Please teach me how to solve them.

1. Suppose a firm, in a perfectly competitive market, with production function F(K, L) = min[4L, 3/] has only 6 units of capital. What is their cost minimizing level of labor in the short run USR(q)? (5 points) (a) 1 (b) 4 -4 (c ) (d) 9 -63. A production function F(K, L) = KLIK+L has increasing returns to scale. Remember the scale factor is a > 1 (5 points) (a) TRUE (b) FALSE Explanation:5. Since Midterm 1, we have solved many optimization problems: . Short Run Cost Minimization . Profit Maximization over Quantity . Long Run Cost Minimization . Profit Maximization over Inputs . Utility Maximization (a) Write the 5 problem types above into the appropriate cell in the table below according to what type of optimization problem they are. (Note: Constrained us Unconstrained is talking about constraints other than non-negativity constraints. ) (5 points) Number of Types of Optimization Problems Variables + Unconstrained Constrained 1 Variable Multi- Variable (b) Write out the steps to solve a profit maximization problem over inputs (K,L) for K*(p, w, r), L*(p, w, r), a* (p, w, r). Write out the equation for profit over inputs with a general production function F(K, L) and be detailed when describing the steps. Note, profit maximization over inputs is the same as "one step method. " (5 points)?. Now consider a market that has an aggregate demand function {2\"[p}== Bp. Every firm in this market is identical and has the cost function C(qj = 5472 + 2D. is) Define a short run competitive equilibrium. [4 points} {b} Setup and solve the profit maximization problem for the firm over quantity to find q*{1n}. Remember this is o perfect cowet-tton environment, so 31: is o constant. [3 points). (c) In the short run, there are N = 5. Solve for the short run equilibrium firm quantity, aggregate quantity, price, and firm profits. You do not need to simplify your answer here (d) Use your answer from part c to explain if firms enter or exit in the long run? (Hint: 12 - 3 = 36 and 21-5 122 102 5 = 28.8) (3 points)