I finished the previous section arguing, among other things, that, within a general equilibrium economy, changes in household preferences, increasing or decreasing quantities of output demanded at all prices, solicit immediate adjustments by firms in output quantities produced, accompanied by changes in financial allocations from households, such that firms will produce the exact quantities demanded by households. Provided firms conform to our assumptions regarding the homotheticity of production functions and constant returns to scale/constant average total costs, such changes may generate no changes in output prices, if transformations of household preferences do not simultaneously affect the relative prices of labor and capital. In this section, I will consider the possibility that a transformation of household preferences or changes in production technologies will impact relative factor prices. In order to reinforce the conclusions made here, I will develop a quantitative example, utilizing a Cobb-Douglas type production function to evaluate the effect of a change in factor prices.
Reiterating one of our larger assumptions about general equilibrium economies, firms not only operate as price takers in output markets among large numbers of other firms, but approach factor markets as price takers because they draw from labor and capital markets in competition with all other firms in their industry and with all other firms in all other industries. In this light, the relative intensities of labor and capital employment across all industries must impact, in part, the relative compensation rates for the production factors. If, at some moment in time, households demand relative large quantities of labor-intensive goods and services, then a sudden shift in preferences toward goods and services utilizing capital more intensively in their production should be expected to raise the relative price of capital across all markets. Given the integrated nature of a general equilibrium economy, such a change would impact the equilibrium levels of output in every market.
Playing this scenario out graphically, figure 20 illustrates the effects of a shift in aggregate demand for production factors, raising the relative price of capital.
Elaborating on figure 20, the increase in the relative price of capital is illustrated by the pivoting of isocost IC11 downward into IC11', with a common maximum quantity of labor and a reduced maximum quantity of capital for the same level of total cost TC11 (thus, TC11 = TC11'). The increase in capital costs shifts the firm from isoquant curve I1(x11) to I0(x10), where x11 > x10, without altering its total costs. In this manner, the average total cost per unit of output for the firm increases.
I attempt to show two separate effects of the transformation of factor market prices here. First, holding output levels constant, an increase in the relative price of capital generates a factor substitution effect evident in the movement along isoquant curve I1 from point A to C, a point of tangency with isocost curve IC12(TC12), where TC12 > TC11. Isocost IC12 is parallel to IC11', denoting a common set of relative prices defining the slope of the curve (-Pl/Pk). It, thus, lies on an expansion path EP2 constituted as a locus of all profit maximizing/cost minimizing factor combinations at the new set of relative factor prices. At point C, the firm selects lC labor (lC > lA) and kC capital (kC < kA). This factor combination reflects the fact that, as the relative price of capital increases, the firm has an incentive to substitute labor for capital in order to arrive at a lower level of total costs relative to its factor combination at point A. Thus, holding the quantity of output constant, the firm maximizes profit/minimizes costs by renting more labor and less capital.
The factor substitution effect becomes evident when we ask how much of each factor of production would the firm rent at the new set of relative prices if it were to continue to produce the same quantity of output. By contrast, the scale effect of a change in factor prices measures the extent to which, given a substitution between factors, the firm may be compelled to reduce its employment of both factors if it continues to produce at the same initial level of total costs. As such, the scale effect measures the degree to which a change in factor prices will affect output levels. This effect is illustrated by the movement along expansion path EP2 from point C to point B, resulting in the decrease in output of (x11 -x10). At output level x10, the firm selects lB labor (lB < lC) and kB capital (kB < kC). Thus, accounting for the inability of the firm to maintain its previous level of output under the elevated capital prices, the firm reduces its employment of both labor and capital in order to maximize profit/minimize costs under the same initial total cost level.
Proceeding to the market level, we must account for the fact that the change in fact prices has reduced output from all representative firms in the industry (because, given our conclusion in the last section derived from the general equilibrium assumption of perfect information, all firms operate with identical production functions). Moreover, firms continue to operate with constant average total costs per unit of output except that the level of average total costs increases relative to the initial level. Hence, we get an upward shift in the market level supply function, shown in figure 21.
Arguing out the connection between figures 19 and 20, it might make sense to interpret X11' as the aggregation of x10s in figure 19, implying that the effect of an increase in the relative price of capital, barring any other change in household preferences, is a decrease in the quantity of commodity 1 supplied and demanded by households in competitive equilibrium. The higher output price P11' likewise reflects the elevated price of capital as a component to the cost functions of all representative firms within the industry.
Having laid out this graphical example of an increase in the relative price of capital, certain outcomes remain unclear beyond the specification of factor substitution and scale effects. For example, it is certain that the change in relative prices will place the firm on expansion path EP2 and that, given the increase in the relative price of capital, such a shift between expansion paths will be accompanied by an increase in average total costs, leading to shift of the market supply function. On the other hand, the exact level of output and total cost level at which the firm will actually produce remains unknown. That is to say, the specification of the above effects proceeds as if the firm will be constrained by its level of household financing to produce at the same initial level of total costs. It may be the case that households demand higher or lower quantities of output than x10 from the firm. It may be the case that households seek quantities of output higher than x11. This is ultimately a question that must be answered by households through their continuous negotiation of output quantities and factor supply levels, and determinations of the relative prices that will bring everything into equilibrium across all markets (in Walras' French terminology tâtonnement, implying jostling/groping along between competing agents toward a resolution). The one thing that we know for a fact is that the previous level of output is now more costly for households to obtain from firms and that, subsequently, if households seek to consume a constant level of output from this industry, then, ceteris paribus (all other things being equal), they will need to supply larger levels of both labor and capital, reducing the relative prices of both factors, in order to achieve the same output. I will approach this dimension in the next section discussing factor market supply decisions directly.
Proceeding beyond our graphical example, I want to develop a quantitative example here that lays out both the methodology heretofore presented on profit maximization/cost minimization in relation to the household decisions on output quantities demanded and responds to speculative changes in factor prices. Thus, let us say that we have a firm operating in a single, perfectly competitive industry, say coffee roasting, in an economy that includes a large number of different industries competing with one another for relatively scarce quantities of labor and capital. The firm operates with the following production function:
xcoffee = 4 (k^.6)(l^.4)
This production function is a Cobb-Douglas form. Such production functions take the generalized form:
x1 = b (k^α)(l^β)
Where b is an adjustment factor for total factor productivity, and α and β are output elasticities, measuring the effect of a change in output in relation to a change in quantities of capital and labor respectively. This construction is particular useful within the larger framework of general equilibrium analysis because, assuming each of the output elasticities take on a value between 0 and 1, any isolated change in one of the factors, holding the other factor constant, will manifest diminishing marginal productivity for the adjusted factor. Conversely, if the sum of the two output elasticities is equal to 1, then any proportionate change of both production factors will result in a proportional increase in output (i.e. constant returns to scale/constant average total costs). Cobb Douglas production functions are, thus, homothetic.
In regard to factor markets, the firm faces prices of $20 per hour of capital usage and $10 per labor hour. Therefore, we can express the firm's cost function accordingly:
TCcoffee = $20k + $10l
Following our methodolgy to determine profit maximizing/cost minimizing combinations of capital and labor by taking total differentials, the production function yield the following first order partial derivatives for capital k and labor l:
δxcoffee/δk = 2.4[(l^.4)/(k^.4)] (Marginal Product of Capital)
δxcoffee/δl = 1.6[(k^.6)/(l^.6)] (Marginal Product of Labor)We can define the total differential as:
δxcoffee = (δxcoffee/δk)Δk + (δxcoffee/δl)Δl
Plugging in our first order partial derivatives/marginal products and setting the change in output equal to 0 (again, because we are attempting to define a relationship between the marginal products for each factor of production at fixed levels of output), we get:
0 = 2.4[(l^.4)/(k^.4)]Δk + 1.6[(k^.6)/(l^.6)]Δl
Algebraically manipulating this equation, we get:
Δk/Δl = -(2/3)(k/l) = MRTSk
Where the marginal rate of technical substitution of capital (MRTSk) expresses the slope of the isoquant surfaces for each given fixed level of output. Again, the term is negative, reflecting the negative slope of the isoquant surfaces and, thus, the necessity to substitute more of one factor of production as we reduce the other in order to maintain constant levels of output. It is, therefore, the first piece in the determination of a profit maximizing/cost minimizing combination of labor and capital in the production of coffee. In order to arrive at this combination, we need to further account for the firm's cost function and the effect of factor market prices on the firm's decision to rent labor and capital. Following the same methodology, we need to define the total differential for the cost function. The cost function yields the following first order partial derivatives:
δTCcoffee/δk = $20
δTCcoffee/δl = $10
Simply speaking, the effect of any unit change for either factor of production on total cost will be equal to its market price per unit. Utilizing, again, the first order partial derivatives from the cost function in order to define the total differential and setting the change in total costs equal to 0, we get:
0 = $20Δk + $10Δl
Algebraically manipulating the equation, we derive the following:
Δk/Δl = -(1/2) = -Pl/Pk
Expressing the slope of isocost surfaces as a ratio of factor prices.
From here, we need to relate the marginal rate of technical substitution of capital (slope of the isoquant) to the ratio of factor prices (slope of the isocosts) in order to define a relationship at which the firm will enjoy a profit maximum/cost minimum (where the two slopes are equal). Setting the two rates of change above equal, we arrive at:
Δk/Δl = -(1/2) = -(2/3)(k/l)
Algebraically manipulating, we arrive at the following profit maximizing/cost minimizing relation between capital and labor:
k = (3/4)l
Thus, the firm maximizes its profits and minimizes its costs along an expansion path defined by this constant relation between capital and labor for all quantities of output. Graphically, this can be illustrated as a ray from the origin with a slope of 3/4.
This concludes our arithmetic analysis of the production conditions faced by the firm. As I have argued so far, our assumption of perfect information assures us that every firm in the coffee roasting industry produces coffee using this production function because it must be the most efficient possible way to produce coffee. Therefore, our analysis of the production relations of this one firm mirrors the production relations for every firm in the market. If all firms realize that, in order to maximize profits, they must rent 3 units of capital for every 4 units of labor they rent, then the sole question that remains for firms concerns where they are along their expansion paths. This is question answered not by the firms themselves but, jointly, by the households that demand coffee and by the households that supply the finances to firms that produce coffee. We can begin this stage of the analysis from either side (the consumption side or the financing side) because, ultimately, in a general equilibrium economy, both sides must be harmonized. For purposes of argument, I am going to begin with the financiers. Let us say that, given perfect knowledge of how much coffee households want to consume and perfect knowledge of how much each coffee roasting firm would be able to contribute to the market while maximizing profits, these households supply the firm with $900 to rent labor and capital for some given period of production. Incorporating this information into our cost function, along with our profit maximizing/cost minimizing condition, we get:
$900 = $20[(3/4)l] + $10l
Solving separately for capital k and labor l, we arrive at the following combination:
l = 36
k = 27
Plugging these quantities into our production function, we get:
xcoffee = 4 (27^.6)(36^.4)
Yielding the (rounded off) total output:
xcoffee = 121
We can illustrate this result graphically through figure 22.
Let us say that there are 3000 identical firms in the market for coffee roasting. If each produces 121 units/pounds of coffee for this production period, then the market will consume 363,000 pounds of coffee. Moreover, each of these firms faces an identical average total cost for producing each pound of coffee at all quantities along its expansion path:
ATCcoffee = TCcoffee/x1 = $900/121 = $7.44
With this information in mind, given our assumption that price competition between firms will force all firms to charge their average total costs per unit of output, we can illustrate the market outcome graphically through figure 23.
Where I have penciled in an arbitrary non-linear downward sloping demand schedule to speculate on the nature of household consumption demand for roasted coffee under assumptions about diminishing marginal utility in coffee consumption that seem, at least in my mind (as an enthusiastic caffeine addict), to be fairly reasonable.
We can also arrive at this outcome by proceeding in the opposite direction (and, in a sense, given the knowledge possessed by household financiers, we already have!). If across the economy consuming households demand, for any given period, 363,000 pounds of roasted coffee, then, given 3,000 identical firms producing an identical/homogeneous product, each firm must produce 121 pounds of roasted coffee. Plugging this quantity and our profit maximizing condition into our production function, we get:
121 = 4 [(3/4l)^.6](l^.4)
Solving this equation separately for capital and labor, we arrive at the same factor combination that we find by proceeding in the other direction (i.e. 36 units of labor and 27 units of capital). Finally, plugging these quantities into our cost function, we would arrive at a total cost of $900, the financing for which households would need to supply to firms.
Now, to evaluate the effect of a change in factor prices, let us say that something happens in another market, say the market for meat products. This market has been affected by a rapid change in technologies that allows firms to produce meat products with much smaller quantities of labor (i.e. automation of production lines). Such a change has led firms to substitute larger quantities of capital for labor and, as a consequence, a relative abundance of labor now exists within labor markets, driving the price for labor down, ceteris paribus. For purposes of our example, let us say the price of labor goes down to $7.50 per labor hour from $10.
Beginning our analysis of the effects of this price change on production levels and total costs for the firms in this industry, we need to first assess the effect of a reduction in labor costs on the cost function of our representative firm. In contrast to my initial graphical example, where capital costs were increasing, the firm in this case is enjoying lower labor costs and, therefore, a reduction in its average total costs at all quantities of output. With this in mind, I will, first, ask how much labor and capital our representative firm will rent under the new set of relative factor prices and how much output this new combination will produce under the assumption that the firm receives the same level of financing from households. To do this, I will need to repeat my total differential analysis with respect to the cost function. I do not need to repeat the procedure for the production function because the technological conditions that govern coffee production have not changed. Thus, the firm now faces the cost function:
TCcoffee' = $20k + $7.50l
Differentiating, I get the following first order partial derivatives:
δTCcoffee'/δk = $20
δTCcoffee'/δl = $7.50
The new total differential for the cost function is:
δTCcoffee' = $20Δk + $7.50Δl
Setting the change in total cost to 0 and algebraically manipulating this equation, we get the following ratio of factor prices:
Δk/Δl = -.375 = -Pl'/Pk
Reintroducing the marginal rate of technical substitution of capital (MRTSk) derived from the production function and equalizing our two rates of change, we get:
Δk/Δl = -.375 = -(2/3)(k/l)
Algebraically manipulating, we arrive at the new profit maximizing/cost minimizing condition for the firm, defining the slope of its new expansion path:
k = (.5625)l
Now, working under the assumption that the firm receives the same quantity of financing from households, we get:
$900 = $20[(.5625)l] + $7.50l
Solving for capital k and labor l, we get:
l = 48
k = 27
Inserting these quantities into our production function, we get:
xcoffee' = 4 (27^.6)(48^.4)
Solving for Xcoffee', our (rounded off) new output quantity:
xcoffee' = 136
Finally, dividing our fixed level of total costs $900 by our new output quantity of 136, we get the following average total cost:
ATCcoffee' = TCcoffee'/x1' = $900/136= $6.62
This solution represents the net effect of a change in the price of labor under the assumption that the firm will continue to receive a constant level of financing from households to undertake production. In order to complete my analysis and enable myself to disaggregate the factor substitution and scale effects of the price change, I now have to ask, conversely, how much labor and capital the firm would rent and what level of total costs it would incur if households demanded a constant quantity of output from each firm of 121 pounds of coffee. Thus, starting this time from the production function, I set the quantity produced by the firm at 121 pounds, incorporating our new profit maximizing condition:
121 = 4 ([(.5625)l]^.6)(l^.4)
Solving for capital k and labor l, we get (rounding off):
l = 43
k = 24
Inserting this combination into our cost function, we get:
TCcoffee'' = $802.50
And, finally, dividing the new total cost by the fixed output quantity, we get the average total cost:
ATCcoffee'' = TCcoffee''/x1 = $802.50/136= $6.63
Confirming (more or less, after rounding calculations off!) that our two calculations (the first holding total costs constant, the second holding output quantities constant) are both on the same expansion path and that, at every quantity along this path, the firm is minimizing costs/maximizing profits subject to the new set of factor prices that it faces. Figure 24 attempts to capture this change in factor prices graphically.
Where the dashed isocost curve IC11' represents the financing/cost constraint of the firm when we hold the quantity produced fixed at 121 pounds of coffee with reduced labor costs and the parallel solid isocost curve IC12 represents its financing/cost constraint when we hold the financing level constant at $900 with reduced labor costs. Before evaluating the effect of this price change on the market as a whole, I want to measure the factor substitution and scale effects based on the calculations presented above.
First, holding output levels constant at 121 pounds of coffee and asking the effect of a change in labor prices on the amount of labor and capital rented by the firm, we see that the firm substitutes an additional 7 units of labor (43 - 36 = 7) for 3 units of capital (24 - 27 = -3). This is the factor substitution effect of the change in the price of labor. If we proceed from this amended combination of labor and capital at new factor prices (i.e. l = 43, k = 24) and allow the financial/cost constraint faced by the firm to increase to its original level of $900, the firm adds another 5 units of labor (48 - 43 = 5) and 3 units of capital (27 - 24 = 3). This is the scale effect of the price change for labor. The net effect of the price change on labor rented by the firm is an increase of 12 units (addition of 7 units measured by the factor substitution effect plus additional 5 units measured by the scale effect). The net effect of the price change on capital rented by the firm is zero change (loss of 3 units measured by the factor substitution effect plus addition of 3 units by the scale effect).
The market level effects of a decrease in the price of labor may be more uncertain. Let me explain why. The one thing that we know here is that the average total costs faced by firms will decrease to (around) $6.62 per pound of coffee. This is what consuming households will pay per pound of coffee. We do not, however, know how much coffee households will want to consume at this price because I have not specified the outcomes of multifarious, diverse individual household utility functions, aggregating to define the market demand function. We can assume predictively that households will experience various degrees of diminishing marginal utility as they continue to consume larger quantities of coffee. In this manner, the market demand function will slope downward, but we do not know the extent to which market demand will exhibit elasticity with respect to a change in the price of coffee beyond a general assumption that demand will not be perfectly elastic (i.e. a horizontal demand function). In any case, within the larger structure of a general equilibrium economy, we will hold to the conclusion that the market level demand for coffee will, in some manner, be harmonized with the market supply of money capital/financing to coffee roasting firms. This means that, whatever consuming households are willing to consume will be communicated to households supplying money capital to firms, such that the total cost constraint of firms along their expansion paths will enable the 3,000 identical firms roasting coffee in the market to produce exactly what households are willing to consume. Figure 25 illustrates such an outcome without assigning a number to the speculative equilibrium quantity.
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