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.



(ii) Find the coordinates of the equilibrium point.



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.



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



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