Question

Palpa's Company manufactures bookbags, when the bookbags price is $75 the demand is 25; when the price is $20 the demand is 100 bookbags. But when the price is $50 the supply is 100 and for $25 the supply is 20.a. What is the demand function? b. What is the supply function c. What are the equilibrium price and quantity?

Answer 1

Summarizing the information:

Price: $75 => Demand: 25

Price: $20 => Demand: 100

Price: $50 => Supply: 100

Price: $25 => Supply: 20

a.

To calculate the demand function, we identify at least two points of the demand line. Using the information given by the problem, these two points are:

`\begin{gathered} D_1=(75,25) \\ D_2=(20,100) \end{gathered}`

Given the slope formula:

`m=(y_2-y_1)/(x_2-x_1)`

Where:

`\begin{gathered} (x_1,y_1)=(75,25)_{} \\ (x_2,y_2)=(20,100) \end{gathered}`

Then:

`\begin{gathered} m=(100-25)/(20-75)=(75)/(-55) \\ \Rightarrow m=-(15)/(11) \end{gathered}`

Now, by the slope-intercept form of the line equation (and using the point D₁):

`\begin{gathered} y-25=-(15)/(11)(x-75) \\ y=-(15)/(11)x+(1125)/(11)+25 \\ \Rightarrow f(x)=-(15)/(11)x+(1400)/(11) \end{gathered}`

And that is the demand function.

b.

We identify two points of the supply line:

`\begin{gathered} S_1=(50,100) \\ S_2=(25,20) \end{gathered}`

The slope of the line is:

`m=(20-100)/(25-50)=(-80)/(-25)=(16)/(5)`

Now, by the slope-point form of the line (using S₁):

`\begin{gathered} y-100=(16)/(5)(x-50) \\ y=(16)/(5)x-160+100 \\ \Rightarrow g(x)=(16)/(5)x-60 \end{gathered}`

Where g(x) is the supply function.

c.

To find the equilibrium point, we solve the equation f(x) = g(x):

`\begin{gathered} -(15)/(11)x+(1400)/(11)=(16)/(5)x-60 \\ (1400)/(11)+60=(16)/(5)x+(15)/(11)x \\ (2060)/(11)=(251)/(55)x \\ \Rightarrow x=(10300)/(251)\approx\text{ \$}41.04 \end{gathered}`

Now, we find the corresponding quantity:

`\begin{gathered} f((10300)/(251))=-(15)/(11)x+(1400)/(11) \\ \Rightarrow f((10300)/(251))\approx71 \end{gathered}`

The equilibrium point is:

`(\text{\$}41.04,71)`