A1. Milk or meat production Y per day is best explained by the inputs: feed needed per day in kg, X₁, and (number of) milk cows or calves, X2. Based on fitting the data for milk or meat production the revenue is modelled by a Cobb-Douglas function R(X₁, X2) = pX₁ X2 constrained by relation h(X₁, X₂) = C₁X₁+C₂X2 = C3. For milk production, the constraint expresses a relation between the average sale price c3 of milk (per day), the average price c₁ of a kg of feed and the price C2 of a milk cow (cost per day and based on the annual cost of a milk cow). For meat production, the constraint expresses a relation between the average sale price c3 of meat (per day), the average price c₁ of a kg of feed and the price c2 of a calf (cost per day and based on the annual cost of a calf). The parameters for the two cases have been established by linear regression of data to be the following: regarding the milk case: p = 445.69, a = 0.346, b = 0.542, C1 = £4.00, c₂= £1.36/d, c3 = £5.38; and, - regarding the meat case: p= 2.348, a = -0.205, b = 1.118, c₁ = £4.00, c2 = £500.0, c3 = £6.02. (i) Consider the general case first (without substituting any parameter values). Determine the Lagrangian. Find the FOCs, stationary point and analyse the bordered Hessian to classify the critical point. (Hint: Do not eliminate either X₁ or X₂ till you need to.)