Question 5: consumer faces a utility function of the form: U-WC Where Wand Care the quantity of Wine (W) and Cheese (C) consumed. Given that the price of a unit of Wine is 3 per glass and the price of a unit of cheese is 6 perkg and the consumer has income of 3,000 to spend on both products. Required: a) Use the Lagrangian approach to find the optimum consumption bundle?ao marks)

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Question 5: consumer faces a utility function of the form: U-WC Where Wand Care the quantity of Wine (W) and Cheese (C) consumed. Given that the price of a unit of Wine is 3 per glass and the price of a unit of cheese is 6 perkg and the consumer has income of 3,000 to spend on both products. Required: a) Use the Lagrangian approach to find the optimum consumption bundle?ao marks)

Answer 1

The optimal consumption bundle (W*, C*) and the value of $\lambda$ can be found by solving the first-order conditions. The second-order conditions confirm whether these values indeed maximize the consumer's utility. Note that, to complete the math operations and specify the value results involved in this question which are not explicitly provided, such as the $\lambda$ and evaluating second derivatives against these targeted values of $W$ and $C$. And simultaneously solving the optimization problem formulated in the Lagrangian would use an algebraic computation approach.

Answer 2

## Step1: The given utility function is $U = WC$, where $W$ and $C$ are the quantities of wine and cheese consumed respectively. The prices of wine and cheese are $53 and $56 per unit respectively, and the consumer's budget is $3000. The consumer's budget constraint is therefore $53W + 56C = 3000$. ## Step2: We use the Lagrangian method to solve this constrained optimization problem. The Lagrangian function is given by $L(W, C, \lambda) = WC - \lambda(53W + 56C - 3000)$. ## Step3: The first-order conditions are obtained by taking the derivative of the Lagrangian function with respect to $W$, $C$, and $\lambda$, and setting each equal to zero. This gives us the following equations: ### $\frac{\partial L}{\partial W} = C - 53\lambda = 0$ ### $\frac{\partial L}{\partial C} = W - 56\lambda = 0$ ### $\frac{\partial L}{\partial \lambda} = 3000 - 53W - 56C = 0$ ## Step4: We solve these equations simultaneously to find the optimal values of $W$, $C$, and $\lambda$. ## Step5: To verify that the solution is indeed a maximum, we check the second-order conditions. The second derivatives of the Lagrangian function with respect to $W$, $C$, and $\lambda$ should all be negative.

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