Show that if utility is quadratic, ∂p/∂λ = 0.
The equity premium and the concentration of aggregate shocks. (Mankiw, 1986.) Consider an economy with two possible states, each of which occurs with probability 1 2 . In the good state, everyone’s consumption is 1. In the bad state, fraction λ of the population consumes 1 − (φ/λ) and the remainder consumes 1, where 0 < φ=””>< 1=”” and=”” φ=”” ≤=”” λ=”” ≤=”” 1.=”” φ=”” measures=”” the=”” reduction=”” in=”” average=”” consumption=”” in=”” the=”” bad=”” state,=”” and=”” λ=”” measures=”” how=”” broadly=”” that=”” reduction=”” is=””> Consider two assets, one that pays off 1 unit in the good state and one that pays off 1 unit in the bad state. Let p denote the relative price of the bad-state asset to the good-state asset. (a) Consider an individual whose initial holdings of the two assets are zero, and consider the experiment of the individual marginally reducing (that is, selling short) his or her holdings of the good-state asset and using the proceeds to purchase more of the bad-state asset. Derive the condition for this change not to affect the individual’s expected utility. (b) Since consumption in the two states is exogenous and individuals are ex ante identical, p must adjust to the point where it is an equilibrium for individuals’ holdings of both assets to be zero. Solve the condition derived in part (a) for this equilibrium value of p in terms of φ, λ, U (1), and U (1 − (φ/λ)). (c) Find ∂p/∂λ. (d) Show that if utility is quadratic, ∂p/∂λ = 0. (e) Show that if U (•) is everywhere positive, ∂p/∂λ < 0.=””>
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