Problem 3 Let's consider the dual problem of the restaurant in problem 2. The restaurant is going to be minimizing its cost, subject to the constraint that its production has to be the same amount as you got in problem 2. All prices are the same as in problem 2. (a) Setup the Lagrangian of this problem. Make sure to specify the function and the variables we're minimizing with respect to. (b) Derive the first order conditions of the problem. (c) Given the fact that 40, find the optimal values of the choice variables. (Why is = 40? A firm in perfect competition maximizes its profit where the market price equals its marginal cost. What is the interpretation of in this problem?) (d) Find the optimal levels of output, revenue, cost, and profit. (e) Derive the second partials of the minimization problem and setup the bordered Hessian. (f) Using the bordered Hessian are the sufficient second order conditions for a minimum satisfied? Problem 2 Consider a restaurant that is deciding to open in Manhattan. The owner knows that he can sell a meal, q, for $40, that every waiter/bartender he hires, L, costs him $10 a month, that he will have to pay $20 a month rent per appliance, M, and that rent and taxes are $10 per square foot, S, a month. The restaurant's production function is q = 2L1/4M1/2 + 1851/2, and it operates in perfect competition in the restaurant and factors markets. (a) What is the restaurant's problem? Setup the problem it has, specifying the function what we need to do and with respect to the variables that we need to do it. (b) Derive the first order conditions of the problem and find the optimal values of the variables we were deciding on. (c) Based on the values you found in part (b) find the optimal production, revenue, cost, and profit of the restaurant. (d) Find the second partial derivatives of the optimization problem and form the Hessian. (e) Using the Hessian, are the sufficient second order conditions satisfied for this problem? Explain what the rule is, and why they are satisfied. Problem 3 Let's consider the dual problem of the restaurant in problem 2. The restaurant is going to be minimizing its cost, subject to the constraint that its production has to be the same amount as you got in problem 2. All prices are the same as in problem 2. (a) Setup the Lagrangian of this problem. Make sure to specify the function and the variables we're minimizing with respect to. (b) Derive the first order conditions of the problem. (c) Given the fact that 40, find the optimal values of the choice variables. (Why is = 40? A firm in perfect competition maximizes its profit where the market price equals its marginal cost. What is the interpretation of in this problem?) (d) Find the optimal levels of output, revenue, cost, and profit. (e) Derive the second partials of the minimization problem and setup the bordered Hessian. (f) Using the bordered Hessian are the sufficient second order conditions for a minimum satisfied? Problem 2 Consider a restaurant that is deciding to open in Manhattan. The owner knows that he can sell a meal, q, for $40, that every waiter/bartender he hires, L, costs him $10 a month, that he will have to pay $20 a month rent per appliance, M, and that rent and taxes are $10 per square foot, S, a month. The restaurant's production function is q = 2L1/4M1/2 + 1851/2, and it operates in perfect competition in the restaurant and factors markets. (a) What is the restaurant's problem? Setup the problem it has, specifying the function what we need to do and with respect to the variables that we need to do it. (b) Derive the first order conditions of the problem and find the optimal values of the variables we were deciding on. (c) Based on the values you found in part (b) find the optimal production, revenue, cost, and profit of the restaurant. (d) Find the second partial derivatives of the optimization problem and form the Hessian. (e) Using the Hessian, are the sufficient second order conditions satisfied for this problem? Explain what the rule is, and why they are satisfied
