5. Trivially,x∗is a feasible allocation with endowmentsw𝑖=x∗𝑖and𝑚𝑖=p∗w𝑖=p∗x∗𝑖. To show that (p∗,x∗) is a competitive equilibrium, we have to show thatx∗𝑖is the best allocation in the budget set𝑋𝑖(p, 𝑚𝑖) for each consumer𝑖. Supposeto the contrary there exists some consumer𝑗and allocationy𝑗such thaty𝑗≻x𝑗andpy𝑗≤𝑚𝑗=px∗𝑗.By continuity, there exists some𝛼∈(0,1) such that𝛼y𝑗≻𝑖x∗𝑗andp(𝛼y𝑗) =𝛼py𝑗<py𝑗≤px∗contradicting (3.66). We conclude thatx∗𝑖≿𝑖x𝑖for everyx∈𝑋(p∗, 𝑚𝑖)for every consumer𝑖. (p∗,x∗) is a competitive equilibrium.3.229By the previous exercise, there exists a price systemp∗such thatx∗𝑖is optimalfor each consumer𝑖in the budget set𝑋(p∗,p∗x∗𝑖), that isx∗𝑖≿𝑖x𝑖for everyx𝑖∈𝑋(p∗,p∗x∗𝑖)(3.67)
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