Cost minimization

When considering costs it is important to distinguish between:

• Explicit costs that involve a monetary expense (as when a firms buys a machinery)

• Implicit costs that do not involve a monetary expense (as the time an entrepreneur uses to build his company)

When deciding to use resources in a certain way you are deciding in an implicit way how not to use your resources. This links us to the concept of opportunity cost. An opportunity cost is the value of the best alternative you are not pursuing. To understand this concept better consider John. John can decide freely how much to work. If he works he can earn $10 an hour. This means that, if he decides not to work for an hour, he is giving up the opportunity to earn $10. These $10 represent John’s opportunity cost. This distinction in how we value costs is important to separate Economic costs and Accounting costs. When computing economic costs both explicit costs and implicit costs have to be considered, while when computing accounting costs only explicit costs have to be considered. This means that we have also to distinguish between economic profits and accounting profits. Economic profits are computed as the name suggests using the firm’s revenues and economic costs while accounting profits are computed using the firm’s revenues and accounting costs. Because when computing economic profits you are considering higher costs (because you consider also the implicit ones), accounting profit is higher than the economic one. When you are looking at a firm from the inside, for example from the point of view of the owner, you are interested more in economic costs (and profits) because, as an owner, you are dedicating your time to the firm and therefore you are interested if it is convenient for you to continue business considering that you could spend your time doing other things (opportunity cost). If you are looking at a firm from the outside, for example from the point of view of an investor, you are interested in accounting costs (and profits) because you care only about the ability of the firm to continue business and not if it is convenient for the owner.

SUNK AND NONSUNK COSTS

Another way to differentiate costs is dividing them in sunk and nonsunk ones. Sunk costs are costs that can not be recovered in the short term no matter how you behave. Suppose that you have signed a contract to rent a building for 1 year, even if you do not use the building you will not have back the money you payed to rent the place. Nonsunk costs are instead costs that can be avoided. You can decide whether to buy raw materials as inputs and until the transaction has happened you can still decide not to spend the money. When you have to make a decision you should only consider non sunk costs because sunk costs have to be payed in any case.

COST MINIMIZATION

Remember that a firm can use various combinations of inputs in order to obtain the same quantity of output (did you miss out lecture on production?). The firm is obviously interested in producing its output bearing the minimum cost possible. A firm can trying to do so is called a cost minimizing firm. In order to do so the firm has to employ the least expensive combination of inputs. The problem of finding that specific combination of inputs is called the cost minimization problem. When considering a the cost minimization problem we have to distinguish the time frame in which we are working. In the long run a firm can change at will the quantities of all its input and therefore has no sunk costs meaning that if the firm does not produce anything the firm’s costs are 0. In the short run firms generally have sunk costs and therefore are subject to some constraints. This means that firms in the short run can decide only how much to use of some inputs while the use of other inputs is fixed.

HOW TO SOLVE A COST MINIMIZATION PROBLEM IN THE LONG RUN

Suppose that the firms uses as inputs labor (L) and capital (K). Let’s denote the price of labor as w and the price of capital as r. Suppose that a firm decides to produce a certain quantity Q0 of output. Total cost is defined as:

TC = wL + rK. The firm’s objective is to minimize TC while producing Q0. The firm can substitute freely (because we are in the long run) one input for the other according to the prices the market sets for both the inputs. Obviously different combinations of inputs will result in different total costs. An isocost line is a line along which, when substituting capital for labor (or vice versa), the total cost does not change. The equation of a general isocost line is: K =TC/r –(w/r)L. In order to represent graphically an isocost line you have to decide TC in advance while w and r are the market prices and therefore generally are given. The higher is the isocost line the higher is Total cost. An isocost line which is nearer to the origin corresponds to a lower total cost. In the equation TC/r represents the vertical intercept of the isocost line and –(w/r) represents the slope of the line. The vertical intercept of the isocost line represents the maximum amount of capital you can employ by using all the budget (TC) on it. The horizontal intercept instead, which is algebraically TC/w, represents the maximum quantity of labor the firm can employ by spending all the total cost on it. To find the cost minimizing bundle you have to find a tangency condition between the isocost line and the isoquant (which depends on the firm’s possibility to substitute the various inputs, did you miss the lecture on production with more than one input?). The isoquant shows the minimum quantities of inputs a firm can use to produce quantity Q0. This means that only points on the isoquant line are technically efficient. Points above the isoquant are technically inefficient. Point B is technically inefficient, points E, A and F are technically efficient but only A is cost minimizing. In the cost minimizing point the slope of the isoquant and the slope of the isocost line are exactly the same. This means that the cost minimizing condition occurs when:

Slope of the isoquant = slope of isocost line

-MRTS(L,K) = -w/r

MPL/MPK = w/r

Which can be rearranged as:

MPL/w = MPK/r

The last expression can be interpreted as follows: in the cost minimizing point the additional output per dollar spent in labor is equal to the additional output per dollar spent in capital services.

HOW THE SHAPE OF THE ISOQUANT CAN INFLUENCE THE OPTIMAL CHOICE

Remember that not always isoquants have the convex shape described by a Cobb-Douglas function. Isoquants can also be linear or L-shaped.

When dealing with linear isoquants the solution to the cost minimization problem is a corner point meaning that the firm will only use one of the inputs (the ones which provides a better ratio between productivity and costs). In order to understand which one of the input the firm will choose and which one will not you have to compare the ratio of the marginal rate of technical substitution (which remember is constant with a linear function) with the ratio of the prices of the inputs. We can distinguish again three cases:

If MPL/MPK > w/r the company will only use labor.

If MPL/MPK < w/r the company will only use capital.

If MPL/MPK = w/r the isocost and the isoquant overlap and any combination of inputs on the isoquant is cost minimizing. height="177" src="file:///C:/Users/gianm/AppData/Local/Temp/msohtmlclip1/01/clip_image008.jpg" align="right" hspace="12" /

WHAT HAPPENS WHEN PRICES CHANGE?

When the price of an input modifies the isocost line changes in slope. If the price of labor (w) increase as the price of capital (r) stays the same the isocost line becomes steeper. If the price of capital increases as the price of labor stays the same the isocost line becomes flatter. When both prices change by the same proportion, the budget line will not change in slope and therefore the cost minimizing bundle of inputs will not change. To understand why, let’s say that the prices of labor and capital both double. Because the slope of the isocost line is given by -w/r, if you double both w and r you get: -2w/2r, if you cancel out the 2 at the nominator and the denominator you return to the original slope. When only the price of one of the inputs increases (or they increase by different percentages) the firm will try to substitute it if it has the opportunity to do so. This means that, if you consider a Cobb-Douglas function and the price of labor goes up, the tangency point between the isocost line and the isoquant will be more to the left (because the firm is using less labor). if you are considering a linear isoquant, then the firm will decide to employ only capital if thanks to the price change MPL/MPK < w/r. if you are dealing with a L-shaped production function, the firm you are considering can not substitute one input for the other and therefore the optimal bundle will not change.

In all these cases a price increase in at least one of the inputs implies an increase in total costs for the firm. To understand why, consider a general isocost line tangent to an isoquant. Imagine an increase in the price of capital. Because the vertical intercept, representing the maximum quantity of capital the firm can employ with its total cost, is defined as TC/r, an increase in r means that the number resulting from that ratio is less than the one before the price increase. This means that the vertical intercept is no longer where it was originally but it is now closer to the origin. If that is the case, as you can see from the picture, the isocost line is no more tangent to the isoquant. Because the isoquant is “fixed” because, even thought the price increase, the firm wants to produce the same Q0, in order to have again the tangency condition we have to shift in a parallel fashion the new isocost line. But in order to do so algebraically we have to increase the vertical intercept of the line which is exactly TC/r. Because r is a price given by the market, the only thing a firm can modify in order to modify the result of that ratio is TC. So, in order to keep a tangency condition, total cost has to be increased. The same reasoning applies also in the cases of L-shaped or linear isoquants even thought in those cases you do not have a tangency condition.

WHAT HAPPENS WHEN A FIRM DECIDES TO PRODUCE MORE?

Until now we have considered the quantity a firm wants to produce as fixed and therefore also the isoquant line has been considered fixed until now. But this needs not to be the case. When a firm decides to produce more output (and so increasing Q0) the cost minimizing quantities will follow an expansion path: a path indicating where on the graph would be located the cost minimizing bundles when changing Q0. We can have two different scenarios depending on the nature of the inputs the firm is using. If both capital and labor are for the firm normal inputs, the expansion path will be sloping upward indicating that, as the quantity of output increases, the quantities employed of both inputs will increase. In for example labor is for the firm an inferior input, the expansion path will be sloping downward indicating that, as the quantity of output increase for the firm is more convenient to reduce the quantity of labor employed while increasing the quantity of capital used. An example of that could be a firm producing jars: if the quantity of jars produced is little, for the firm it is more convenient to employ people rather than buying a large machinery but, if the quantity produced has to be increases drastically, then for the firm will be more convenient to use a large machinery rather than many workers. It is important to stress that only one of the two inputs can be inferior. That is because, if both inputs were inferior, it would be better for the firm using the lowest quantity of inputs possible in order to increase production (but we know that this in reality could never happen: can you imagine a firm producing without any workforce or machinery?).

THE INPUT DEMAND CURVES

The curve that shows the relationship between the cost minimizing quantity of labor depending on the price of labor (w) is the labor demand curve. When you have a change in the price of labor, you move along the curve while, when the firm wants to increase output, there is a shift to the right of the labor demand curve. If you move from point A to point B because of a price increase of labor when you consider the labor demand curve you move from A’ to B’ (notice that on the vertical axis is represented the price of labor). You can find point B’ on the labor demand curve by choosing the point on the line with height equal to the price of labor. If you move from point A to point C because the firm increases the quantity produced, the labor demand curve shifts until it crosses point C’. You can represent the same graph with respect to capita by putting on the horizontal axis the quantity of capital used and on the vertical one the price of capital. Obviously that curve will be called the capital demand curve.

THE COST MINIMIZATION PROBLEM IN THE SHORT RUN

Remember that in the short run a firm is subject to some constraints because it can not change freely the quantity of all its inputs because they represent the firm’s fixed costs. Suppose that in the short run the firm is not able to change the amount of capital employed which is fixed at a quantity K1. The firm’s total cost curve is given by wL + r(K1). The quantity of labor employed by the firm can be freely changed and therefore it represents the firm’s variable cost. Being a variable cost, the firm can decide not to use any labor and therefore spending nothing. This means that labor is for the firm a nonsunk cost. Capital on the other hand is a fixed cost and therefore can not be avoided in any way representing therefore the firm’s sunk costs. We have to make a clear distinction between fixed and sunk costs because they are not always the same. The firm can incur in three types of costs:

>Variable and nonsunk. These are costs that are said to be output sensitive because they depend directly from the output the firm wants to produce. (Raw materials are an example for that)

>Fixed and nonsunk. These costs are output insensitive but avoidable if the firm does not produce anything. Think about heating: a firm can turn on heating into its factory (so it is avoidable and therefore nonsunk) but the amount spent to heat the factory does not depend on how many units the factory is able to produce because the factory has to be heated up (and therefore is output insensitive).

>Fixed and sunk (output insensitive and unavoidable) height="207" src="file:///C:/Users/gianm/AppData/Local/Temp/msohtmlclip1/01/clip_image014.jpg" align="right" hspace="12" /Because in the short run the firm can change only labor (supposing capital is fixed) you can not have a tangency condition anymore. This means that the cost minimizing bundle of inputs is the one laying on the isoquant with y-value equal to the fixed amount of capital. In the example of the picture point A represents the cost minimizing quantity in the long run while point F represents the cost minimizing quantity in the short run. As you can see the isocost line passing through F is higher than the one passing through A meaning that the former is the isocost with a total cost higher than the latter.

In the short run, because the firm can not adjust the quantities of inputs freely, the total cost in order to produce Q0 will be higher than the one it would incur in the long run. The only case in which the short run and the long run total costs are equal happens when the quantity of capital is fixed exactly at the level of the cost minimizing bundle in the long run meaning, in the graphical example, that K1 is equal to the y-value of A. in the short run, when a firm wants to increase the quantity produced, the expansion path is not downward sloping nor upward sloping because as said, the quantity of capital employed can not be changed, and therefore there can not be any vertical movement when considering cost minimizing quantities.