Consider three consumers with demand functions:

Consumer 1: q = 10 p

Consumer 2: q = (5 p )/2

Consumer 3: q = 10 2p

1. Draw these three demand curves in a single graph and label each curve.

2. For each consumer, give the expression for the inverse demand function, p as a function of q. Using these expressions, or your earlier graph, which consumer has the highest Marginal Value for any q between 0 and 10? (This property will prove useful in part c below.)

3. Assume that output is perfectly divisible (you can allocate it in any increment, not just full units). Aggregate Total Value (TV) is the sum of the total values across all 3 consumers:

TV(q) = TV1 (q1 ) + TV2 (q2 ) + TV3 (q3 )

where TVi is the Total Value function for person i, and q i is the quantity allocated to person i. Suppose you have 3 units (q = 3) to allocate among these consumers. How should you allocate this output among the consumers so as to maximize TV? Explain briefly. (Hint: Dont mechanically apply the equate-Marginal-Values condition!)

4. Now suppose you have 10 units (q = 10). How should you allocate this output among the consumers so as to maximize TV? (Hint: Youll need to satisfy two conditions: q 1 + q2 + q3 = 10, and another relevant condition.)