Microeconomic Analysis

Microeconomic Analysis
Hal R. Varian Third Edition Microeconomic Analysis

CHAPTER 5 COST FUNCTION

Dr. Yarmohamadian Art University of Isfahan
COST FUNCTION 5-1- Average and marginal costs Ex: short-run Cobb-Douglas cost functions Ex: Constant returns to scale and cost function 5-2- geometry of costs Example: Cobb-Douglas cost curves 5-3- Long-run and short-run cost curves 5-4- Factor prices and cost functions 5-5- envelope theorem for constrained optimization Ex: Marginal cost revisited 5-6- Comparative statics using cost function Dr. Yarmohamadian Art University of Isfahan

intro cost function measures min cost of producing a given level of output for some fixed factor prices summarizes information about technological choices available tell us a lot about nature of firm's technology. describing economic possibilities of a firm As prod function describing technological possibilities of production Dr. Yarmohamadian Art University of Isfahan

5.1 Ac and MC structure of cost function It can always be expressed simply as value of conditional factor demands This says: minimum cost of producing y units of output is cheapest way to produce y. Dr. Yarmohamadian Art University of Isfahan

5.1 Ac and MC In short run, some of factors of production are fixed. xf be vector of fixed factors, xv vector of variable factors, w break up into w = (wv, wf), short-run conditional factor demand functions will generally depend on xf, so we write m as xv (w, y, xf). Dr. Yarmohamadian Art University of Isfahan

5.1 Ac and MC Dr. Yarmohamadian Art University of Isfahan

5.1 Ac and MC When all factors are variable, firm will opt choice of xf. Hence, long-run cost function only depends on factor prices and level of output this long-run function in terms of short-run cost function in following way. xf (w, y) be opt choice of fixed factors, xv(w, y) = xv (w, y, xf (w, y)) be long-run opt choice of variable factors. Dr. Yarmohamadian Art University of Isfahan

5.1 Ac and MC LAC equals LAVC since all costs are variable in long-run; LFC are zero Long run and short run are relative concepts Dr. Yarmohamadian Art University of Isfahan

EX: short-run Cobb-Douglas cost functions
Dr. Yarmohamadian Art University of Isfahan

EX: CRS and cost function
If production function is CRS cost function should exhibit linear in output if you want to produce twice as much output it will cost you twice as much. n cost function may be written as c(w, y) = y.c(w, 1). Let x* be a cheapest way to produce one unit of output at prices w so that c(w,1) = wx*. n I claim that c(w, y) = w.y.x* = y.c(w, 1). Dr. Yarmohamadian Art University of Isfahan

EX: Constant returns to CRS
Notice y.x* is feasible to produce y since technology is CRS. Suppose y.x* does not minimize cost; instead let x' be cost-min bundle to produce y at prices w so that w.x' < w.y.x*. n w.x'/y < w.x* and x'/y can produce 1 since technology is CRS. This opposes definition of x*. AC=AVC=MC Dr. Yarmohamadian Art University of Isfahan

Constant returns If technology exhibits increasing returns to scale, n cost will increase less than linearly with respect to output, so AC will be decreasing in output If technology exhibits decreasing returns to scale, n cost will increase more than linearly with respect to output, so AC will be increasing in output Dr. Yarmohamadian Art University of Isfahan

Constant returns A given technology can have regions of increasing, constant, decreasing returns to scale, Similarly cost function can increase less rapidly, equally rapidly, or more rapidly than output, So AC may decrease, remain constant or increase over different levels of output Dr. Yarmohamadian Art University of Isfahan

5.2 geometry of costs some of properties of cost function in two stages First, under assumption of fixed factor prices write cost function as c(y). Second, when factor prices are free to vary Dr. Yarmohamadian Art University of Isfahan

5.2 geometry of costs Since factor prices to be fixed, costs depend only on level of output and useful graphs can be drawn that relate y & C TC curve is always assumed to be monotonic in output AC curve, can increase or decrease with output, depending on wher total costs rise more than or less than linearly. Dr. Yarmohamadian Art University of Isfahan

5.2 geometry of costs most realistic case, in short run, AC curve first decreases and n increases. Because In short run cost function has two components: fixed costs variable costs. So how costs factors will change as output changes Dr. Yarmohamadian Art University of Isfahan

5.2 geometry of costs AVC may initially decrease if there is initial region of economies of scale. However, it seems reasonable to suppose that variable factors required will increase more or less linearly until we approach some capacity level of output determined by amounts of fixed factors. When we are near to capacity, we need to use more than a proportional amount of variable inputs to increase output. Thus, AVC function should eventually increase as output increases, Dr. Yarmohamadian Art University of Isfahan

5.2 geometry of costs AFC must decrease with output, Adding together AVC and AFC curves gives us U-shaped AC curve (Fig 5.1C) initial decrease in AC is due to decrease in AFC; eventual increase in AC is due to increase in AVC. level of output at AC is minimized known as minimal efficient scale. Dr. Yarmohamadian Art University of Isfahan

5.2 geometry of costs Dr. Yarmohamadian Art University of Isfahan

5.2 geometry of costs In long run all costs are variable costs; in such situation increasing AC seems unreasonable since a firm could always replicate its production process. Hence, it is constant or decreasing On or hand, Some firms may not exhibit a long-run CRS technology because of long-run fixed factors. appropriate long-run AC should most probably be U-shaped, (for same reasons given in short-run ) Dr. Yarmohamadian Art University of Isfahan

5.2 geometry of costs What is MC relationship to AC curve? y* denote point of Min AC; then to left of y*, AC are declining so for y<=y* Dr. Yarmohamadian Art University of Isfahan

5.2 geometry of costs Dr. Yarmohamadian Art University of Isfahan

5.2 geometry of costs What is relationship of MC to AVC curves? by changing notation in above argument, we can show that MC lies below AVC curve when AVC curve is decreasing, and lies above it when it is increasing. It follows that MC curve must pass through minimum point of AVC curve Dr. Yarmohamadian Art University of Isfahan

5.2 geometry of costs also MC must equal AC for first unit of output since both numbers are equal to VC(1)-VC(0) A more formal demonstration If y = 0, it becomes 0/0, which is indeterminate. However, limit of Cv(y)/y can be calculated using 'Hopital's rule: It follows that AVC at zero output is just MC Dr. Yarmohamadian Art University of Isfahan

5.2 geometry of costs All of analysis just discussed in both long and short run. If production CRS in long run, so that cost function is linear in level of y, then AC, AVC, and MC are all equal Dr. Yarmohamadian Art University of Isfahan

EX: Cobb-Douglas cost curves
generalized Cobb- Douglas technology has a cost function of form If a + b <1, cost curves exhibit increasing AC if a + b =1, cost curves exhibit constant AC Dr. Yarmohamadian Art University of Isfahan

EX: Cobb-Douglas cost curves
short-run cost function for Cobb-Douglas technology has form Dr. Yarmohamadian Art University of Isfahan

EX: C(y) = y2+1 VC,FC,AVC,AFC,AC,MC?? Two plants C1(y1), C2(y2) How much should produce in each plant? Min C1(y1) + C2(y2) S.t: y1+y2 = y Dr. Yarmohamadian Art University of Isfahan

5.3 Long-run and short-run cost curves
long-run cost curve must never lie above any short-run cost curve, since short-run cost min problem is just a constrained version of long-run cost min problem. If long-run cost function as c(y) = c(y, z(y)). we have omitted factor prices since y are assumed fixed, z(y) be cost-min demand for a single fixed factor y* be some given level of output, and z* = z(y*) be associated long-run demand for fixed factor. Dr. Yarmohamadian Art University of Isfahan

Short-run cost, c(y,z*), must be at least as great as long-run cost, c(y, z(y)) , for all levels of output and short-run cost will equal long-run cost at output y*, so c(y*, z*) = c(y*,z(y*)). Hence, long- and short-run cost curves must be tangent at y*. This is just a geometric restatement of envelope theorem. Dr. Yarmohamadian Art University of Isfahan

slope of long-run cost curve at y* is But since z* is optimal choice of fixed factors at output level y*. we must have Thus, long-run marginal costs at y* equal short-run marginal costs at (y*, z*) if long- and short-run cost curves are tangent, LAC& SAC curves must also be tangent. Dr. Yarmohamadian Art University of Isfahan

c(y,z*)/y c(y)/y Dr. Yarmohamadian Art University of Isfahan

Another way is to start with family of SAVC curves. Suppose, we have a fixed factor that can be used only at three discrete levels: z1, z2, z3. (Fig 5.3) What would be long-run cost curve? It is lower envelope of se short-run curves since optimal choice of z to produce output y will simply be choice that has minimum cost of producing y This envelope operation generates a scalloped-shaped long-run average cost curve. If there are many possible values of fixed factor, se scallops become a smooth curve. Dr. Yarmohamadian Art University of Isfahan

5.3 Long-run and short-run cost curves (discrete levels of plant size)

LAC Dr. Yarmohamadian Art University of Isfahan

5.3 Long-run and short-run cost curves (discrete levels of plant size)
MC1 MC4 MC2 MC3 MC5 SAC1 SAC2 SAC5 SAC3 SAC4 Dr. Yarmohamadian Art University of Isfahan

LMC SMC Dr. Yarmohamadian Art University of Isfahan

5.4 Factor prices and cost functions
price behavior of cost functions. Several interesting properties follow directly from definition of functions. Note close similarity with properties of profit function. Dr. Yarmohamadian Art University of Isfahan

Nondecreasing in w. If w' >=w, n c(w’,y) >= c(w,y). This is obvious, but a formal proof may be instructive. Let x and x‘ be cost-min bundles associated with w and w'. if w<=w' then wx<=wx‘ by min and wx'<= w'x' since w<=w' w’x’<=w’x Putting these inequalities together gives wx<=w'x' as required. Dr. Yarmohamadian Art University of Isfahan

2) Homogeneous of degree 1 in w. c(tw, y) = tc(w, y) for t >0 We show that if x is cost-minimizing bundle at prices w, n x also minimizes costs at prices tw. Suppose not, and let x' be a cost minimizing bundle at tw so that twx' < twx. But this inequality implies wx' < wx, which contradicts definition of x. Hence, multiplying factor prices by a positive scalar t does not change composition of a cost minimizing bundle, and, thus, costs must rise by exactly a factor of t: c(tw,y) = twx = tc(w,y). Dr. Yarmohamadian Art University of Isfahan

3) Concave in w. c(tw+(1-t)w’,y)>=tc(w,y)+(1- t)c(w’,y) for 0<=t<=1 Let (w,x) and (w',x') be two cost-minimizing price-factor combinations and w" =t.w + (1-t)w’ for any 0<=t<=1. Now, Dr. Yarmohamadian Art University of Isfahan

4) Continuous in w. c(w,y) is continuous as a function of w, for w >> 0. Consider a parameterized maximization problem of form M(a) = max f(x,a) (27.11) such that x is in G(a) Existence of an optimum. If constraint set G(a) is nonempty and compact, and function f is continuous, n re exists a solution x* to this maximization problem. Uniqueness of optimum. If function f is strictly concave and constraint set is convex, n a solution, should it exist, is unique Dr. Yarmohamadian Art University of Isfahan

only property that is surprising here is concavity. Suppose we graph cost as a function of price of a single input, with all or prices held constant. If price of a factor rises, costs will never go down (property I), but y will go up at a decreasing rate (property 3). Why? Because as this one factor becomes more expensive and or prices stay same, cost-minimizing firm will shift away from it to use or inputs. Dr. Yarmohamadian Art University of Isfahan

Figure 5.4. x* be a cost minimizing bundle at prices w*. Suppose price of factor 1 changes from w1* to w1 If we just behave passively and continue to use x*, our costs will be minimal cost of production c(w, y) must be less than this "passive" cost function; thus, graph of c(w,y) must lie below graph of passive cost function, with both curves coinciding at w*. this implies c(w,y) is concave with respect to w1. Dr. Yarmohamadian Art University of Isfahan

same graph can be used to discover a very useful way to find conditional factor demand. Shephard's lemma ( derivative property) xi(w,y) be firm's conditional factor demand for input i then if cost function is differentiable at (w,y), and wi > 0 for i = 1,. . . , n then proof is very similar to Hotelling's lemma. Dr. Yarmohamadian Art University of Isfahan

x* be a cost-minimizing bundle that produces y at prices w*. n define function g(w) = c(w,y) - wx*. Since c(w,y) is cheapest way to produce y, this function is always non-positive. At w = w*, g(w*) = 0. Since this is a maximum value of g(w), its derivative must vanish: Hence, cost-minimizing input vector (x*) is just given by vector of derivatives of cost function with respect to prices. Dr. Yarmohamadian Art University of Isfahan

Since this proposition is important, we will suggest four different ways of proving it. First, cost function is by definition equal to c(w,y) == w.x(w,y). Differentiating this expression with respect to wi and using F.O.C give us result. (Hint: x(w,y) also satisfies identity f(x(w, y))== y. You will need to differentiate this with respect to wi) Second, repeating derivation of envelope theorem (described in next section). Dr. Yarmohamadian Art University of Isfahan

Third, re is a nice geometrical argument that uses same Figure 5.4 that we used in arguing for concavity of cost function. Recall in Figure 5.4 that line lay above c = c(w,y) and both curves coincided at w1 = w1*?. Thus, curves must be tangent, so that Dr. Yarmohamadian Art University of Isfahan

Finally, we consider basic economic intuition behind proposition. If we are operating at a cost-minimizing point and price wi increases, re will be a direct effect, in that expenditure on first factor will increase. re will also be an indirect effect, in that we will want to change factor mix. But since we are operating at a cost-minimizing point, any such infinitesimal change must yield zero additional profits. Dr. Yarmohamadian Art University of Isfahan

EX: Marginal cost revisited
As another application of envelope theorem, consider derivative of cost function (Lagrangian) with respect to y. Lagrangian for cost minimization problem is So, Lagrange multiplier in cost minimization problem is marginal cost. Dr. Yarmohamadian Art University of Isfahan

5.6 Comparative statics using cost function
Cost functions have certain properties that follow from structure of cost minimization problem; conditional factor demand functions are derivatives of cost functions. Hence, properties of cost function will translate into certain restrictions on its derivatives ( factor demand functions). se restrictions will be same of restrictions we found earlier using or methods, but their development using cost function is quite nice. Dr. Yarmohamadian Art University of Isfahan

1) cost function is nondecreasing in factor prices. n dc(w,y)/dwi = xi(w,y) >= 0. 2) cost function is homogeneous of degree 1 in w. therefore, factor demands, are homogeneous of degree 0 in w. 3) cost function is concave in w. therefore, matrix of second derivatives of cost function ( matrix of first derivatives of factor demand functions) is a symmetric negative semidefinite matrix. This is not an obvious outcome of cost-minimizing behavior. Dr. Yarmohamadian Art University of Isfahan

since concavity of cost function followed only from hypothesis of cost minimization, symmetry and negative semi definiteness of first derivative matrix of factor demand functions follow only from hypothesis of cost minimization and do not involve any restrictions on structure of technology. Dr. Yarmohamadian Art University of Isfahan

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