A consumer spends time t searching for a good, the price of which is p(t).

Assume the longer the search goes, the lower price the consumer would pay for the good. Furthermore, assume there are diminishing returns to the search since it is harder to find even lower prices as the search continues; that is: p?(t) > 0. Without search the consumer would pay the current going price of the good p 0 . While searching, the consumer loses income at a constant rate w .

i) Find the condition for an optimal search time t, if q number of units of the good are bought. Explain what your condition is saying in plain English.

ii) Check to make sure your second-order condition is satisfied.

iii) What is the impact of a change in the consumer’s wage rate w on the optimal search time if nothing else changes? Explain why your answer makes sense.

iv) What is the impact of a change in the number of units’ q on the optimal search time if nothing else changes? Again, provide the necessary intuition.