Please refer to the example solution to answer the following question

Example?

"Determine which production process the firm will choose as a function of the rental price of capital r" The rental price of capita r is a variable, and it will be a function of the parameters of the model. So, it is the value of r relative to the parameters that will determine your answer.2. Let there be two factors, L and K, with factor prices w and r, respectively. Let the production function be y = LIK? where y is output. a. Does this firm have DRS, CRS, or IRS technology (or a mixture of them)? CRS b. Solve the firm's long run cost minimization problem to find i. the long run demand function for the factors, L'(y) and K* (v). ii. the long run total cost function C(y). Note: the factor demands, and cost function depend not just on the choice variable, firm output y, but also on the parameters, input prices w and r, so we could also write them as L'(y; w, r) and K*(y; w, r) and C(y; w, r). Note: we will always assume the global optimum exists and that the constraint qualification holds there (See the Method of Lagrange pdf). Given that, if the global optimum is an interior solution (Vi, x, > 0), then it a critical point of the Lagrangian. Otherwise, the global optimum is a boundary point of the inequality constraints (3i, x( = 0). Since f (L, K) = LIK , output can never equal y when L = 0 or K = 0: f (0, K) = f(L, 0) = 0 # y. Therefore, the boundary points of the inequality constraints L 2 O and K 2 0 - all the points of the form (0, K) or (L, 0) - cannot be the solution to our cost minimizing way to produce y units of output. The solution is an interior solution and so it is a critical point (L', K') of the Lagrangian. Firm's long run cost minimization problem: Choose (L, K) tominwl + rk subject to LzKz = y First, MP, = of (L, K) =SL EKZ aL ANI of ( L, K ) MPK OK MPL K So, MPK 1LIK-2 We know the firm's long run demand for labour and capital (L', K*) is a critical point of the Lagrangian which occurs at the tangency condition (slope of isocost line, - - equals the slope of the isoquant curve, - MPk' -L, and has output y = f(L, K): MPL W MPK r 1 LIKE = y Solving: MPL W K W - = K = -L MPK Sub into production constraint LIKE = y: LZ ( W L ) ? = y L'(V ) =y- K ( y ) = = L'(V) =>W C( V) = WL'( V) + rk () = wy utry *=zyvow c. Derive the LRAC and LMC curves. What do the long run cost curves look like? C(V) 2yvrw LRAC(y) = = 2vrw y V dc (y) LMC(y) = = 2vrw dy LRAC(y) = LMC(y) curve is horizontal - but you knew that because there is CRS technology.A firm produces output 1; using two factors of production [inputs], laioour L and capital it. It is considering whether to use one of two production processes. Production process A has the 1 1 production function fAIIL, K) = oLEKi. Production process B has the production function fBifL, K] = 1613K i The wage rate w = 1 and the rental price of capital is r. The input markets are competitive, so w and r are taken as parameters [xed]: by the firm. As aiwavs, we assume firms want to maximize prots. Determine which production process the firm Win choose as a function of the rentai price of capital r. [12 marks