LONG RUN SHORT RUN COST MINIMIZATION. Labor is variable Capital is fixed Solve for: labor only

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1 SHORT RUN Labor is variable Capital is fixed Solve for: labor only LONG RUN Labor is variable Capital is variable Solve for: labor and capital COST MINIMIZATION Conceptual Goal: 1. Find the cheapest way to produce a fixed amount of output (L SR Q ; L LR Q and K LR Q) Total Cost = wl + rk Short Run Method: To solve for L, simply isolate L directly from the production function, regardless of the production function type (one exception is perfect complements; see practice problem 2b) Long Run Method: Part 1: Always determine the type of production function before solving for Cost Min 1. Cobb-Douglas à Set TRS equal to w/r Main Characteristics Write down the main characteristic of each production function type (hint: think back to the characteristics of utility functions) 2. Perfect Subs à Choose all of one input and none of the other If TRS > w/r then choose all labor If TRS < w/r then choose all capital If TRS = w/r then indifferent between all input bundles that produce fixed output 3. Perfect Comps à Set inside of min function equal to itself 4. Quasi-convex à Set TRS equal to w/r Part 2: Isolate K from the main characteristic and substitute the isolated K into the production function, simplify à L LR Q Part 3: Substitute L LR Q back into the isolated K, simplify à K LR Q

2 PROFIT MAXIMIZATION Conceptual Goals: 1. Choose the amount of output that maximizes profits (Q SR ; Q LR ); also known as firm supply Π = Revenue Total Cost = pq Total Cost = pf(l, K) wl rk 2. Find the cheapest way to produce that amount of output (L SR ; L LR and K LR ) Total Cost = wl + rk Normal Cases (Short Run and Long Run): Normal cases are those that fit all of the following characteristics: Short Run: Diminishing MP L ; constant price, wage, rental rate Long Run: Cobb-Douglas; Decreasing Returns to Scale; constant price, wage, rental rate For normal cases, simply follow the steps listed on the Profit Max Cobb-Doug Shortcut worksheet. Abnormal Cases (Short Run and Long Run): Abnormal cases are those that violate at at least one of the normal cases characteristics. You cannot follow the shortcut steps for abnormal cases. Instead, you must create a solving process to fit the specific situation given in the prompt. Use the checklist below as a guide to creating a solving process for each unique abnormal case you are faced with. Abnormal Cases Checklist Set up profit function (plug in price, wage, rental rate, fixed capital, and production function) àobserve interconnectedness of inputs; are both inputs needed to produce output or can output be produced with one input? *If inputs are not interconnected, analyze each individually *If inputs are interconnected, use the main characteristic to put profit in terms of one input à Observe the relationship between inputs and profit; what happens to profit as each input increases? *If profit always increases as the input increases, the firm will demand as much as possible of the input *If profit always decreases as the input increases, the firm will demand zero of the input *If the relationship is ambiguous, use the first order condition to determine demand First order condition (first derivative of profit function set equal to zero; finds a maximum point) Second order condition (second derivative of profit function; negative answer verifies that the first order condition yielded a maximum point rather than a minimum point) Shutdown condition Short Run: Operate if profit is greater than negative fixed costs Long Run: Operate if profit is greater than zero

3 1. F(L, K) = 2L 1/4 K 1/2 a. Solve for short run profit max Normal or Abnormal: Normal b. Solve for long run profit max Normal or Abnormal: Normal c. Solve for short run cost min d. Solve for long run cost min Type of function: Cobb-Douglas 2. F(L, K) = min [2L, 5K] a. Solve for long run profit max. Assume p = $6, w = $10, and r = $10 Normal or Abnormal: Abnormal b. Solve for short run cost min. Assume Q = 50, K = 12, w = $5, and r = $5 c. Solve for long run cost min. Assume Q = 50, w = $5, and r = $5 Type of function: Perfect Complements 3. F(L, K) = L + 10K a. Solve for long run profit max. Assume p = $10, w = $15, and r = $200 Normal or Abnormal: Abnormal b. Solve for short run cost min. Assume Q = 100, K = 8, w = $5, and r = $30 c. Solve for long run cost min. Assume Q = 100, w = $5, and r = $30 Type of function: Perfect Substitutes 4. F(L, K) = min [L 1/2, K 1/2 ] a. Solve for long run profit max Normal or Abnormal: Abnormal b. Solve for long run cost min Type of function: Perfect Complements (not technically perfect complements, but can be treated as such for cost min) 5. F(L) = 4L 3/4 a. Solve for short run cost min b. Solve for long run cost min Type of function: Single input c. Solve for short run profit max. Assume p = $1, w = $3, and there is a licensing fee of $10 that the firm has already paid Normal or Abnormal: Abormal d. Solve for long run profit max. Assume p = $1, w = $3, and there is a licensing fee of $10 that the firm must pay to operate, but has not yet paid Normal or Abnormal: Abnormal

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