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Over the past decade, medical costs have increased more rapidly than other prices. In order to illustrate how rising medical costs have affected consumer alternatives, let X represent the quantity of medical services, and let Y represent the quantity of other goods. Furthermore, let income (M) be measured in hundreds of dollars, the price of medical services and other goods in terms of dollars per minute, with M = 100, Px = 4, and Py = 5.

a) Graph the budget line, and determine the market rate of substitution.

b) Illustrate the budget set.

c) Show in your graph what happens to the budget constraint if Px increases to $10.

d) What is the meaning of the slope of the two budget constraints?

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Answer:

Part a: The graph is as attached in the figure.

Part b: The budget sets are

[tex]S_1= (0,20)\\S_2= (5,16)\\S_3= (10,12)\\S_4= (15,8)\\S_5= (20,4)\\S_6= (25,0)[/tex]

Part c: The graph is as shown in the figure

Part d: For the 1st constraint, a 1 unit change in X, there is a change of Y is 0.8 unit.

For the 2nd  constraint, a 1 unit change in X, there is a change of Y is 2 units.

Explanation:

Part a

As from the data

M=100

Px=4

Py=5

X=quantity of medical services

Y=quantity of other goods this the equation is given as

M=PxX +PyY

100=4X+5Y

The graph is as attached in the figure.

Part b:

For the budget sets

[tex]x,\, y=\frac{100-4x}{5}\\x=0,\, y=\frac{100-0}{5}=y=\frac{100}{5}=y=20, S_1= (0,20)\\x=5,\, y=\frac{100-20}{5}=y=\frac{80}{5}=y=16, S_2= (5,16)\\x=10,\, y=\frac{100-40}{5}=y=\frac{60}{5}=y=12, S_3= (10,12)\\x=15,\, y=\frac{100-60}{5}=y=\frac{40}{5}=y=8, S_4= (15,8)\\x=20,\, y=\frac{100-80}{5}=y=\frac{80}{5}=y=4, S_5= (20,4)\\x=25,\, y=\frac{100-100}{5}=y=\frac{0}{5}=y=0, S_6= (25,0)\\[/tex]

So the budget sets are

[tex]S_1= (0,20)\\S_2= (5,16)\\S_3= (10,12)\\S_4= (15,8)\\S_5= (20,4)\\S_6= (25,0)[/tex]

Part c

If Px is 10 so the equation now becomes

M=PxX +PyY

100=10X+5Y

Its graph is given on the same graph to indicate the difference.

Part d:

The slope of 1st constraint is given as

[tex]MSR_1=\frac{P_x}{P_y}\\MSR_1=\frac{4}{5}=0.8[/tex]

The slope of 2nd constraint is given as

[tex]MSR_2=\frac{P_x}{P_y}\\MSR_2=\frac{10}{5}=2[/tex]

For the 1st constraint, a 1 unit change in X, there is a change of Y is 0.8 unit.

For the 2nd  constraint, a 1 unit change in X, there is a change of Y is 2 units.

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