# Consumer Theory and Budget Line

Week 6 Chapter 4: The theory of Individual Behavior

Question 1. Page 154 ( with some modifications)

A consumer has $300 to spend on goods X and Y. The market prices of these two goods are Px = $15 and Py = $5.

a. Draw the budget constraint. i.e provide a carefully labeled diagram b. What is the market rate of substitution? Give an interpretation. c. Illustrate the consumer’s opportunity set in part a) above. d. Show how the consumer’s opportunity set changes if income increases by $300. e. Does the increase of income by $300 in part d) above alter the market rate of substitution between goods X and Y? Answer:

a. The total budget is $300. Px and Py are used to stand for market prices of two goods x and y. Hence, 300=15X+5Y. Now if the entire budget money is spent on good X, then Y=0, X=20. Also, if the entire budget money is spent on good Y, then X=0, Y=60. So the maximum amount of good X is 20 and the maximum amount of good Y is 60. Steps:

1. 300=15X+5Y

2. Y=-3X+60, if X=0, then Y=60

3. X=-(1/3)Y+20, if Y=0, then X=20

4. So Points (0, 60) and (20, 0) are on the curve

Therefore, The budget constraint diagram can be drawn and is shown as follow:

b. The market rate of substitution is the rate at which good X (or good Y) may be traded for good Y (or good X) in the market. It is the slope of the budget line. So if we calculate the slope of the budget line above, we can get the market rate of substitution in this case. Steps:

1. The Slope of budget line= (M/Py)/(M/Px)= Px/Py

2. Px/Py=15/5=3

3. Thus, the market rate of substitution in this case is 3.

Interpretation: This means that in order to get one unit of good X, we have to give up three units of good Y. This reflects the meaning of substitution. c. The opportunity set (also called the budget set) is th bundles of goods a consumer can afford. This means that in this case consumer can buy different quantity combinations of good X and good Y within the limit of $300. So in the diagram the opportunity set is the shadow area below the budget line.

d. If income changes by $300, then the entire budget of the consumer goes to $600. Then the equal changes to 3X+Y=120. So the diagram may change to as follow:

The budget line shifts out parallel to its self. The area under the new budget line is greater than before as shown by the extra area in brown. e. No, it will not change the market rate of substitution because an increase in budget can only change income, it does not change the price of goods. So the slope of the budget line will not change. Thus, the market rate of substitution in this case will not change.

Question 3 (with some modifications to the question). On page 155 1. Bottom of Form

A consumer must spend $600 between the consumption of product X and Y. The relevant market prices are Px = $10 and Py = 40. a. Write the equation for the consumer’s budget line

b. Illustrate the consumer’s opportunity set in a carefully labeled diagram. c. Show how the consumer’s opportunity set changes when the price of good X increases to$20. d. Does the change in the price of good X in part c alter the market rate of substitution between goods X and Y?

Instructions: For each part of the above question, show your work. Graphs may be useful. Explain and show all steps of how you have reached your answer. Don’t simply state your answer. Answer:

a. Now we know that the entire amount of income is $600. And the price of good X is $10, the price of good Y is $40. So The equation for the consumer’s budget line is given as: 600=10X+40Y

Also, we can simplify the equation and then we get X+4Y=60

b. To show the opportunity set, we can simply maximum each amount of good X and Y as shown below. Steps:

1. X+4Y=60

2. X=-4Y+60, if Y=0, then X=60

3. Y=-(0.25)X+15, if X=0, then Y=15

4. So the points (0, 15) and (60, 0) are on the curve

Thus, we can draw the diagram of budget line as follow:

So the opportunity set in this case is the shadow area shown above....

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