a. The Lagrangian for the Randolph's utility maximization problem is:
L(h,t,r,λ) = 20h^(0.1)t^(0.3)r^(0.4) + λ(800 - 10h - 25t - 50r)
Where λ is the Lagrange multiplier for the budget constraint.
b. To find the numbers of home-cooked, take-out, and restaurant dinners that the Randolphs should consume per month to maximize their prandial utility, we need to solve the following optimization problem:
max 20h^(0.1)t^(0.3)r^(0.4)
s.t. 10h + 25t + 50r <= 800
We can use the Lagrangian from part (a) to find the first-order conditions:
∂L/∂h = 2h^(-0.9)t^(0.3)r^(0.4) - 10λ = 0
∂L/∂t = 6h^(0.1)t^(-0.7)r^(0.4) - 25λ = 0
∂L/∂r = 8h^(0.1)t^(0.3)r^(-0.6) - 50λ = 0
∂L/∂λ = 800 - 10h - 25t - 50r = 0
Solving these equations simultaneously, we get:
h = 3.16, t = 6.32, r = 4.00
Therefore, the Randolphs should consume approximately 3 home-cooked dinners, 6 take-out dinners, and 4 restaurant dinners per month to maximize their prandial utility. Their maximum utility is:
U(h,t,r) = 20(3.16)^(0.1)(6.32)^(0.3)(4.00)^(0.4) ≈ 41.63
c. To find how much the Randolphs need to increase their monthly dinner budget to increase their prandial utility by 1 util, we can use the marginal utility of income:
MU(I) = ∂U/∂I = 2h^(0.1)t^(0.3)r^(0.4)/(10h + 25t + 50r)
where I is the monthly dinner budget and ∂U/∂I is the marginal utility of