Question

An ice-cream lover has a total of $30 to spend one evening. The price of ice-cream is $p per litre. The person's preferences for buying a litres of ice-cream, leaving a nonnegative amount $(30-pg) to spend on other items, are represented by the utility function: U(q) = va+ 30-1 (a) Find the first-order condition for a utility maximizing quantity of ice-cream. (Hint: the square root function is a power function with exponent 1/2.) (b) Solve the first-order condition derived in (a) in order to express the utility maximizing quantity q* as a function of p. (c) What guarantees that your quantity q' is really a maximum? (d) Express the elasticity of demand for ice-cream as a function of the price $p per litre. When the price is $3 per litre, calculate the price elasticity of the person's demand for ice-cream.

Answer 1

a) First order says, necessary condition for maximum of function is where first deriv =0

dU/dq=0

1/(2*sqrt(q))-p/(2*sqrt(30-pq))=0

1/q=p^2/(30-pq)

30=pq+p^2

q=30/(p^2+p)

b)

Now, putting value q=30/(4*p^2+p) in eq1 will give max

So,

q*=sqrt(30/(p^2+p))+sqrt(30-(30/(p+1)))=sqrt(30/(p^2+p))+sqrt((30p/(p+1)))

c)

Second derivative d^2U/dQ^2 says that it is less than 0. So, it is maximum

d)

E=(p/q)*dq/dp

E=-(p/(30/(4*p^2+p)))*(30*(2*p+1)/(p^2+p)^2)=-91/16