Hi.. :)
27) Its given that Agent 1 is a first mover. So, just maximise his utility subject to the constraint given. You'll get his demand. Now, whatever left in the economy will be consumed by Agent 2 as his utility is increasing in both x and y. So, no matter what he would like to consume whatever is available.
31) None of the four options is correct.
Given: P_e = P
Solve the two equations, you'll get μ=λ/α(1+λ).
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