Chapter 8 Cost Functions Done

1 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 L. Total cost , Revenue, and Profit Total cost = C = wL + vK (with only 2. inputs, capital and labor). TR = pq (with only 1 output). Then, economic profit is: pq wL vK pf (K, L) wL vK. Thus, economic profit is simply a function of K and L, 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 q0 and the firm wishes to minimize its cost .

2 That is, the firm must choose a specific point on the q0 isoquant. K.. q0. L. cost will be minimized by choosing the point where RTSLK (-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 = wL + vK is the increase in C. st: q0 = f(K, L) or q0 f(K, L) = 0 when q0 increases by = wL + vK + (q0 f(K, L)) one unit. is marginal cost , MC. f w 0. L L The SOC require diminishing f RTSLK or q=f(K, L) strictly quasi- FOC v 0. K K concave or K = f(L,q0) strictly convex. q 0 f (K , L) 0.. w f L MPL dK. Then RTS L , K . v f K MPK dL. So, RTSLK should equal w/v for the minimum cost combination of inputs, or the slope of the isoquant (dK dL) equals the slope of the isocost line ( w v). MPL MPK 1.. w v the marginal product per dollar spent is equal for all inputs. Also, = marginal cost is the inverse of the above, w v MC. MPL MPK. K. Graphically C3 > C2 > C1 Minimum cost is C1, which is the minimum cost to C1 achieve q0.

3 V . The isocost line shows K* q0 combinations of K and L. C1 C2 C that can be purchased with 0 3. L* L fixed total cost . C1. w C w K L isocost line. v v The solution can be a corner point, but not usually unless the inputs are close substitutes (close to linear isoquants). Assuming perfect substitutes and RTSLK > w/v, which input would not be used? (K!). Dual: Output maximization subject to a cost constraint: Max: q=f(K, L). : C1 = wL + vK or C1 wL vK = 0. = f(K, L) + D(C1 wL vK). D represents the marginal product of one additional dollar of expenditure on inputs. It equals 1/ . Solving the FOC yields K* and L* as did the primal. C1. K Output Maximum K* qo q-1. 0 L. L*. Can we derive a demand curve for L. by changing price (w) and looking at the resulting change in L*? The answer is yes , but the logic is somewhat different from the consumer's demand for a good. As w changes and L*. changes, the output level changes, which will change the market for q, which will change p (price of q).

4 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 L, 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.). py U for consumer w q for firm p in market for q. Demand for L is a derived demand from the market for q. Expansion Path As the firm expands q, the cost minimization points trace out the expansion path. The Expansion Path shows how optimal input usage changes as output expands with K C4 v, w, and technology constant (isocost lines C3.. C2 are parallel because v and w are constant). C1. q2. q3. q4. If the production function is homothetic, the Expansion Path will be linear. The shape of the isoquants determines the shape of the q1 L Expansion Path. The Expansion Path shows 0.

5 Points of equal RTSLK on the isoquants because the isocost lines are parallel. An Expansion Path that slopes toward an axis indicates an K. 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 0. L. inferior inputs. 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*, L*) for any given q. A total cost function is analogous to an expenditure function in consumer theory. Someone define an expenditure function. Average cost C AC is the cost per unit of output;. AC AC = AC(v, w, q).

6 Q Marginal cost C MC is the change in C as output MC changes. It is the added cost for q 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 C. C '' 0 and then 0. C is concave; C convex $ Inflection point AC = the slope of a cord from the origin to any point on C. Based on C, can draw MC C q . Thus, when C is concave, MC is q declining; when C is convex, MC is $ increasing; and MC is at a minimum at the C inflection point. MC. AC 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 Slope of the cord = slope of C.

7 MC of later units of q. MC crosses AC at minimum AC. q 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, L*, 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 C1 = vK1 + wL1 and v and w are multiplied by m, C2 = mvK1 + mwL1 = mC1. K If v, w, and C double, optional point C2/2v=C1/v remains at A.. A If w and v double, this isocost line changes from C1 to 2C1 = C2, but L*, K* and q* do not change. 2w w RTS. LK. C1/w = 2C1/2w =C2/2w is the intercept. 2v v L. Changes in a Single Input Price (relative prices of K and L change). The slope of the isocost line will change, resulting in changes in the optimal input combination and changes in the expansion path.

8 Input Substitution (q constant) Deals with how the optimal combination of K and L (or K/L) 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 ( q) 2 2 ( v) k MC k MC. ; , so if k is inferior, 0, v v q v v q q q v q v where from the Envelope Theorem q MC, v k, and is the optimal Lagrangian function from the cost minimization problem subject to an output constraint. K . w % K. L. An alternative elasticity of substitution is s v L. w K % w with q, v, and w constant (Partial Elasticity of Substitution). v L 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 L w dK. RTSLK . K/L RTS. LK. v dL RTS K/L. LK.

9 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). w Large s implies a w Small s implies a K large change in K/L K . v for a change in w/v - v small change in K/L for a change . close substitutes. q0 in w/v not close q0 substitutes. L. L. For many inputs xi w . w j This formula gives a partial x s ij . j i approach for use where many w j xi inputs are involved.. wi xj Other input usages (other than inputs i and j) are not held constant in sij but they are in ij. sij does not have to be non-negative. Other inputs' usages may change to give a net negative sign on sij; q is constant. Contingent Demand for Inputs and Sheppard's Lemma 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.

10 Contingent demand for an input (output-constant input demand) holds output constant similar to compensated demand for a good. Sheppard's Lemma is a result of the Envelop Theorem for constrained optimization. Sheppard's Lemma 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 Lemma says that when the Lagrangian expression is at its optimum, C/ v */ v and C/ w */ w. Use the Lagrangian method to find K*. If * vK* wL* *[q f(K* , L* )], and L*. Or if you are given C, just take the partial derivative of C with respect C(v,w,q) * (v,w,q, ) to w and v to get the contingent demand Kc (v,w,q). v v Functions for L and K, respectively. C(v,w,q) * (v,w,q, ) c L (v,w,q). w w 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.