chooses the consumption bundle of:

x* = I/2Px

y* = I/2Py

We can derive this by setting the consumer maximization problem, the consumer will consume good x and y quantity where he maximizes his consumption. That is where the marginal rate of substitution is equal to the ratio of prices.

That is,

MRS = Px/Py

Marginal rate of substitution is the ratio of marginal utilities of both the goods.

therefore,

the consumer maximises his utility where,

Putting it in the budget constraint, we get the optimal consumption of y and the x.

Budget constraint:

Putting the value of y,

similarly, the optimal consumption of y is,

Now solving for the given questions,

a.) Optimal bundle:

Given:

Price of good x (Px) = $1

Price of good y (Py) = $2

Income (I) = $100

Therefore putting price of goods and income, we get the optimal bundle.

Optimal consumption of good x is:

Putting Px =1 and I = 100

Now, doing the same for Y's consumption,

Optimal consumption of good x is:

Putting Py =2 and I = 100

Therefore the optimal bundle is 50 units of good x and 25 units of good y.

That is, Optimal bundle = (50 units of good x, 25 units of good y)

b.) Optimal bundle at different income levels:

The income changes and increases with an increment of one hundred. We can find the new optimal bundle at each income by setting the Income in the optimal bundle equation of both the goods, and prices being the same.

i.) At Income of $200.

I= $200, Px = $1, Py = $2

Solving for the optimal bundle by using the equal of the optimal number of good x and good y.

Optimal consumption of good x is:

Putting Px =1 and I = 200

Now, doing the same for Y's consumption,

Optimal consumption of good x is:

Putting Py =2 and I = 200

That is, Optimal bundle at income of $200 = (100 units of good x, 50 units of good y)

ii) At Income of $300.

I= $300, Px = $1, Py = $2

Solving for the optimal bundle by using the equal of the optimal number of good x and good y.

Optimal consumption of good x is:

Putting Px =1 and I = 300

Now, doing the same for Y's consumption,

Optimal consumption of good x is:

Putting Py =2 and I = 300

That is, Optimal bundle = (150 units of good x, 75 units of good y)

Doing the same for all the values of income level, that is:

using the formula for consumption bundle as:

here, price of good x is fixed as $1. Therefore, the final formula to calculate optimal x at variable income is.

For good y, the optimal consumption is:

As the price of good y doesn't change and is $2. Therefore, final formula to calculate optimal of y at variable income is.

Using, these formulas we can calculate the optimal bundle from income level $100 to $900.

c.) The income consumption curve is the curve showing different optimal bundles at a various income level. Here, we can see that as the income rises, the optimal bundle shifts upwards. As the consumer now consumed on a higher indifference curve as the new budget line is shifts outward. The increase in income shifts the budget constraint outward.

We can derive the income consumption curve from the optimal bundles at different prices. Where good x is on the x-axis and good y is on the y-axis.

ICC is the Income consumption curve at different levels of income, from an increment of 100. From income level $100 to $900.

Description of the diagram:

The combinations of good x and good y shows the optimal bundle at various levels of income. We can see that as income rises, the consumption bundle increases. As more income means that the consumer can now consume more with the given prices. He moves at a higher indifference curve and the optimal bundle is determined at the new budget constraints with the new income.