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How to find utility maximizing bundle/ optimal bundle

- A consumer if better off if he can reach to a higher indifference curve.
- Due to the limited resource he has to be on the budget line.
- So our purpose is to find a combination/bundle of goods on the budget line that allows us to attain maximum utility. This bundle of goods is known as optimal bundle

Necessary Condition of Utility Maximization

- Tangency Condition: Utility will be maximized if the budget line is tangent to IC. For any bundle where budget line cuts the IC will not be an optimal solution because there is a way to improve the utility by moving to higher IC.
- Tangency condition implies that the slope of IC and budget line are same.
- Slope of IC= MRS
- Slope of budget line= Price Ratio= Px/Py
- So the tangency condition of utility maximization is
MRS= Px/Py

So A is the optimal bundle that is the solution to the utility maximization

Impact of price change on budget line

- If the price of one of the goods changes then the budget line will rotate.
Case 1: If the price of one good is increased ( price of other good remains the same) then the budget line will rotate inward.

Example: Suppose the budget line is 2X+ 2Y= 400. So the price of X is Px=2. Now if the price of X increases to Px=4 then the budget line will rotate inward.

- Case 2: If the price of one good is decreased ( price of other good remains the same) then the budget line will rotate outward.
- In the previous example if the price of good X decrease Px=1 then the budget line will rotate outward

Impact of Price Change on Utility other good remains the same) then the budget line will rotate outward.

- Case 1: Price Increase will Lead to a Decrease in Utility: If the price of one good is increased then the budget line rotates inward. To find the impact of price increase on utility we need to find the new optimal bundle after the price increase.
- First we draw the initial budget line and the indifference curve to get the initial optimal bundle ( IC1). Then we draw the new budget line which is found by rotating the initial budget line inward. Finally we shift the initial indifference curve inward to find the new optimal bundle. Usually in the new optimal bundle the amount of both goods will be decreased. So the utility level will decrease as the consumer is consuming less of both of the goods.

- Case 2 other good remains the same) then the budget line will rotate outward. : Price Decrease will Lead to a Increase in Utility: If the price of one good is decreased then the budget line rotates outward. To find the impact of price decrease on utility we need to find the new optimal bundle after the price decrease.
- First we draw the initial budget line and the indifference curve to get the initial optimal bundle ( IC1). Then we draw the new budget line which is found by rotating the initial budget line outward. Finally we shift the initial indifference curve outward to find the new optimal bundle. Usually in the new optimal bundle the amount of both goods will be increased. So the utility level will increase as the consumer is consuming more of both of the goods.

Impact of Income Change on Utility other good remains the same) then the budget line will rotate outward.

- Case 1: Increase in Income will Increase Utility
When there is an increase in income the budget line shifts to the right. So we can find the optimal bundle by shifting the indifference curve to the right and finding the tangency point. Usually in the new optimal bundle the amount of both of the goods will be increased. So the utility level will be increased as the consumer is consuming more of both of the goodswhen the income increases.

Example: In our example the budget line was 2X+ 2Y=400. So the income is 400. Now if the income increases to 800 then the budget line shifts to the right.

- Case other good remains the same) then the budget line will rotate outward. 2: Decrease in Income will Decrease Utility
When there is an decrease in income the budget line shifts to the left. So we can find the optimal bundle by shifting the indifference curve to the left and finding the tangency point. Usually in the new optimal bundle the amount of both of the goods will be decreased. So the utility level will be decreased as the consumer is consuming less of both of the goods when the income decreases.

- Example: In our example the budget line was 2X+ 2Y=400. So the income is 400. Now if the income decreases to 200 then the budget line shifts to the left.

Decompositio other good remains the same) then the budget line will rotate outward. n of Price Effect

- We see that when there is a price change of a good then the optimal bundle changes. That means the amount of goods that we consume changes. This is known as price effect.
- Price Effect: It is the change in consumption of a good due to the change in its price.
- Example: When the price of a good decreases the consumption of that good usually increases. This is the price effect.
- We can separate the impact of a price change into two components:
- substitution effect ( SE)
- income effect ( IE)
So we can write

Price Effect = Substitution Effect + Income Effect

Or, PE = SE + IE

THE IMPACT OF A PRICE CHANGE other good remains the same) then the budget line will rotate outward.

- The substitution effect involves the substitution of one good for another good due to a change in relative prices of the two goods.
- The income effect results from an increase or decrease in the consumer’s real income or purchasing power as a result of the price change.
- The sum of these two effects is called the price effect.

Example other good remains the same) then the budget line will rotate outward.

Suppose, we have two goods: Fine rice and Coarse rice. We exhaust our budget by consuming this two goods. When the price of fine rice decreases there are two sorts of effects:

Substitution Effect: As the price of fine rice has increased so we will want to substitute fine rice with coarse rice. That means as fine rice becomes relatively expensive we will want to replace fine rice with coarse rice. So we will decrease fine rice consumption and increase coarse rice consumption. This decrease in fine rice consumption due to the increase in relative price of fine rice is known as substitution effect.

Income Effect: As the fine rice becomes expensive we can now buy less fine rice with the same level of income. That means our purchasing power has decreased. This decrease in consumption of fine rice due to decrease in purchasing power is known as income effect.

Here our budget line is 2X + 2Y= 400 and initially our optimal bundle is X=100 and Y=100. Now suppose the price of X ( Fine Rice) increases to 4 ( initially it was 2). So in the new optimal bundle we consume 60 fine rice. We know that the price effect is the change in consumption due to change in price. So here price effect PE= 60-100= -40.

Now we can break down this price effect into substitution effect and income effect.

Suppose we decrease our consumption of fine rice by 30 ( kg) because we substitute fine rice with coarse rice. So substitution effect , SE= -30.

And we decrease our consumption of fine rice by 10(kg) due to the decrease in purchasing power ( as price of fine rice is increased we cannot buy the same amount of fine rice). So income effect , IE=-10

Thus PE= SE+ IE

-40= -30-10

Slutsky optimal bundle is X=100 and Y=100. Now suppose the price of X ( Fine Rice) increases to 4 ( initially it was 2). So in the new optimal bundle we consume 60 fine rice. We know that the price effect is the change in consumption due to change in price. So here price effect PE= 60-100= -40. Equation

- The decomposition of price effect into substitution effect and income effect is known as Slutsky equation. So
- PE= SE + IE
is the Slutsky equation

- Substitution Effect is always negative optimal bundle is X=100 and Y=100. Now suppose the price of X ( Fine Rice) increases to 4 ( initially it was 2). So in the new optimal bundle we consume 60 fine rice. We know that the price effect is the change in consumption due to change in price. So here price effect PE= 60-100= -40.: Suppose the price of fine rice increases. How does a rational consumer react?
He will try to substitute fine rice with its alternative, like coarse rice. That is he will decrease fine rice consumption and increase coarse rice consumption. So we can write

Price of Fine Rice↑ => Quantity of Fine Rice ↓

So there is negative relation between price and quantity which implies that substitution effect is always negative.

- Income Effect can be either positive or negative: optimal bundle is X=100 and Y=100. Now suppose the price of X ( Fine Rice) increases to 4 ( initially it was 2). So in the new optimal bundle we consume 60 fine rice. We know that the price effect is the change in consumption due to change in price. So here price effect PE= 60-100= -40.
- Negative Income Effect: In our example price of fine rice increases which leads to a decrease in purchasing power. That means our real income has decreased. So we will buy less fine rice now. So we can write
- Price of fine rice ↑ => Purchasing power ↓ => Quantity of fine rice ↓
- So here price increase leads to an decrease in purchasing power/ real income and so quantity has decreased. That’s why income effect is negative ( because price and quantity moves in the opposite direction)

- Positive Income Effect: optimal bundle is X=100 and Y=100. Now suppose the price of X ( Fine Rice) increases to 4 ( initially it was 2). So in the new optimal bundle we consume 60 fine rice. We know that the price effect is the change in consumption due to change in price. So here price effect PE= 60-100= -40.Sometimes income effect will be positive. Suppose price of coarse rice has decreased significantly. In this case purchasing power/ real income will go up. Now as real income is increased we can buy more rice. But as coarse rice is a bad ( inferior) quality rice so we will want to buy more good quality rice like fine rice. In such case even if real income has gone up we do not increase the consumption of coarse rice, instead we decrease the consumption of coarse rice and increase consumption of fine rice. So we can write
- Price of coarse rice ↓=> Purchasing power ↑ => Quantity of fine rice ↓
- So here income effect is positive ( because price and quantity moves in the same direction)

- Types of good depending on income effect: optimal bundle is X=100 and Y=100. Now suppose the price of X ( Fine Rice) increases to 4 ( initially it was 2). So in the new optimal bundle we consume 60 fine rice. We know that the price effect is the change in consumption due to change in price. So here price effect PE= 60-100= -40.
- Normal Good
- Inferior Good
Normal Good: If the consumption demand of a good increases when the income increases and vice- versa, then the good is called normal good. For normal good income effect is negative. Example: Fine rice. Fine rice is a good quality rice and when income increases we want to increase its consumption

Inferior Good: If the consumption demand of a good decreases when the income increases and vice- versa, then the good is called inferior good. For inferior good income effect is positive. Example: Coarse rice. Coarse rice is a bad quality rice and when income increases we want to decrease its consumption because we replace it with good quality rice like fine rice.

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