Question

1. You're assembling the perfect trail mix and need to determine the optimal number of nuts to add. You have the option between Almonds, Brazil Nuts, and Cashews, which you buy in quantities A B. C. at prices PA, pB, and po, respectively. Your nut budget is M dollars. You really like Brazil nuts, so your happiness (utility) with the mix is dictated by UA. B. C) = AB2C. Find the nut mixture which optimizes your utility subject to your budget constraints. 2. (Question 1 Continued) Suppose that your budget is allowed to change, so your solution to the above problem depends upon M; namely, you have three functions (A(M), B(M), C(M)) Recall that the marginal utility of income is the rate of change of your ty with respect to your budget. Show that the marginal utility of income is λ. Hint: This is always true, so you can either show this is true for the example above, or you can do this abstractly

Answer 1

1. Prices of Goods like Almonds, Brazil Nuts and Cashews are given as p_A, p_B and p_C.

Budget of Pricing = M dollars. Let us assume I = Income. I is the cost of consumption. Actually it will restrict the customers to handle optimal utilization of consumption of all goods.

It is given that f(A,B,C) = AB²C. As B² indicates the most prefered goods liked by the particular customer.

So in order to have the utility for all the products in one single shell, We have to frame the Budget Constraint as,

P(A) + P(B²) + P(C) $\leq$ I (income). If the customer need to mix all the preferenes optimally then the situation will be as

U (P(B²)) = ((P(A) + P(C))²/I*M. Preference of P(A) + P(C) will be divided with income and again multiplying with M (actual budget) will give equal Optimal Mix.

2. Marginal Utility of income refers to the additonal benefit of procuring the particular goods with quite preference.

   In this scenario, the three functions A(M), B(M) and C(M) are depicted here in order to show the marginal utility of income as $\lambda$. We can Evidently show the instance by grouping all the functions of preferences in the single order of Utility.

U(I) refers to Utility of Income which has to spended for goods and services.

i.e. U(I) = A(M) + B(M) + C(M)/$\lambda$*M, where M separately indicate the price preference for all the prices separately.

A(M) $\leq$ B(M)$\leq$C(M)/U(I) will clearly give the picture of Marginal Utility of Income ($\lambda$) .