Intermediate Microeconomics

                                                                                                                                                Professor Yongmin Chen

 

Topic 6: Input Choices and the Cost of Production

The Optimal combination of Inputs

     We have seen that in order to produce certain amount of output, it is often possible to use different combinations of inputs, as is represented by the isoquant.  Now the question is: what input combination should a firm choose to produce the desired output?  This is the problem of determining the optimal combination of inputs.

     Suppose the production uses two inputs: labor (L) and capital (K).  The price of labor is w, and the price of K is r.  That is, each unit of labor costs w, and each unit of capital costs r.  If L units of labor and K units of capital are used, the total cost would be

     wL + rK = C

     Clearly, given w , r, and C, there are different combinations of (L,K) that would require the same total cost C.  The line that represents all the input combinations with a total cost C is called the isocost line.  For different C, there corresponds to different isocost lines.  An isocost line that is closer to the origin represents a lower level of total cost.  The slope of the isocost line is -w/r.

     To produce a certain amount of output, one essentially selects a point on an isoquant.  Of course, different points on an isoquant will generally have different costs.  To choose the optimal input combination is to choose the input combination that requires least cost.  This is achieved by finding out the point on the isoquant that is tangent to an isocost line.  See illustration in class.

     Since MRTSLK = -slope of the isoquant, and since the slope of the isocost line is -w/r, the conditions for cost minimization in producing certain output X* can be written as:

           MRTSLK = w/r

           X(L,K) = X*

     Since MRTSLK = MPL/MPK, the above condition can also be  written as

            MPL/MPK = w/r, and X(L,K) = X*.

Example 6-1. Suppose the production function is X = L0.5K0.5.  Suppose w = 1 and r = 9.  If the firm wants to produce X = 9, what will be the cost-minimizing input combination?

Example 6-2. Suppose the production function is X =  5LK.  Suppose w = 2 and r = 8.  If the firm wants to produce X = 80, what will be the cost-minimizing input combination?

 

Example 6-3. At the current level of output for the Fellows Corporation, the marginal product of labor is 15 while the marginal product of capital is 45.  Each unit of labor costs $3 and each unit of capital costs $4.  Based on this information, we know that the firm is employing

A. too much labor and too little capital

B. too much capital and too little labor.

C. too little of both labor and capital.

D. too much of both labor and capital.

E. none of the above.

 

     It is obviously very important for a firm to know the cost of production.  The proper way to measure costs for economic analysis is to use opportunity costs, the highest value that can be achieved by putting the resources in alternative uses.

     Recall that when we discussed production, we distinguished short run and long run.  In the short run, at least one input is fixed; while in the long run all inputs are variable.  We shall discuss costs in a similar way.

 

Short-run Costs

     In the short run, total cost is the sum of variable cost and fixed cost:

         TC = VC + FC

     If labor is the variable input, the quantity of labor is denoted by L, and wage (price of labor) is w, then VC = wL.

     If capital (K) is the fixed input, and the price of capital is r, then FC = rK.

     Recall that when variable input L changes, the maximal amount of output X that can be produced changes.  This is how we derived the total product curve of labor: X = X(L).  Now we want to pose the question in a different way.  We want to know for different levels of output, what is the smallest quantity of variable input L that is required to produce the output.  This relationship is described by the input requirement curve: L = L(X).   The point where APL is maximized on the X(L) curve corresponds to the point where L/X is minimized on the L(X) curve.

     Variable cost can be obtained by:

           VC(X) = wL(X)

     See a graph of VC(X).

     Total cost as a function of output is:

           TC(X) = VC(X) + FC

     See a graph of TC(X).

     Average variable cost is variable cost per unit of output.

           AVC = VC/X

     Since VC = wL(X) and APL = X/L, we have:

           AVC = w/APL

     Therefore AVC and APL are inversely related.  When APL is maximized, AVC is minimized.  See a graphic illustration.

     When output changes, cost often also changes.  Marginal cost is the change in cost divided by an incremental change in output.

           MC = DVC/DX = DTC/DX

     MC is equal to the slope of the VC curve (or the TC curve).

     When labor is the variable input, we can express MC as:

           MC = w/MPL

 Therefore MC and MPL are inversely related.  When MPL is maximized, MC is minimized.

     Relations between MC, VC, and AVC.

     Total average cost, AC, is:

           AC = TC/X = AVC + AFC

        A graph of short-run cost curves.  Notice the MC curve passes through the minimum points of the AVC and AC curves.   At the point where AVC is minimized,  MC = AVC.

Example 6-4. Suppose the production function is given by X = L0.5K0.5, and for this production function, MPL = 0.5L-0.5K0.5.  Assume w=1, r=4, and K=1 in the short run.  Derive the functions and sketch the graphs of the following:

(a) Marginal and average product curves for labor

(b) The input requirement curve for labor: L = L(X)

(c) VC(X) and TC(X)

(d) AVC, AC, MC.

    

Example 6-5.  The break-even point.  Suppose a firm's average variable cost is a, its fixed cost is F, the price of output is p.  What is the minimum amount of output that will enable the firm to break even?

 

Long-run Cost Functions

     The long-run cost functions describe the relationships between costs and output levels in the long-run, that is, in such a time period that all inputs can be changed.

     The long-run total cost function, LTC(X),  can be derived from the isoquant map.  See illustration in class.

     The long-run average cost: LAC = LTC/X.

     The long-run marginal cost:  LMC = dLTC/dX

     At the point where LAC is minimized, we have LAC = LMC.

Relationships between long-run cost and short-run cost:  Each short-run cost curve will be tangent to the long-run cost function from above at one point.

 

Example 6-6. According to economic analysis, the ratio of marginal products  of  labor over capital should be equal to w/r, the ratio of the prices of labor over capital, in order to minimize production cost.

(a) According to this analysis, what will happen to the number of workers employed if wage rate goes up? 

(b) A labor union representative disagrees with the conclusion from the economic analysis.  He states that the theory is wrong because according to his observations, in several occasions there is no drop in employment following wage increases.  Explain why his observations might indeed be true but there is nothing wrong with the theory.

 

Economies of Scale and Economies of Scope

     When all inputs are varied, production scale is changed. 

     Economies of scale are said to occur when average costs fall as production scale is increased.  Economies of scale may be caused by increasing returns to scale, but these two need not be the same.

     If average cost rises as production scale increases, then we say diseconomies of scale occur. 

     Empirically, it is often the case that average cost first falls as production scale increases, and then either becomes constant or rises.  That is, the LAC curve tends to be L-shaped or U-shaped.

     When the cost of producing two products jointly is lower than the cost of producing them separately, we say there exist economies of scope.  Suppose the cost of producing X and Y is C(X,Y).  Economies of scope occur if C(X,Y) < C(X,0) + C(0,Y).