Question

The demand function for a particular good in competitive market is given by y = a + bP. What are the associated direct and indirect utility functions?

Answer 1

For the given function of demand  $y=a+bp$

 The function for "indirect utility" can be written as 

$V(y,p,m)=max u(y,z)$

where, 

z=other goods

m=income/wealth

Indirect utility function$=−\frac{\frac{∂V}{∂p}}{\frac{∂V}{∂m}}$

Indirect utility function$=y(p,m)$

Thus, the consumer can only get as much utility as limited to the price and also to their income/wealth. 

Now, for the utility function of "direct" nature, 

$U(y,z)=Min V(y,p,m)$