Question

6. A person consumes two goods, bacon (B) and tofu (T). They get utility from consumption in the following way U(B,T)-4B2T. The person has $50 to spend on the two goods and plans to spend all $50. Bacon costs $1 and tofu costs $2. Set up your objective function of the form (YOU DO NOT HAVE TO SOLVE, JUST SET IT UP): Max U such that budget constraint is satisfied.

Answer 1

6. The utility function is given as $\dpi{80} U = 4B^2 T$ . The income is $50, and price of B is $1 and T is $2. The budget constraint would be $\dpi{80} 2T + B = 50$ . The maximization problem (in terms of objective function and constraint) would be as below.

$\dpi{80} Max \; \; U = 4B^2 T$

such that $\dpi{80} 2T + B = 50$ .

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Note : The solution of the problem can be done via Lagrange's multiplier method as below.

The Lagrangian function of the maximization problem is $\dpi{80} L = 4B^2 T + \lambda (50 - 2T - B)$ . The FOCs are as below.

$\dpi{80} \frac{\partial L}{\partial B} = 0$ or $\dpi{80} \frac{\partial }{\partial B}(4B^2 T + \lambda (50 - 2T - B)) = 0$ or $\dpi{80} 8BT + \lambda (- 1) = 0$ or $\dpi{80} 8BT = \lambda$ .

$\dpi{80} \frac{\partial L}{\partial T} = 0$ or $\dpi{80} \frac{\partial }{\partial T}(4B^2 T + \lambda (50 - 2T - B)) = 0$ or $\dpi{80} 4B^2 + \lambda (- 2) = 0$ or $\dpi{80} 4B^2 = 2\lambda$ or $\dpi{80} 2B^2 = \lambda$ .

$\dpi{80} \frac{\partial L}{\partial \lambda} = 0$ or $\dpi{80} \frac{\partial }{\partial \lambda}(4B^2 T + \lambda (50 - 2T - B)) = 0$ or $\dpi{80} 2T + B = 50$ .

Solving for the first two FOCs, we have $\dpi{80} 8BT = 2B^2$ or $\dpi{80} 4T = B$ , which is the utility maximizing combination of products.

Putting this in the budget constraint, we have $\dpi{80} 2T + (4T) = 50$ or $\dpi{80} T^* = 25/3$ , and as $\dpi{80} B^* = 4T^*$ , we have $\dpi{80} B^* = 100/3$ . These are the optimal combination of goods which would maximize the utility given the budget constraint.