Isocost Line
What is Isocost Line?
An isocost line shows every combination of two inputs a firm can buy for the same total cost, with slope equal to minus the input price ratio.
An isocost line is the firm's version of a budget line. Its equation is w·L + r·K = C, where w is the wage, r is the rental price of capital, and C is total spending, so the slope is negative w/r and the intercepts are C/w and C/r. A larger budget draws a new isocost line parallel to and above the old one, while a change in the wage or the rental rate rotates the line. Cost minimization for a target output happens where an isoquant touches the lowest attainable isocost line, and at that tangency MRTS = w/r, which is the same condition as MPL/w = MPK/r. Keep the roles straight: the isocost holds spending fixed, the isoquant holds output fixed.
Isocost Line: a worked example
A workshop pays $20 an hour for labor and $40 an hour to rent a machine, with $800 to spend. All $800 on labor buys 40 labor hours; all $800 on machines buys 20 machine hours, so the isocost line runs between those intercepts with a slope of -0.5 (one labor hour costs half a machine hour). If the wage rises to $25 while the rental rate holds at $40, the labor intercept falls to 32, the capital intercept stays at 20, and the line rotates inward and steepens to -0.625, which pushes the least-cost mix toward capital.
The mistake students make with isocost line
Many students reverse the two curves and call the isocost the one that holds output constant. The isocost holds total spending constant; output is constant along the isoquant. A second slip is thinking a wage increase shifts the isocost line inward in parallel. It rotates it, pivoting on the capital intercept, because only the labor intercept changes.
Isocost Line questions
What is the slope of an isocost line?
The slope of an isocost line is negative w/r, the wage divided by the rental price of capital, made negative. It shows how many units of capital the firm must give up to hire one more unit of labor at constant total cost. Only relative input prices set the slope; the size of the budget sets the position.
How do you find the least-cost combination of inputs?
The least-cost combination is where the isoquant for the target output is tangent to the lowest isocost line the firm can reach. At that point the marginal rate of technical substitution equals the input price ratio w/r. Equivalently, the marginal product per dollar is equal across inputs: MPL/w = MPK/r.
What shifts an isocost line?
A change in the firm's total cost budget shifts the isocost line parallel to itself, out for a bigger budget and in for a smaller one. A change in the wage or the rental rate instead rotates the line around the intercept of the input whose price did not change. Both input prices changing by the same percentage shifts the line without changing its slope.
Formula / Example
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