This chapter focuses on how the firm chooses its optimal input combination for a given level of output.
9.1 Cost in the Short Run
The short-run economic cost is the minimum level of total expenditures (measured in opportunity-cost terms) needed to produce a given amount of output in the short run.
The short-run variable cost is an abbreviated name for the short-run total economic cost. Economic cost and variable cost are synonyms because if the firm cannot vary the level of expenditure, then it is sunk and is not an economic cost.
To find costs in the short run, follow these steps:
1. Draw the isoquant associated with the output level whose short-run variable cost we are trying to find. (The x0-isoquant.)
2. Label the level at which capital is fixed in the short run, say Kf.
3. Read across from this fixed level of capital to find the point on the x0-isoquant associated with Kf robots. (Point a.)
4. Multiply the quantity of labor found in step (3), La, by the wage rate to get the short-run variable cost, VCSR(x0)
Properties of Short-Run Costs
Variable Cost: When the amount of capital is fixed at Kf, the firm must employ La units of labor to produce x0 units of output. Hence, the short-run variable cost of producing x0 units of output is w*La, when w is the wage per unit of labor. The short-run variable cost curve must slope upwards. This is because a firm operating on its production function must hire more inputs in order to produce more output, and more inputs cost more money.
Marginal Cost: The short-run marginal cost is the change in the short-run variable cost due to the production of one more unit of output. The marginal cost is derived from the variable cost, and the variable cost is derived from the production function.
The cost of one additional unit of the variable input is known as the marginal factor cost (MFC) of that input.
The short-run marginal cost = (additional amount of the variable input needed to produce one more unit of output: 1/MPPL)*(marginal factor cost of the variable input: MFC)
Whenever a firm is a price taker in an input market, the marginal factor cost is equal to the price of the factor. (For labor, this is the wage, w)
MFCL=w therefore: MCSR=w/MPPL
This tells us that the higher the marginal product of labor, the lower the marginal cost of output. If a single new worker can produce a lot of output, then not many new workers are needed, and the extra output does not cost very much to make.
If there are diminishing returns to labor, increasingly large increments of labor are needed to produce additional output. As the quantity of labor employed rises, the marginal physical product of labor falls. (MPPL falls when MCSR rises). So when the production function exhibits a diminishing marginal product of labor, the short-run marginal cost curve is upward sloping. (And the MPPL curve slopes downward)
When the marginal physical product of labor is constant at m, each additional unit of output requires 1/m additional units of labor. Because each laborer receives a wage of w, short-run total cost rises by w/m. Hence the short-run marginal cost is w/m at every output level.
Short-run average variable cost is the short-run variable cost divided by the number of units being producted.
The short-run average total cost is the short-run total cost divided by the number of units being produced.
The short-run average fixed cost is the short-run fixed cost divided by the number of units produced.
The Relationship between Short-Run Marginal Cost and Short-Run Average Variable Cost
Given that marginal and average variable cost curves are derived from the same variable cost curve, they are related in these ways:
1. Whenever marginal cost is below average variable cost, average variable cost falls.
2. Whenever marginal cost is above average variable cost, average variable cost rises.
3. The short-run marginal cost curve crosses the short-run average variable cost curve at the point where average variable cost is at a minimum.
9.2 Cost in the Long Run
Since all factors are variable in the long run, the (explicit and imputed) expenditures on all factors are economic costs in the long run.
Since the levels of more than one factor can be varied, it may be possible for the firm to substitute quantities of one factor for another.
To maximize profit, the firm must choose the cheapest combination of inputs that can produce the desired level of output. It must make an economically efficient input choice – an input combination is economically efficient when it has the lowest opportunity cost of those input combinations that can be used to produce the desired output.
Isocost Lines: An Isocost line is the line representing all input combinations that cost the firm the same amount. An isocost map is the whole family of isocost lines that exists for a given set of factor prices.
Finding the Economically Efficient Input Mix: The firm minimizes the total cost of producing a given number of outputs by using the input bundle that is on the isoquant representing that level of output and on the lowest possible isocost line.
The isocost line and isoquant line have the same slope where they are tangent (at the equilibrium). The slope of an isocost line is equal to the factor price ratio, w/r, in absolute value. By definition, the negative of the slope of the isoquant is the marginal rate of technical substitution between capital and labor. So, at the point of tangency: MRTS=w/r
Also, along an isoquant, MRTS=MPPL/MPPK, so we can say: MPPL/MPPK=w/r
A firm that takes factor prices as given should operate at a point where, at the margin, the inputs’ marginal physical products are proportional to their prices.
How does a change in the price of one factor affect the firm’s long term decisions? When the price of labor rises, the isocost lines pivot inward. To produce a given level of output, the firm substitutes away from the factor whose price has risen. Total cost must always rise in response to an increase in the price of one of the factors that the firm uses in production.
As a result of technological improvement, the firm can continue to produce the same amount of output while using fewer inputs. The isoquant shifts inward, and the total cost falls.
The Nature of Output
A firm may decide to improve the quality of its output. This change shifts the isoquant outward – it takes more inputs to produce the same amount of outputs.
The expansion path of a firm is the long-run set of least-cost input levels traced out as the level of output changes, all else equal.
Summary of Comparative Statics Analysis
1. Sketch the equilibrium that holds before there are any changes in the underlying market conditions
2. Given a change in underlying market conditions (Such as factor price change or a shift in technology), determine whether the isoquant map or the isocost map is affected.
3. Ascertain the direction of the shift in the affected set of curves
4. Find the new equilibrium by locating the tangency between the relevant isoquant curve and the relevant isocost line.
Deriving the Long-Run Total Cost Curve
The long-run total cost is the minimal level of total expenditures (measured in opportunity-cost terms) needed to produce a given amount of output in the long run. To find the entire long-run total cost schedule, just follow these steps:
1. Choose an output level.
2. Find the optimal input combination by taking the tangency of the isoquant and isocost lines.
3. Compute the cost of this input combination by multiplying the price of each input times the quantity used and then adding up the dollar total.
4. Graph this point.
5. Repeat steps (1) through (4) for each output level.
Properties of Long-Run Costs
Long-Run Marginal Cost: The long-run marginal cost is the change in long-run total cost due to the production of one more unit of output.
Long-Run Average Cost: The long-run average cost is the total cost divided by the number of units being produced.
Economies of Scale: When long-run average costs fall as output rises, costs are said to exhibit economies of scale. When the production function exhibits increasing returns to scale, the long-run total cost function exhibits economies of scale. The average cost curve is downward sloping.
When the production function exhibits constant returns to scale, long-run average cost remains constant as the level of output changes. The average cost curve is flat.
When long-run average costs rise with the output level, costs are said to exhibit diseconomies of scale. When the production function exhibits decreasing returns to scale, the long-run total cost function exhibits diseconomies of scale. The average cost curve is upward sloping.
Economies of Scope: When it is cheaper to produce two products together in one firm instead of separately in two specialized firms, costs are said to exhibit economies of scope. Economies of scale and scope both arise when a fixed facility can be “shared” to produce a large number of units of output, either of a single product, or a mix of products.
Long-Run Costs Compared to Short-Run Costs
1. In the short run, fixed factors are fixed. Since these fixed factors have no alternative uses, any expenditures made by the firm on these factors are sunk expenditures and thus are not economic (opportunity) costs. In the long run, however, all input levels are variable and there are no sunk expenditures; hence, everything counts as an economic cost. Because more things get counted as costs, this effect raises long-run economic cost relative to short-run economic cost.
2. Since the levels of more than one factor can be varied in the long run, it may be possible to substitute quantities of one factor for another. For example, in the long run but not the short run, National Motors can adjust the number of robots that it uses on its assembly line. This increased flexibility tends to lower the cost of producing a given amount of output.
These act in opposite directions on the economic cost curves, so in some instances short-run variable cost of producing a given level of output may be greater than the long-run total cost, while in other cases the relationship will be the reverse. However, short-run total cost is always at least as large as long-run total cost.