Question

1. Consider the following demand function q1 = 10 - .1p₁ + .2p₂ + .10Y
   1. Graph the demand function for q1.  What happens to the demand for q1 when p2 increases? When Y increases?
   2. What is price elasticity of demand (in general and for this demand function)? What is price elasticity of demand when p1 = 2, p2 = 10, and Y =100. What does that mean?  Is the demand for  elastic, inelastic, or unit elastic with respect to price?
   3. What is income elasticity of demand (in general and for this demand function)? When p1 = 2, p2 = 10, and Y = 100. What type of good is q1, respect to income?
   4. Calculate consumer surplus when p1 = 2, p2 = 10, and Y =100.  Now, suppose p1 increases to 50. What happens to consumer surplus?

Answer 1

Solution:

Given,

The demand funcition is q₁ = 10 - .1p₁ + .2p₂ + .10Y

Here, the demand function for good 1 shows the linear relationship among the quantity demanded of good 1, q₁ and p₁, p₂ and Y

where, p₁: The price of good 1

q₁ : The quantity demanded of good 1

p₂: The price of good 2

Y: The income of the consumer

The following graph shows the plot of demand curve for q₁, assuming ceteris paribus. DD is the demand curve.

When p2 increases, the demand for q1 increases. This is because good 1and good 2 are substitute goods and this claim is based on the observation from the demand curve that the quantity demanded, q₁ is positively related to p₂. So, when p₂ rises, q₁ will also increase. This happens only in the case of substitute goods.

The above graph was plotted based on the assumption of ceteris paribus, that is, the price of related goods, income of the consumer, taste and preferences remain constant but there p₂ increases. So the demand curve dor good 1 will shift upward from DD to D'D'. This is because with the increase in the price of the substitute good, good 1 will now be attractive for the consumers, so at all price levels the quantity demanded, q₁ will rise, leading to a upward shift in the demand curve.

Similarly, when Y increases, the demand curve for q₁ will shift upwards from DD to D"D". This is because an increase in the income of the consumer will mean an increase in the purchasing power of the consumers. So they will now tend to spend more on consumption of good 1 and will increase the quantity demanded at each price level. This can also be verified from the demand function as q₁ and Y are positively related. Thus, an increase in Y will lead to a rise in q_1.

The following graph shows the shift of demand curve DD to D'D' in case of increase in p₂ and to D"D" in case of increase in Y.

Price elasticity of demand is the degree of responsiveness of the quantity demanded of a commodity to the change in its price. Price elasticity always attains a negative value for normal goods because as per the Law of Demand, a rise in the price of a commodity leads to a fall in its quantity demanded. For this demand function, the price elasticity is the percentage change in q₁ as a response to the change in p₁.

Price elasticity of demand is calculated as follows:

Price elasticity of demand =

=

=

When p₁ = 2, p₂ = 10, and Y =100, then from the demand equation

q₁ = 10 - .1p₁ + .2p₂ + .10Y

=> q₁ = 10 - (0.1x2) +( 0.2x10) + (0.10x100)

=> q₁ = 10 - 0.2 +2 + 10

=> q₁ = 21.8

Therefore, price elasticity of demand =

=

= -0.00917

The demand for good1 is price inelastic rather it is almost inelastic as the price elasticity of demand is very small in magnitude along with being less than one and the good 1 is a normal good as the elasticity attains a negative value. The good1 is price inelastic whih means that for a large change in price, the quantity demanded wil change by a negligible amount.

Income elasticity of is the degree of responsiveness of the quantity demanded of a commodity to the change in the income of the consumer. Income elasticity of demand always attains a positive value for normal goods because as the income rises, the consumers tend to increase their consumption by demanding more goods. Here, the good is income inelastic, so even if the income changes by a large value, the change in the quantity demanded will be small.

Income elasticity of demand =

=

=

When p₁ = 2, p₂ = 10, q₁ = 21.8 and Y =100,

the income elasticity of demand =

   = 0.46

The demand for good1 is income inelastic as it i as the income elasticity of demand is less than one and the good 1 is a normal good as the elasticity attains a positive value.

The consumer surplus is defined as the difference between the consumer's willingness to pay for a commodity and the actual price paid by them, or the equilibrium price. It is the area of the triangle is the area between the demand curve, price, and the y-axis. So when p1 = 2, p2 = 10, and Y =100, we can calculate the consumer surplus in the following way,

Consumer Surplus = (1/2)x Base x Height

= (1/2)x Price x Quantity

= (1/2)x 2 x 21.8

= 21.8

Now, supposing that p₁ increases to 50 then the q₁ = 10 - .1p₁ + .2p₂ + .10Y

=> q₁ = 10 - (0.1x50) +( 0.2x10) + (0.10x100)

=> q₁ = 10 - 5 +2 + 10

=> q₁ = 17

Therefore, New Consumer Surplus = (1/2)x Base x Height

= (1/2)x ( New Price - Old price) x Quantity

= (1/2)x (50-2) x 17

= 408

Theoretically, for an increase in the price of a commodity, the consumer surplus falls but here the opposite has happned because the good is price inelastic. So even though the price has changed by 48 units but the quantity demanded has changed by only 3.8 units. So instead of falling the consumer surplus has increased in this case.