Dixit Exercise 3.2: Distribution Between Envious Consumers

There is a fixed totalYof goods at the disposal of society. There are two consumers who envy each other. If consumer 1 getsY1and consumer 2 getsY₂, their utilities are

U₁=Y₁?kY₂², U₂=Y₂?kY₁²

wherekis a positive constant. The allocation must satisfyY₁+Y₂?Y, and maximizeU₁+U₂.

Show that ifY >1/k, the resource constraint will be slack at the optimum. Interpret the result.

Exercise 3.2: Distribution Between Envious Consumers There is a fixed total Y of goods at the disposal of society. There are two consumers who envy each other. If consumer 1 gets Y, and consumer 2 gets Y2, their utilities are U1 = Y1 - KY2, U2 = Y2 - KY? where k is a positive constant. The allocation must satisfy Y1 + Y2 1/k, the resource constraint will be slack at the optimum. Interpret the result