Question

Question 2 Suppose the demand for DVD players (good X) is given by * = 1200 - 22+ 3P– 82, +16M Are goods Y and Z substitutes or complements of good X? Is good X an inferior or a normal good? What is the quantity demanded of good X, if research shows that P,= 500, P,= 400, P=10, M=10,000? Determine the demand function and the inverse demand function for good X. Draw the demand curve. If the price level is $3440, what is the consumer surplus?

Answer 1

Answer

The demand for DVD players is given by;

Q_x^d = 1200 - (1/2)P_x + (1/4)P_y - 8P_z + (1/10)M......(1)

Let us see the relationship between good X and good Y.

Differentiating equation(1) with respect to P_y, keeping other variables, i.e., P_x, P_z, and M constant.

d(Q_x^d) / d( P_y) = 1/4

From the above value, we see that the demand for good X, and the price of good Y are directly related, i.e., if the price of good Y rises, then the demand for good X rises.Thus good Y is a substitute of good X.

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Now, let us see the relationship between good X and good Z.

Differentiating equation(1) with respect to P_z, keeping other variables, i.e., P_x, P_y, and M constant.

d(Q_x^d) / d( P_z) = -8

From the above value, we see that the demand for good X, and the price of good Z are inversely related, i.e., if the price of good Z rises, then the demand for good X falls.Thus good Z is a complement of good X.

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Let us now see the relationship between good X and income M.

Differentiating equation(1) with respect to M, keeping other variables, i.e., P_x, P_y, and P_z constant.

d(Q_x^d) / dM = 1/10

From the above value, we see that the demand for good X, and income M are directly related, i.e., if the income rises, then the demand for good X also rises.Thus good X is a normal good.

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Now, let us see what will be the quantity demanded of good X, if

P_x = 500

P_y = 400

P_z = 10

M = 10,000

Putting the value of all the above variables in equation(1), we get,

Q_x^d = 1200 - (1/2)*500 + (1/4)*400 - 8*10 + (1/10)* 10,000

Or, Q_x^d = 1200 - 250 + 100 - 80 + 1,000

Or, Q_x^d = 1200 + 100 + 1,000 - 250 - 80

Or, Q_x^d = (1200 + 100 + 1,000) - ( 250 + 80)

Or, Q_x^d = 2300 - 330

Or, Q_x^d = 1970

Thus quantity demanded of good X is 1970

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The demand function for a product shows the relationship between the quantity demanded of the product and its price.

Thus the demand function for good X is;

Q_x^d = 1200 - (1/2)P_x , where 'Q_x^d' is the demand for good X, and 'P_x' is the price of good X.

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We can write the above demand function for good X as;

1200 - (1/2)P_x = Q_x^d

Or, - (1/2)P_x = Q_x^d - 1200

Or, (1/2)P_x = 1200 -  Q_x^d

Or, P_x = 2400 - 2Q_x^d

Thus the inverse demand function for good X is;

P_x = 2400 - 2Q_x^d

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We get the demand function for good X as;

Q_x^d = 1200 - (1/2)P_x

Price(Px)	Quantity demanded for good X(Qxd)
0	1200
400	1000
800	800
1200	600
1600	400
2000	200
2400	0

The demand curve for good X is shown below;

Demand Curve for Good X 2600 24000,2400 2200 Price 200, 2000 1500 400, 1600 1400 1200 600, 1200 1000 800, 800 1000,400 1200,0 0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 Quantity of goodx

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From the above figure, we see that the maximum price, the consumer is willing to pay is $2,400. Thus when price is $3,440, there will be no consumer in the market. Thus, we can not consider about consumer surplus.

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