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PEAMCQEasy-Moderate

2024 PEA Q28

AA has a pond which yields ff fish at t=1t=1 and nothing at t=2t=2. He consumes only fish. Storage is costless and undepreciated; no discounting. Utility in any period tt is u(ct)=ct nu(c_t)=c_t^{\,n} with 0<n<10<n<1. Optimal consumption?

Reveal answer and solution

Answer

B

Solution

  1. 1

    The problem is

  2. 2
    max⁡c1,c2≥0 c1 n+c2 ns.t.c1+c2=f. \max_{c_1,c_2\ge 0}\ c_1^{\,n}+c_2^{\,n}\quad\text{s.t.}\quad c_1+c_2=f.
  3. 3

    Substituting c2=f−c1c_2=f-c_1 and differentiating with respect to c1c_1:

  4. 4
    nc1 n−1−n(f−c1)n−1=0 ⟹ c1 n−1=(f−c1)n−1 ⟹ c1=f−c1 ⟹ c1=c2=f2. nc_1^{\,n-1}-n(f-c_1)^{n-1}=0\ \Longrightarrow\ c_1^{\,n-1}=(f-c_1)^{n-1}\ \Longrightarrow\ c_1=f-c_1\ \Longrightarrow\ c_1=c_2=\tfrac{f}{2}.
  5. 5

    The second-order condition holds because u(c)=cnu(c)=c^{n} with 0<n<10<n<1 is strictly concave, so the agent strictly prefers a smooth path.

Answer structure / marking notes

Concave utility + no discount ⇒\Rightarrow perfect consumption smoothing.

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Content note

Imported from public/resources/isi/msqe/solutions/pea/2024/ISI_MSQE_PEA_2024_Solutions.tex. Question wording is retained from the available local TeX source; incomplete option blocks or ambiguous source status are flagged for review.