Isocost Lines PDF

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Detailed information/expansion on isocost lines....

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Isocost Lines The cost of producing a given level of output depends on the price of labour and capital. The firm hires L hours of labour services at a wage of w per hour, so its labour cost is wL. The firm rents K hours of machine services at a rental rate of r per hour, so its capital cost is rK. ● If the firm owns the capital, r is the implicit rental rate. The firm’s total cost is the sum of its labour and capital costs. For an isocost it is:

C(¿)=wL +rK Where wL is labour costs (L hours of labour at a wage of w per hour) and rK is the capital costs (K hours of capital at a rental rate of r per hour). It is not always fixed but along an isocost, it always is at a particular level. Rewrite the isocost equation for easier graphing as:

C w K= − L r r The firm can hire as much labour and capital as it wants at these constant input prices. The firm can use many combinations of labour and capital that cost the same amount. These combinations of labour and capital are plotted on an isocost line, which indicates all the combinations of inputs that require the same (iso) total expenditure (cost). The graph below shows three isocost lines for the Japanese beer manufacturer where the fixed cost is C = $1,000, $2,000 or $3,000 and w = $24 per hours and r = $8 per hour. Using isocost isocost

the rearranged equation, the lines are

C 24 C K= − L= −3 L 8 8 8 First, the point where the isocost lines hit the capital and labour axis depend on the firm’s cost and the input prices. The C isocost line intersects the capital axis where the firm uses only capital.

By setting L = 0, we find that the firm buys K = C/r units of capital. Similarly, the intersection of the isocost line with the labour axis is at C/w, which is the amount of labour the firm hires if it uses only labour. Second, isocosts that are farther from the origin have higher costs than those closer to the origin. Because the isocost lines interest the capital axis at C/r (y axis) and the labour axis at C/w, an increase in the cost shifts these intersections with the axes proportionately outward. Third, the slope of each isocost line is the same. By differentiating the equation

C w K= − L r r

we find that the slope of any isocost line is:

dK −w = dL r Therefore, the slope of the isocost line depends on the relative prices of the inputs. Because all isocost lines are based on the same relative prices, they all have the same slope, so they are parallel. The role of the isocost line in the firm’s decision making is similar to the role of the budger line in a consumer’s decision making. Both an isocost line and a budget line are straight lines with slopes that depend on relative prices. However, they differ in an important way. The single budget line is determined by the consumer’s income. The firm faces many isocost lines, each of which corresponds to a different level of expenditures that the firm might make. A firm may incur a relatively low cost by producing relatively little output with few inputs, or it may incur a relatively high cost by producing a relatively large quantity. Minimising Cost By combining the information about costs contained in the isocost lines with information about efficient production that is summarised by an isoquant, a firm determines how to produce a given level of output at the lowest cost. In the graph above, we examine how our beer manufacturer picks the combination of labour and capital that minimizes its cost of producing 100 units of output. The graph shows the isoquant for 100 units of output and the isoquant lines where the rental rate of a unit of capital is $8 per hour and the wage rate is $24 per hour. The firm can choose any of three equivalent approaches to minimise its cost: 1. Lowest Isocost Rule - Pick the bundle of inputs where the lowest isocost line touches the

isoquant. 2. Tangency Rule - Pick the bundle of inputs where the isoquant is tangent to the isocost line. 3. Last Dollar Rule - Pick the bundle of inputs where the last dollar spent on one input gives as much extra output as the last dollar spent on any other input. Using the lowest isocost rule, the firm minimizes its cost by using the combination of inputs on the isoquant that lies on the lowest isocost line to touch the isoquant. The lowest possible isoquant that will allow the beer manufacturer to produce 100 units of output is tangent to the $2,000 isocost line. ● This isocost line touches the isoquant at the bundle of inputs x, where the firm uses L = 50 workers and K = 100 units of capital. How do we know that x is the least costly way to produce 100 units of output? We need to demonstrate that other practical combinations of inputs produce fewer than 100 units or produce 100 units at a greater cost. If the firm spent left than $2,000, it could not produce 100 units of output. Each combination of inputs on the $1,000 isocost line lies below the isoquant, so the firm cannot produce 100 units of output for $1,000. The firm can produce 100 units of output using other combination of inputs besides x, but using these other bundles of inputs is more expensive. For example, the firm can produce 100 units of output using the combinations y(L = 24, K = 303) or z(L = 116, K = 28). However both of these combinations cost the firm $3,000. If an isocost line crosses the isoquant twice, as the $3,000 isocost line does. There must be another lower isocost line that also touches the isoquant. The lowest possible isocost line to touch the isoquant, the $2,000 isocost line, is tangent to the isoquant at a single bundle, x. ● Therefore, the firm may use the tangency rule: the firm chooses the input bundle where the relevant isoquant is tangent to an isocost line to produce a given level of output at the lowest cost. We can interpret the tangency or cost minimisation condition in two ways. At the point of tangency, the slope of the isoquant equals the slope of the isocost. The slope of the isoquant is the marginal rate of technical substitution (MRTS). The slope of the isocost is the negative of the ratio of the wage to the cost of capital, -w/r. Therefore, to minimise its cost of producing a given level of output, a firm chooses its inputs so that the marginal rate of technical substitution equals the negative of the relative input prices:

MRTS=

−w r

The firm chooses inputs so that the rate at which it can substitute capital for labour in the production process, the MRTS, exactly equals the rate at which it can trade capital for labour in input markets, -w/r. We can interpret the MRTS equation above in another way. The marginal rate of technical substitution equals the negative of the ratio of the marginal product of labour to that of capital: MRTS = -MPL / MPK. Therefore, the cost minimising condition in the equation above when taking the absolute calue of both sides we get:

MP❑L w = MP ❑K r The above equation may also be rewritten as:

MP ❑L MP ❑K = w r The above equation is the last dollar rule: cost is minimised if inputs are chosen so that the last dollar spent on labour adds as much extra output as the last dollar spent on capital. To summarise, the firm can use three equivalent rules to determine the lowest cost combination of inputs that will produce a given level of output when isoquants are smooth: the lowest-isocost rule, the MRTS and ●

MP ❑L w = MP ❑K r

equation and the last dollar rule,

MP❑L MP ❑K . = w r

The lowest isocost rule always works, even when isoquants are not smooth.

Minimising Costs Using Calculus Formally, the firm minimizes its cost subject to the information about the production function that is contained in the isoquant expression: q = F(L, K). Seeking the least cost way of producing a given level of output. Minimising cost subject to a production constraint yields the Lagrangian and its first order condition:

min よ= wL+rK +λ [q − f (L , K )] L,K , λ Assuming that we have an interior solution where both L and K are positive, the first order conditions are:

∂よ ∂f =0 =w−λ ∂L ∂L

∂よ =r−λ ∂ f =0 ∂K ∂K ∂よ =q−f (L , K )=0 ∂λ Rearranging the terms reveals the last dollar rule, the earlier equation

∂f w ∂ L MP ❑L = = r ∂ f MP❑K ∂K

MP ❑L w = : MP❑K r

MP ❑L MP ❑K = w r

Maximising Output An equivalent or dual problem to minimizing the cost of producing a given quantity of output is maximising output for a given level of cost. Here, the Lagrangian problem is:

minよ=f (L , K )+ λ(C− wL −rK ) Rearranging the terms reveal the ‘tangency rule’:

∂よ ∂ f = −λ w=0 ∂L ∂ L ∂よ ∂ f −λ r =0 = ∂K ∂K ∂よ =C−wL−rK =0 ∂λ

By examining the above equations and the ratio of the first two conditions, we obtain the same condition as when we minimized cost by holding output constant: MPL/MPK = (df/dL)(df/dK) = w/r. That is, at the output maximum, the slope of the isoquant equals the slope of the isocost line. The beer manufacturer maximises its production at a cost of $2,000 by producing 100 units of output at x using L = 50 and K = 100. The q = 100 isoquant is the highest one that touches the $2,000 isocost line. The firm operates where the q = 100 isoquant is tangent to the $2,000 isocost line. The graph shows that the firm maximises its output for a given level of cost by operating where the highest feasible isoquant, q=100, is tangent to the $2,000 isocost line....