Question

Suppose that there are two goods, X and Y. The utility function is ?(?, ?) = 5?² ?. The price of Y is $2 per unit, and the price of X is P. Income is $2,400.

A.) Derive the demand curve and state the law of demand in relation to your product or service.

B.) ? = 800 − 10?^0.5 . Calculate elasticity of demand when Q=100, Is the good elastic?

C.) ? = 800 − 10?^0.5 − 0.5?. Calculate elasticity of income when Q=4 and I=100. Is the good normal or inferior?

D.) ?_A = 80 − 10?^0.5 − ?B Calculate cross price elasticity when Q= 16 and ?B =10. Are the goods complementary or are they substitutes?

Answer 1

A.) Demand function and curve of good X:

Given:

Utility function: U(X,Y) = 5X²Y

Price of good X= P

Price of good Y = 2

Income (I) = 2400

Therefore, budget constraint:

where X and Y are the quantities of good X and good Y, putting the values in the formula for budget constraint.

Now the demand is derived by setting the consumption maximising problem,

Consumer wants to maximise his utility given the budget constraint.

Therefore he will demand the goods where the marginal rate of substitution equals the ratio of prices of two goods or the ratio of marginal utilities of two goods is equal to the ratio of prices.

Finding the marginal utilites of both the goods.

Now the marginal utility of good Y,

Therefore the consumer maximize his utility where:

Putting the values of marginal utilities and prices,

Now, this is the maximizing condition, putting this condition into the budget constraint we will get the demand curve for good X.

Therefore this is the demand function for good X.

The law of demand states that there is a negative or inverse relation between price of a good and the quantity demanded of the good, ceterius peribus. Becuase of this inverse relation, the demand curve slope downward.

Here, we can see that as the price will increase, the quantity demanded of good X will decrease and vice versa. Therefore this demand function for good X draws an inverse relationship between the price of good X and quantity demanded of it. Therefore it satisfied the law of demand.

This is the demand curve that slopes downward as the law of demand suggests.

B.) Finding Price elasticity of demand:

Given:

Q = 100

Therefore putting this quantity into the inverse demand function, we get price at his quantity.

Price elasticity of demand is the responsiveness of the quantity demanded as the price of the good changes.

Since, slope of the demand function is the derivative of the price with respective to quantity.

Therefore, putting Q = 100

Therefore putting these values into the formula of price elasticity of demand;

Slope of the demand function = -1/2

Price (P) = 700

quantity demanded (Q)= 100

Therefore the price elasticity of demand is -14

As the price elasticity of demand's modulus is greater than 1, therefore the demand for this good is elasic. That is, |PED|>1, therefore the demand is elastic.  

C.) Given:

Where,

I is the income = 100

Q is the quantity demanded = 4

Income elasticity of demand:

Income elasticity of demand is the responsiveness of the quantity demanded as the income of the consumer changes.

First of all rewriting it as a function of quantity;

Now income elasticity of income is,

Therefore,

Putting I = 100

Now plugging these values into the income elasticity of demand formula.

Therefore income elasticity of demand is -2.4, The minus sign shows that there is a negative relationship between income and quantity demanded. And therefore this good is an inferior good. As only inferior good have a negative relation between income and quantity demanded. As income increases the quantity demanded would decrease and vice versa. The good is an inferior good.

The income elasticity is elastic as the modulus of income elasticity is greater than 1.

D.) Finding cross elasticity of demand:

Given:

where P_B is the related good's price.

And P_A is the price of the own good.

Price of related good P_B = 10

Quantity demanded Q = 16

Cross-price elasticity of demand is the responsiveness of quantity demanded as the price of related good changes.

Therefore,

rewriting the given function in the form of Q,

Now,

Now plugging these values into the cross price elasticity of demand formula,

Therefore the cross price elasticity of demand for good A is -0.0625. The negative sign shows that there is a negative relationship between price of good B and the quantity demanded of good A. And as we know that only complementary goods have a negative relations. Therefore good A and good B are complementary goods.

As the the modulus of cross-price elasticity of demand is less than 1, (i.e. |cross PED|<1) therefore is shows the cross price elasticity is inelastic, And the changes in price of good B doesn't affect the quantity demanded of good A to a greater degree.