ProdCost

Production and costs

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• Production functions: short and long run
• Marginal product and diminishing returns
  • Marginal product (MP) is the addtional output per period when there's one additional unit of input (holding other inputs constant). This is the derivative of the total product (TP).
  • Average product is

{$$ \text{AP} = \frac{\text{Total Product}}{\text{Quantity of Input}} $$}

• Can show that the AP and MP curves intersect at a point where the AP obtains its maximum.

• Short-run costs
  • Fixed cost (in terms of units of output) does not change when there is more output; variable cost do change. Total cost is the sum. We can define average cost for all 3 quantities, but the marginal cost is defined as derivative of total cost (which equals the derivative of the variable cost, because the fixed cost is constant with respect the x variable (units of output).)
  • The above scenario describes short-run costs -- We assume at least one input is fixed (and contributes to the fixed cost).
• Long-run costs and economies of scale
  • For long-run costs we assume that every input is variable, so there is no fixed cost.
  • Suppose the current amount of capital is fixed and we are to graph the corresponding short-run average cost curve (Quantity vs Cost in $ per unit). Then
  • 1. The short-run curve will always be above the long-run curve (i.e., higher short-run cost);
  • 2. If the amount of capital is the cost-minimizing amount to produce {$q$} items, then the two curves touch at the point where the quantity equals {$q$}. If a different short-run curve is drawn for a different amount of capital, then the short-run and long-run curves will always touch at the quantity for which the amount of capital is the cost-minimizing amount.
  • Economies of scale occurs in the range of output where the long-run average cost has negative slope. Diseconomies of scale occurs when the same curve has positive slope.
• Cost minimizing input combination
  • Increasing returns (to scale) occurs when the output increases more than increases in all inputs. [vs marginal returns, involving increase of only one input.] Decreasing returns (to scale) occurs when the output increases less than increases in all inputs. Constant returns (to scale) occurs when the output increases in proportion to increases in all inputs.
  • Diminishing (marginal) returns (to scale) occurs when an additional unit of input increases total output by less than the previous unit of input, holding all other input constant.
  • An increasing cost firm faces decreasing returns to scale; a decreasing cost firm faces increasing returns to scale.
  • An increasing cost industry experiences increases in average production costs as output increases. This results in a postively sloped long-run supply curve. A constant cost industry experiences no change in average production costs as output increases. This results in a horizontal long-run supply curve. A decreasing cost industry experiences decreases in average production costs as output increases. This results in a negatively sloped long-run supply curve.
  • Productive efficiency occurs when MC = AC (also min of AC).
  • Economies of scope occurs when average production costs decrease because multiple (e.g., complementary) products are being produced.