Question

Consider the market for sugary drinks. The demand and supply curves are given by:

?? = ? −???

?? = ? +???

(a) Find the equilibrium prices and quantities.

(b) Suppose that the government wants to lower consumption of sugary drinks by 50% and has considered setting a minimum price for sugary drinks. Find the minimum price ???? which should be set.

(c) Discuss whether a maximum price would achieve the same objective.

(d) Now suppose the government is considering implementing an ad valorem sugar tax instead. Assuming the values ? = 1 and ? = 0, find the tax rate ? which would reduce consumption of sugary drinks by 50%.

Answer 1

Answer a

At Equilibrium, QS=QD=Q* and pS=pD=p*

Hence a-bp* = c+dp*

or, dp* + bp* = a-c

or, p*(d+b)= (a-c)

or, p* = (a-c)/(d+b)

q* = a-b*(a-c)/(d+b)

=[a*(d+b)-b*(a-c)]/(d+b)

= (ad+ab-ab+bc)/(d+b)

q*=(ad+bc)/(d+b)

Hence Equilibrium Price = (a-c)/(d+b) and Equilibrium Quantity = (ad+bc)/(d+b)

Answer b

To reduce demand by 50% new quantity , QD = 50% of q* = 50%* (ad+bc)/(d+b)= (ad+bc)/(2(d+b))

Hence this QD = a-bPmin

or, (ad+bc)/(2(d+b)) = a-bPmin

or, bPmin = a- (ad+bc)/(2(d+b))

or, bPmin = [a*2*(d+b) - (ad+bc)]/2(d+b)

or, Pmin = (2ad+2ab -ad-bc)/(2bd+2b²)

or, Pmin = (2ab +ad-bc)/(2bd+2b²)

Answer c

Minimum Price is usually set higher than equilibrium hence it tends to reduce the demand as Sellers are not allowed to go below minimum price to reach equilibrium. A Maximum Price if set above the equilibrium price will have no effect as market is already operating at a lower price level. A Maximum Price if set below the equilibrium price will further increase the demand as sellers will not be allowed to increase the price to equilibrium level.

Answer d

QD = a-bpD after tax t, price will be pD*(1+t)

Hence QD = a-bpD(t+1)= 1-bpD(t+1) (a=1)

we know QD at pDt should be half of q* =  (ad+bc)/(2(d+b) )

if a=1, c=0, QD= (1*d+b*0)/(2(d+b)) = d/(2(d+b)

Hence d/(2(d+b) = 1-bpD (t+1)

Since Tax will be imposed at p* pD should be replaced by p* = (a-c)/(d+b) = (1-0)/(d+b)=1/(d+b)

Hence d/(2(d+b) = 1-b*(t+1)/(d+b)

or, d/(2(d+b) = (d+b-bt-b)/(d+b)

or, d/2 = d-bt

or, bt = d-d/2

or bt = d/2

or, t= d/2b

Hence tax rate shall be d/2b