IsocostFrom Wikipedia, the free encyclopedia

In economics an isocost line shows all combinations of inputs which cost the same total amount.[1][2] Although

similar to the budget constraint in consumer theory, the use of the isocost line pertains to cost-minimization in

production, as opposed to utility-maximization. For the two production inputs labour and capital, with fixed unit

costs of the inputs, the equation of the isocost line is

where w represents the wage rate of labour, r represents the rental rate of capital, K is the amount of

capital used, L is the amount of labour used, and C is the total cost of acquiring those quantities of the

two inputs.

The absolute value of the slope of the isocost line, with capital plotted vertically and labour plotted

horizontally, equals the ratio of unit costs of labour and capital. The slope is:

The isocost line is combined with the isoquant map to determine the optimal production point at any

given level of output. Specifically, the point of tangency between any isoquant and an isocost line

gives the lowest-cost combination of inputs that can produce the level of output associated with that

isoquant. Equivalently, it gives the maximum level of output that can be produced for a given total

cost of inputs.

[edit]The cost-minimization problem

The cost-minimization problem of the firm is to choose an input bundle (K,L) feasible for the output

level y that costs as little as possible. A cost-minimizing input bundle is a point on the isoquant for

the given y that is on the lowest possible isocost line. Put differently, a cost-minimizing input bundle

must satisfy two conditions:

1. it is on the y-isoquant

2. no other point on the y-isoquant is on a lower isocost line.

[edit]The case of smooth isoquants convex to the origin

If the y-isoquant is smooth and convex to the origin and the cost-minimizing bundle involves a

positive amount of each input, then at a cost-minimizing input bundle an isocost line is tangent to

the y-isoquant. Now since the absolute value of the slope of the isocost line is the input cost

ratio w / r, and the absolute value of the slope of an isoquant is the marginal rate of technical

substitution (MRTS), we reach the following conclusion: If the isoquants are smooth and convex to

the origin and the cost-minimizing input bundle involves a positive amount of each input, then this

bundle satisfies the following two conditions:

It is on the y-isoquant (i.e. F(K, L) = y where F is the production function), and

the MRTS at (K, L) equals w/r.

The condition that the MRTS be equal to w/r can be given the following intuitive interpretation. We

know that the MRTS is equal to the ratio of the marginal products of the two inputs. So the condition

that the MRTS be equal to the input cost ratio is equivalent to the condition that the marginal

product per dollar is equal for the two inputs. This condition makes sense: at a particular input

combination, if an extra dollar spent on input 1 yields more output than an extra dollar spent on

input 2, then more of input 1 should be used and less of input 2, and so that input combination

cannot be optimal. Only if a dollar spent on each input is equally productive is the input bundle

optimal.

In economics, an isoquant (derived from quantity and the Greek word iso, meaning equal) is a contour

line drawn through the set of points at which the same quantity of output is produced while changing the

quantities of two or more inputs.[1][2] While an indifference curve mapping helps to solve the utility-

maximizing problem of consumers, the isoquant mapping deals with the cost-minimization problem of

producers. Isoquants are typically drawn on capital-labor graphs, showing the technological tradeoff

between capital and labor in the production function, and the decreasing marginal returns of both inputs.

Adding one input while holding the other constant eventually leads to decreasing marginal output, and this

is reflected in the shape of the isoquant. A family of isoquants can be represented by anisoquant map, a

graph combining a number of isoquants, each representing a different quantity of output. Isoquants are

also called equal product curves.

An isoquant shows the extent to which the firm in question has the ability to substitute between the two

different inputs at will in order to produce the same level of output. An isoquant map can also indicate

decreasing or increasing returns to scale based on increasing or decreasing distances between the

isoquant pairs of fixed output increment,as you increase output. If the distance between those isoquants

increases as output increases, the firm's production function is exhibiting decreasing returns to scale;

doubling both inputs will result in placement on an isoquant with less than double the output of the

previous isoquant. Conversely, if the distance is decreasing as output increases, the firm is experiencing

increasing returns to scale; doubling both inputs results in placement on an isoquant with more than twice

the output of the original isoquant.

As with indifference curves, two isoquants can never cross. Also, every possible combination of inputs is

on an isoquant. Finally, any combination of inputs above or to the right of an isoquant results in more

output than any point on the isoquant. Although the marginal product of an input decreases as you

increase the quantity of the input while holding all other inputs constant, the marginal product is never

negative in the empirically observed range since a rational firm would never increase an input to decrease

output.

[edit]Shapes of Isoquants

If the two inputs are perfect substitutes, the resulting isoquant map generated is represented in fig. A; with

a given level of production Q3, input X can be replaced by input Y at an unchanging rate. The perfect

substitute inputs do not experience decreasing marginal rates of return when they are substituted for each

other in the production function.

If the two inputs are perfect complements, the isoquant map takes the form of fig. B; with a level of

production Q3, input X and input Y can only be combined efficiently in the certain ratio occurring at the

kink in the isoquant. The firm will combine the two inputs in the required ratio to maximize profit.

Isoquants are typically combined with isocost lines in order to solve a cost-minimization problem for given

level of output. In the typical case shown in the top figure, with smoothly curved isoquants, a firm with

fixed unit costs of the inputs will have isocost curves that are linear and downward sloped; any point of

tangency between an isoquant and an isocost curve represents the cost-minimizing input combination for

producing the output level associated with that isoquant.