NYJC/2021/JC2/Prelim/P1/Q11

In economics, a supply and demand chart is made up of two curves: the supply curve and demand curve. The supply curve is a function that shows how the price of a product, $P$, is related to the quantity, $q$, supplied during a given period of time. The demand curve is a function that shows how the price of the same product is related to the quantity demanded during a given period of time. Due to the nature of the curves, they will intersect at a point which is known as the equilibrium point.

For a particular product, the demand curve is given by the equation $D(q)=75-1.22^q$ where $0 \leq q \leq 21$ and the supply curve is given by the equation $S(q)=2(1.22)^q-1$ using the same domain.
(i) Sketch both curves on thle same diagram. Your sketch should indicate the axial intercepts of both curves.

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(ii) Find the coordinates of the equilibrium point.

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Let $p_e$ be the price of the product at equilibrium point. The area between the demand curve, the line $P=p_e$ and the line $q=0$ is defined as the consumer surplus. In a similar fashion, the area between the supply curve, the line $P=p_e$ and the line $q=0$ is defined as the producer surplus.
(iii) Without using a graphing calculator, determine the consumer surplus and producer surplus, leaving both answer to 2 decimal places.

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A global increase in production lead to a shift of the supply curve to the right. The equation of the new curve is given by $S(q-a)$ where $a$ is a positive constant. The price of the product at the new equilibrium point is denoted by $p_e^*$.

(iv) Show that $\frac{p_e^+1}{p_e+1}=\frac{3}{(1.22)^a+2}$.

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(v) For the case when $a=4$, find the increase in the consumer surplus, leaving your answer to the nearest whole number.

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