Chapter 8 Costs Functions The economic cost of an input is the minimum payment required to keep the input in its present employment. It is the payment the input would receive in its best alternative employment. This cost concept is closely related to the opportunity cost concept (not talking about accounting costs). We usually assume that inputs are hired in perfectly competitive markets. The firm can get all the input it wants without affecting prices. The supply curve for an input is horizontal at the prevailing price. w Supply of the input = price of the input. Firm s demand for the input
Total Cost, Revenue, and Profit Total cost = C = w + vk (with only 2 inputs, capital and labor) TR = pq (with only 1 output) Then, economic profit is: pqwvk pf(k,) wvk Thus, economic profit is simply a function of K and, given that all prices (p, w, and v) and technology are fixed.
Cost Minimizing Input Choices (for given q) Assume for now that output has been determined to be q 0 and the firm wishes to minimize its cost. That is, the firm must choose a specific point on the q 0 isoquant. K q 0
FOC Cost will be minimized by choosing the point where RTS K (-slope of isoquant) equals the ratio of input prices (w/v). This happens when the rate of substitution in production equals the rate of substitution in the market. Min: C = w + vk st: q 0 = f(k, ) or q 0 f(k, ) = 0 = w + vk + (q 0 f(k, )) f w 0 f v 0 K K q 0 f (K, ) 0 is the increase in C when q 0 increases by one unit. is marginal cost, MC. The SOC require diminishing RTS K or q=f(k, ) strictly quasiconcave or K = f(,q 0 ) strictly convex.
Then w v f f K MP MP K RTS,K dk d So, RTS K should equal w/v for the minimum cost combination of inputs, or the slope of the isoquant (dk d) equals the slope of the isocost line (w v). MP w v the marginal product per dollar spent is equal for all inputs. Also, = marginal cost is the inverse of the above, w MP MP K v MP K 1 MC.
K C 1 v K * 0 Graphically C 3 > C 2 > C 1 * C 1 w C 1 C 2 C 3 q 0 K Minimum cost is C 1, which is the minimum cost to achieve q 0. The isocost line shows combinations of K and that can be purchased with fixed total cost. isocost line. The solution can be a corner point, but not usually unless the inputs are close substitutes (close to linear isoquants). Assuming perfect substitutes and RTS K > w/v, which input would not be used? (K!) C v w v
Dual: Output maximization subject to a cost constraint: Max: q=f(k, ) s.t.: C 1 = w + vk or C 1 w vk = 0 = f(k, ) + D (C 1 w vk) D represents the marginal product of one additional dollar of expenditure on inputs. It equals 1/. Solving the FOC yields K * and * as did the primal. K K * 0 * C 1 Output Maximum q o q -1
Can we derive a demand curve for by changing price (w) and looking at the resulting change in *?
The answer is yes, but the logic is somewhat different from the consumer s demand for a good. As w changes and * changes, the output level changes, which will change the market for q, which will change p (price of q). We cannot investigate the demand for an input without also considering the interaction of supply and demand for the output. The demand for the input is derived from the output market. Along the demand curve for, v and p are held constant. Therefore, the analogy between consumer and firm optimization is not exact. (Isoquants are not directly interpretable as revenue whereas indifference curves represent utility.) p y ΔU for consumer w q for firm p in market for q. Demand for is a derived demand from the market for q.
K 0 Expansion Path As the firm expands q, the cost minimization points trace out the expansion path. C 4 C 3 C 2 C 1 q 2 q 1 q 3 q 4 The Expansion Path shows how optimal input usage changes as output expands with v, w, and technology constant (isocost lines are parallel because v and w are constant). If the production function is homothetic, the Expansion Path will be linear. The shape of the isoquants determines the shape of the Expansion Path. The Expansion Path shows points of equal RTS K on the isoquants because the isocost lines are parallel. An Expansion Path that slopes toward an axis indicates an inferior input on the other axis; ie., use of the input actually declines as q increases. For example, as you produce larger and larger acreages of vegetables, your use of hand-held implements would decrease; hoes and unskilled labor are inferior inputs. K 0
Cost Functions come directly from the production function and prices. Total cost: C = C(v, w, q) Minimum Total Cost is a function of input prices and output quantity. Thus, the C function represents the minimum cost necessary to produce output q with fixed input prices. C represents the minimum isocost line for any level of q. It reflects the cost minimizing combination of inputs (K *, * ) for any given q. A total cost function is analogous to an expenditure function in consumer theory. Someone define an expenditure function.
Average Cost AC C q Marginal Cost AC is the cost per unit of output; AC = AC(v, w, q). MC C q MC is the change in C as output changes. It is the added cost for producing an additional unit of output. MC = MC(v, w, q).
Initially, we will hold v and w constant and look at how cost varies as q changes. This will give us standard two-dimensional graphs. With a production function that shows constant returns to scale (homogeneous of degree 1, or linear homogeneous), C will be linear with fixed input prices. MC will be constant and equal to AC. $ C $ AC = MC q q
However, typical (in theory) cost curves are ' sloped as follows: C 0 $ C is cubic Inflection point C C '' 0 and then C is concave; C convex AC = the slope of a cord from the origin to any point on C. Based on C, can draw MC C q. 0 $ q MC AC Slope of the cord = slope of C. q Thus, when C is concave, MC is declining; when C is convex, MC is increasing; and MC is at a minimum at the C inflection point. AC = MC for first unit of q. AC reflects lower MC of first units of q. Once MC crosses AC, AC turns up reflecting higher MC of later units of q. MC crosses AC at minimum AC.
Changes in Input Prices When input prices change relative to each other, the expansion path changes and the cost curves shift. But C is homogeneous of degree 1 in input prices, so doubling all input prices doubles C. This doubling of input prices would not affect q, *, K * or the Expansion Path,. Because C is homogeneous of degree 1, MC and AC will be also. Pure inflation will not affect input combinations or q, but it will affect C, AC, and MC. Mathematically and Graphically If C 1 = vk 1 + w 1 and v and w are multiplied by m, C 2 = mvk 1 + mw 1 = mc 1. K C 2 /2v=C 1 /v If v, w, and C double, optional point remains at A. A If w and v double, this isocost line changes from C 1 to 2C 1 = C 2, but *, K * and q * do not change. 2w w C 1 /w = 2C 1 /2w =C 2 /2w is the intercept. 2v v RTS K
Changes in a Single Input Price (relative prices of K and change). The slope of the isocost line will change, resulting in changes in the optimal input combination and changes in the expansion path. Input Substitution (q constant) Deals with how the optimal combination of K and (or K/) changes as w/v changes with q constant. Total Effect and its Direction (q changes) An increase in v or w will increase C. AC will also rise. MC will rise if the input is not inferior. From Footnote 9 on page 228 and Footnote 7 on page 226: MC v ( q) v 2 qv 2 vq ( v) q k ; q MC v k, so if q MC k is inferior, v 0, where from the Envelope Theorem q λ MC, v k, and is the optimal agrangian function from the cost minimization problem subject to an output constraint.
K w K %Δ An alternative elasticity of substitution is s v. w K w with q, v, and w constant (Partial Elasticity of Substitution). %Δ v v s is positive in a two-input world, but can be negative if three or more inputs. This elasticity is similar to the elasticity of substitution () developed earlier from the production function if we remember that at the optimal combination of K and w dk RTSK. K/ RTS K v d RTS K K/ In practical application, which would be easier to estimate, or s? A large s means the optimal input combination changes a lot as the price ratio changes, suggesting close to linear isoquants (close substitutes). K w w K v v arge s implies a large change in K/ for a change in w/v - close substitutes. q 0 q 0 Small s implies a small change in K/ for a change in w/v not close substitutes.
For many inputs s ij x x w w i j j i w w x x i i j j This formula gives a partial approach for use where many inputs are involved. Other input usages (other than inputs i and j) are not held constant in s ij but they are in ij.s ij does not have to be non-negative. Other inputs usages may change to give a net negative sign on s ij ; q is constant.
Contingent Demand for Inputs and Sheppard s emma Quantity of output is under the firm s control and actual input demand changes as output changes. However, cost minimization, subject to an output constraint, creates an implicit demand for inputs with quantity of output held constant. Contingent demand for an input (output-constant input demand) holds output constant similar to compensated demand for a good. Sheppard s emma is a result of the Envelop Theorem for constrained optimization. Sheppard s emma is that the partial derivative of C with respect to an input price gives the contingent demand function for that input (q constant). The envelop theorem and Sheppard s emma says that when the agrangian expression is at its optimum, C/ v If * vk * C(v,w,q) v * / v and * * * * w λ [q f(k, )], * C/ w / w. (v,w,q,λ) c K (v,w,q) v * Use the agrangian method to find K* and *. Or if you are given C, just take the partial derivative of C with respect to w and v to get the contingent demand functions for and K, respectively. C(v,w,q) w * (v,w,q,λ) w c (v,w,q)
Size of Shifts in Cost curves two factors: Cost Share The more important the input, the larger the cost curve response to a change in the input price. If the input makes up a large portion of total cost, an increase in its price will raise total cost substantially. Input substitutability High substitutability with other inputs will reduce the effect on the cost curves of a change in the price of single input, ie., large partial elasticity of substitution (s) will reduce the effect on C of a change in v or w.
Short-Run versus ong-run Costs Short run is a time period in which some inputs are variable, but at least one is fixed. This brings about the aw of Diminishing Returns to the variable inputs. ong run is a time period in which all inputs are variable. So far, evaluation of Economies of Scale and cost minimization have assumed the long run. In short run, we have some fixed cost, while in the long run all costs are variable. Not the same time period for all firms and industries. Not a specific time unit.
The text considers K as the fixed input in the short run. SC = vk 1 + w where K is fixed at K 1. Then vk 1 is short-run fixed cost, because it will not change in the short run. w represents short-run variable cost because can be changed in the short run to affect q. SFC vk1 So SC = SFC + SVC SVC w SFC has to be paid regardless of level of q (0 to ). SVC changes as changes to change q. If q = 0, SVC = 0 because = 0. If q = 0, SFC = vk 1.
vk 1 $ 0 Assuming any level of q requires some. vk 1 SC SVC SFC q This SC represents minimum short-run cost for each level of output, but not minimum C for each level of output. SFC same over all q. SVC = 0 if q = 0. SC = SFC + SVC. SFC has no effect on the shape of SC. Only one of the output levels in this graph shows the minimum total cost (C not SC) for K 1. Because K is fixed at K 1 and there is only one level of that is optimal for K 1, there will be only one point on SC where total cost (C) is minimum for K 1.
K K 1 SC 1 = C 1 SC 2 C 0 SC 0 C 2 A q 3 0 1 Expansion Path 2 q 0 q 1 q 2 SC 0 > C 0 SC 1 = C 1 SC 2 > C 2 With K fixed at K 1, only at output q 1 is total cost (C) minimized because only at point A is w/v = RTS. At any other quantity along the line K = K 1, SC is minimized but C is not minimized. For C minimization, we would need to use less K and more for points on K = K 1 to the left of 1 and more K and less for points on K = K 1 to the right of 1.
Per Unit Costs SC SAC q SC SVC SMC q q The concepts and relationships between SC and SAC and SMC are the same as described earlier for C, AC, and MC. Only variable costs affect MC. If SFC = α and SVC = q 2, then SC = α + q 2, SAC = α/q + q, and SMC SC q 2β q SVC q
$ SAC can be separated into fixed and variable parts. SMC SAC SAVC SAFC (Same vertical distance) 0 q SAFC SVC SAVC βq q SFC α SAFC q q SC α SAC βq q q SAC SAFC SAVC SAFC is a rectangular hyperbola that is asymptotic to 0. Thus, as q increases, SAVC approaches SAC, ie., SAVC is asymptotic to SAC. SMC passes through minimum SAVC and SAC. The cost curves (total and variable) reflect the shapes of corresponding physical product curves (production function, etc.).
Switching to the ong Run By changing the fixed factor (K) we can draw a large number $ of SC curves. SC 0 is drawn for K = K 0 SC 0 SC 1SC2 C Optimal q and C for K = K 0 q 0 q 1 q 2 Optimal q and C for K = K 1 q SC 1 is drawn for K = K 1 SC 2 is drawn for K = K 2 For a constant returns to scale production function, C would be a ray from the origin. At the q that corresponds to optimal for the fixed level of K, the SC curve will be tangent to C curve. At all other points, the particular SC is above C, reflecting the fact that SC represents costs higher than would be necessary if K were variable. The long run C is the envelope of the SC curves when K is allowed to vary. Every point on C is taken from a SC at its optimal combination of K and.
$ Average and marginal curves - For every level of K (fixed) there exists different SAC and SMC. SAC 0 SMC 0 SAC 0 SMC 1 SAC SAC ' 1 0 SMC 2 MC SAC 2 AC A evel of K indicates the size of plant. K 2 > K 1 > K 0 ' >K 0 q 0 =q 0 ( 0,K 0 ) q 0' =q 0' ( 0',K 0 ) q 0' =q 0' ( 0',K 0' ) q 1 = q 1 ( 1,K 1 ) q 2 = q 2 ( 2,K 2 ) If the firm has K 0 and is producing q 0, it could reduce cost per unit of output by increasing labor to 0 ' at minimum SAC 0 ; but q 0 ' could be produced at a lower cost with K 0 ' reflected by SAC 0'. Further reduce average cost by increasing to minimum SAC 0, which q could be produced with more K, etc. Each SAC is tangent to AC at the optimal amount of for that level of fixed K, but that point is not the minimum point for that SAC except at point A where SAC is tangent to the AC curve at minimum AC and at minimum SAC 1. q
All SMC curves cross their respective SAC curves at minimum SAC. SMC curves cross MC at the C minimizing level of for the fixed level of K. MC is made up of many optimal points from the SMC curves. At q 1, minimum SAC 1 equals minimum AC, and SMC 1 equals MC. The most efficient scale is K 1 capital with q 1 produced. The slopes of the SMC curves are greater than the slope of the MC curve because of diminishing returns with fixed K. The slopes of the SAC curves reflect diminishing returns also. Only at q 1 will the firm be satisfied in the long run because, if the firm is at a point other than the minimum on any SAC curve, it can reduce costs for that level of fixed K by changing and q to minimum SAC. But for all SAC curves other than SAC 1, the q produced at the minimum point on the SAC curve could be produced with another level of K at a lower cost.