Consider a two-period economy where individuals are endowed with a bundle of horned melons W in
period 0. Horned melons are extremely nutritious, but they depreciate quickly and will rot between
periods 0 and 1. Fortunately, individuals have access to both a production technology and a saving
technology. In period 0 households may choose to consume their wealth (Co), plant it (I) in order to
grow more horned melons in period 1, or sell it at a price of 1 in financial markets in order to buy bonds
(B > 0) or sell bonds (B < 0) that carry an interest rate of r.
Individuals have lifetime utility U(Co, C₁) = u(Co) + du(C₁), where per-period utility has the form
u(c) = c1-7/(1 - y) with y > 0, and 0 < 8 ≤ 1.
The production technology is K₁ = AIβ/β, 0 < β <1. (“Capital” is the quantity of horned melons
yielded in period 1. Individuals may eat their capital in this economy!)
Individuals must choose their consumption paths and quantities of investment and saving to satisfy two
"per-period" budget constraints:
Period 0: I + B + Co ≤ W
Period 1: C₁ ≤ K₁ + B(1+r)
The constrained optimization problem may be solved by setting up the Lagrangian:
L = U(Co, C₁) + λο [W - I - B - Co] + A₁ [K₁ + B(1+r) - C₁]
Here are the four first-order conditions necessary to maximize lifetime utility U:
$$\frac{\partial L}{\partial C_0} = u'(C_0^*) - \lambda_0 = 0$$
$$\frac{\partial L}{\partial C_1} = \delta u'(C_1^*) - \lambda_1 = 0$$
$$\frac{\partial L}{\partial I} = -\lambda_0 + \lambda_1 AI^{\beta-1} = 0$$
$$\frac{\partial L}{\partial B} = -\lambda_0 + \lambda_1 (1 + r) = 0$$