Question

Consider the following demand and supply functionsDemand: D(p) = q = 164 - 17p and Supply: S(p) = 9 = 32 + 10pa.) Assume there are no taxes imposed. Find the equilibrium price and quantity.Equilibrium Price: ___ (Round your answer to the nearest cent.)Equilibrium Quantity:___ (Round your answer to the nearest whole number.)b.) Assume there is a 18% tax on the consumer, find the new equilibrium price and quantity.New Equilibrium Price:$__ (Round your answer to the nearest cent.)New Equilibrium Quantity:__(Round your answer to the nearest whole number.)

Answer 1

A) Equilibrium quantity: 5

Equilibrium price: $72

B) Equilibrium quantity: 6

Equilibrium price: $92

Explanation:

PART A:

Remember that the equilibrium is when the demand is equal to the supply. To find the equilibrium quantity, we equal both functions and solve for p, as following:

`\begin{gathered} D(p)=S(p) \\ \rightarrow164-17p=32+10p \\ \rightarrow164-32=17p+10p \\ \rightarrow132=27p \\ \\ \rightarrow p=(132)/(27)\rightarrow p=(44)/(9) \\ \\ \Rightarrow p\approx4.89 \end{gathered}`

Since we have to round to the nearest whole number, we'll have that:

`p=5`

Now we know that the equilibrium quantity is 5, we can calculate the equilibrium price using this p value in one of the two equations. We'll use D(p) :

`\begin{gathered} D(5)=164-17(5) \\ \rightarrow D(5)=79 \end{gathered}`

Therefore, we can conclude that (for this situation) the equilibrium price is $79 and the equilibrium quantity is 5.

PART B:

We will apply the same reasoning, but this time we have to take into account the 18% tax on the demand function (customer). We'll do so as following:

`\begin{gathered} D(p)=164-17p \\ \rightarrow D_2(p)=164-17p+0.18D(p) \\ \rightarrow D_2(p)=164-17p+0.18(164-17p) \\ \rightarrow D_2(p)=164-17p+29.52-3.06p \\ \\ \Rightarrow D_2(p)=193.52-20.06p \end{gathered}`

What we just did was add 18% of the demand function to the original demand function, thus representing the 18% tax.

Now, we equal this new demand function to the supply function, solve for p and ceil to the nearest whole number, as following:

`\begin{gathered} S(p)=D_2(p) \\ \rightarrow32+10p=193.52-20.06p \\ \rightarrow10p+20.06p=193.52-32 \\ \rightarrow30.06p=161.52 \\ \\ \rightarrow p=(161.52)/(30.06)\rightarrow p\approx5.37 \\ \\ \Rightarrow p=6 \end{gathered}`

This way, we'll have that the new equilibrium quantity is 6. We can use this p-value in S(p) to find the new equilibrium price:

`\begin{gathered} S(6)=32+10(6) \\ \rightarrow S(6)=92 \end{gathered}`

Therefore, we can conclude that (for this situation) the equilibrium price is $92 and the equilibrium quantity is 6.