The production set Production requires inputs - factors of production- for example. labour and capital equipment. A production set is the set of outputs feasible giv articular combination of inputs. The production function escribes the maximum output given a particular combination of inputs. Both are illustrated below y= f(r)= Production function Isoquant Production set Iput =r An isoquant maps all the combinations of the(two)inputs that that are just sufficient to produce a given output. Technology is assumed to be monotonic and conver, producing isoquants like those above Monotonicity(or free disposal) says that more of one input will result in at least as much output. Convexity says that if two different combinations of inputs produce output y, then a weighted average will produce at least iction -Frans Marginal Products When one of the factors is increased a small amount, Ar1, output rises by Ay. The marginal product is MB(xx2)=y=+△m2)-/2≈ Marginal product gives the slope of the production function y= f(zr, I2) How much additional input 2 is required to just continue producing exactly y units of output, given a decrease in RS). If output remains const MP(x1,x2)△r1+MP2(x1,x2)△ This equation can be solved for the change in input 2 divided by the change input 1. It is the slope of the isoquant. TRS=Ar2=_MA(1,r2) Ar1 MP(a1, r2) The "lawof diminishing marginal product states that marginal product will decrease as the amount of the input used is increased, keeping all other inputs constant. Why is this considered a "law"? Diminishing TRS is also assumed -it follows from the convexity of the isoquant
Production — Firms 1 The Production Set • Production requires inputs — factors of production — for example, labour and capital equipment. • A production set is the set of outputs feasible given a particular combination of inputs. The production function describes the maximum output given a particular combination of inputs. Both are illustrated below. . ................................................................................................................. . ............................................................................................... ................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................................................................................................................. 0 Output = y Input = x 0 x2 x1 Production set y = f(x) = Production function Isoquant y = y • An isoquant maps all the combinations of the (two) inputs that that are just sufficient to produce a given output. • Technology is assumed to be monotonic and convex, producing isoquants like those above. Monotonicity (or free disposal) says that more of one input will result in at least as much output. Convexity says that if two different combinations of inputs produce output y, then a weighted average will produce at least y. Production — Firms 2 Marginal Products • When one of the factors is increased a small amount, ∆x1, output rises by ∆y. The marginal product is: MP1(x1, x2) = ∆y ∆x1 = f(x1 + ∆x1, x2) − f(x1, x2) ∆x1 ≈ ∂f ∂x1 • Marginal product gives the slope of the production function y = f(x1, x2). • How much additional input 2 is required to just continue producing exactly y units of output, given a decrease in the amount of input 1? The answer is the technical rate of substitution (TRS). If output remains constant, then: ∆y = MP1(x1, x2)∆x1 + MP2(x1, x2)∆x2 = 0 • This equation can be solved for the change in input 2 divided by the change input 1. It is the slope of the isoquant. TRS = ∆x2 ∆x1 = − MP1(x1, x2) MP2(x1, x2) • The “law” of diminishing marginal product states that marginal product will decrease as the amount of the input used is increased, keeping all other inputs constant. Why is this considered a “law”? • Diminishing TRS is also assumed — it follows from the convexity of the isoquant
Production Finns The Long and Short Run All factors can be varied in the long run. At least one is fixed in the short ru Suppose input 2 is land, and can only be varied in the long run. The short run production function is written f(r1, I2). It looks identical to the earlier function- getting fatter as I increases. Why? When all inputs are increased by a common factor k what happens to production? Returns to scale can be: 1.Constant: Output also increases by a factor of k-f(krl, kr2)= kf(r1, r2) 2. Decreasing: Output increases by less f(kI1, kr2)kf(r1, I2) iction -Frans Short run profit maximisation Profit is total revenue minns total cost. If I z is fixed at I, in the short run and the prices of output and the two uts are given by p, wn and wz respectively, then short run profit is: x=py-wnr1-u2I2. Isoprofit lines are drawn by rearranging to give y as a function of I1. Hence y=(+u2E2)/p+(w/p)r Isoprofit lines =f(x1,z2) (丌+m22)/ lise profits. Pushing the isoprofit lines upward increases profit. However, they ca go higher than the production set as that output combination would be infeasible. Hence: pf(x1,x2)-11-m22= The slope of the isoprofit line is equal to the slope of the production function. This means the value of the marginal product is set equal to the price of the input. pMP(z, I2)=wn-a natural condition. Why
Production — Firms 3 The Long and Short Run • All factors can be varied in the long run. At least one is fixed in the short run. • Suppose input 2 is land, and can only be varied in the long run. The short run production function is written y = f(x1, x2). It looks identical to the earlier function — getting flatter as x1 increases. Why? • When all inputs are increased by a common factor k what happens to production? Returns to scale can be: 1. Constant: Output also increases by a factor of k — f(kx1, kx2) = kf(x1, x2). 2. Decreasing: Output increases by less than k — f(kx1, kx2) kf(x1, x2). Production — Firms 4 Short Run Profit Maximisation • Profit is total revenue minus total cost. If x2 is fixed at x2 in the short run and the prices of output and the two inputs are given by p, w1 and w2 respectively, then short run profit is: π = py − w1x1 − w2x2. • Isoprofit lines are drawn by rearranging to give y as a function of x1. Hence y = (π + w2x2)/p + (w1/p)x1. . ................................................................................................................ ........................................................................................................................................................................................................................................................................................................... ........................................................................................................................................................................................................................................................................................... ........................................................................................................................................................................................................................................................................................... ........................................................................................................................................................................................................................................................................................... ............................ 0 y x1 y = f(x1, x2) Isoprofit lines (π + w2x2)/p • x ∗ 1 y ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • The firm is assumed to maximise profits. Pushing the isoprofit lines upward increases profit. However, they cannot go higher than the production set as that output combination would be infeasible. Hence; max π = max x1 pf(x1, x2) − w1x1 − w2x2 =⇒ w1 p = MP1(x ∗ 1 , x2) • The slope of the isoprofit line is equal to the slope of the production function. This means the value of the marginal product is set equal to the price of the input. pMP1(x ∗ 1 , x2) = w1 — a natural condition. Why?
Production Finns Comparative Static The analysis so far can be used to show what happens when various prices change. The slope of the isoprofit line is w/p and so is affected whenever wl or p change y In the first graph, wi increases to wi and the result is a fall in the factor demand for input I and a fall in y In the second graph, p increases to p and the result is a rise in the demand for input I and a rise in the output y Changing w has no effect upon the slope and so nothing changes in the short run- except profit levels. iction -Frans Long Run Profit Maximisation In the long run all factors are variable. The firm solves the following problem m(1)一m吗→pMBG动=pB动=吗 Given the optimal amount of input 2(r?) the function pAP(r1, r?)=w'i defines the inverse factor demand cu which is the relationship between optimal factor demand and factor price, derived from the previous slide pMA(r1,r:) The factor demand alays downward sloping. There are no"Giffen"production inputs. Why not? A profit maximising firm with constant returns to scale makes zero profit. Why?
Production — Firms 5 Comparative Statics • The analysis so far can be used to show what happens when various prices change. The slope of the isoprofit line is w1/p and so is affected whenever w1 or p change. . ................................................................................................................. . ................................................................................................................. ........................................................................................................................................................................................................................................................................................................... ........................................................................................................................................................................................................................................................................................... ............................ .................................................................................................................. .................................................................................................................. ........................................................................................................................................................................................................................................................................................................... ........................................................................................................................................................................................................................................................................................... ............................ 0 y x1 0 y x1 y = f(x1, x2) y = f(x1, x2) • x1 y • x 0 1 y 0 • • x1 y x 0 1 y 0 w1 p 0 w 0 1 p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • In the first graph, w1 increases to w 0 1 and the result is a fall in the factor demand for input 1 and a fall in y. • In the second graph, p increases to p 0 and the result is a rise in the demand for input 1 and a rise in the output y. • Changing w2 has no effect upon the slope and so nothing changes in the short run — except profit levels. Production — Firms 6 Long Run Profit Maximisation • In the long run all factors are variable. The firm solves the following problem: max x1,x2 pf(x1, x2) − w1x1 − w2x2 =⇒ pMP1(x ∗ 1 , x ∗ 2 ) = w1 and pMP1(x ∗ 1 , x ∗ 2 ) = w2 • Given the optimal amount of input 2 (x ∗ 2 ) the function pMP1(x1, x ∗ 2 ) = w1 defines the inverse factor demand curve which is the relationship between optimal factor demand and factor price, derived from the previous slide. ................................................................................................................................................................................................................................................................................ . pMP1(x1, x ∗ 2 ) = w1 0 w1 x1 • The factor demand curve is always downward sloping. There are no “Giffen” production inputs. Why not? • A profit maximising firm with constant returns to scale makes zero profit. Why?
Production Finns Cost minimisation An equivalent way to think about the firms production decision is cost minimisation. nin w1I1+w242 subject to f(=1, I2)>y That is, the firm minimises their cost of producing at least an output of y. Isocost curves connect all the input combinations of equal cost and are linea Isocost lines WI1+ugI =C f(x1,x2)≥了 0 The second graph is an isoquant showing the constraint in the abowe equation. f(r1, 12)2 at all points above it. ity with consumer theory is striking. The slope of the isocost lines is -w/uz The TRS Condition The isocost line is moved downward as far as it will go giving another tangent condition. =y The isoquant slope(the TRS)is set equal to the isocost slope giving the TRS condition: JP1(x1,r2) .n what profit maximisation had reveale The conditional factor demands are the optimal factor demands in terms of the amount of output the firm wishes to u2,y)
Production — Firms 7 Cost Minimisation • An equivalent way to think about the firm’s production decision is cost minimisation. min x1,x2 w1x1 + w2x2 subject to f(x1, x2) ≥ y • That is, the firm minimises their cost of producing at least an output of y. Isocost curves connect all the input combinations of equal cost and are linear. . ............................................................................................... ................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................................................................................................................. 0 x2 x1 0 x2 x1 f(x1, x2) ≥ y y = y Isocost lines w1x1 + w2x2 = c • The second graph is an isoquant showing the constraint in the above equation. f(x1, x2) ≥ y at all points above it. • The similarity with consumer theory is striking. The slope of the isocost lines is −w1/w2. Production — Firms 8 The TRS Condition • The isocost line is moved downward as far as it will go giving another tangent condition. . ................................................................................................ ................................................................................................................................................................................................................................................................................ . . ..................... 0 x2 x1 x ∗ 2 x ∗ 1 y = y ............. ............. ............. ............. ......... . . . . . • The isoquant slope (the TRS) is set equal to the isocost slope giving the TRS condition: TRS = − MP1(x ∗ 1 , x ∗ 2 ) MP2(x ∗ 1 , x ∗ 2 ) = − w1 w2 • This is exactly what profit maximisation had revealed. The two methods are equivalent. • The conditional factor demands are the optimal factor demands in terms of the amount of output the firm wishes to produce and the factor prices: x1(w1, w2, y) and x2(w1, w2, y)
Production Finns The Cost function The(long run) cost function is the minimum cost required to produce a certain output. It solves the problem: c(y)= min(aT1 +u2I2) subject to f(r1, r2)>y It can be written in terms of the conditional factor demands: c(y)=wiI(u1, w2, y)+u2I2(wn, w2. y) The short run cost function is the solution to the same problem when one of the factors is fixed, e.g. I?=I3. Constant costs that have to be paid no matter which level of output is produced are called fired costs. Constant ts that have to paid no matter which level of output is produced as long as it is not zero are called quasi- fired t&. All other alled variable costs. For example: +y2 ify y2 The above example has fixed costs of 1, quasi-fixed costs of k and variable costs of y There are no fixed costs in the long run. although there could be quasi-fixed costs. iction -Frans Average Cost Costs can always be written as c(y)=c(y)+ F where c()is variable and includes any quasi-fixed costs. Auerage cost is the minimum cost per unit of producing a total output y. Hence AC(= y)=(y)+F=AvC(w)+ AFC()= Average variable cost Average fixed cost AFC slopes downward. AvC will eventually slope upward. So AC is usually drawn with a quadratic shape
Production — Firms 9 The Cost Function • The (long run) cost function is the minimum cost required to produce a certain output. It solves the problem: c(y) = min x1,x2 (w1x1 + w2x2) subject to f(x1, x2) ≥ y • It can be written in terms of the conditional factor demands: c(y) = w1x1(w1, w2, y) + w2x2(w1, w2, y). • The short run cost function is the solution to the same problem when one of the factors is fixed, e.g. x2 = x2. • Constant costs that have to be paid no matter which level of output is produced are called fixed costs. Constant costs that have to paid no matter which level of output is produced as long as it is not zero are called quasi-fixed costs. All other costs are called variable costs. For example: c (y) = 1 + k + y 2 if y > 0 1 + y 2 if y = 0 • The above example has fixed costs of 1, quasi-fixed costs of k and variable costs of y 2 . • There are no fixed costs in the long run, although there could be quasi-fixed costs. Production — Firms 10 Average Cost • Costs can always be written as c(y) = cv(y) + F where cv(y) is variable and includes any quasi-fixed costs. • Average cost is the minimum cost per unit of producing a total output y. Hence: AC(y) = c(y) y = cv(y) y + F y = AV C(y) + AFC(y) = Average variable cost + Average fixed cost . ............................................................................................... ................................................................................................. ............................. . ............................ ................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................................................. 0 0 0 AC y AC y AC y AFC AV C AC • AFC slopes downward. AV C will eventually slope upward. So AC is usually drawn with a quadratic shape
Production Finns Marginal cost measures the change in cost given a small change in output. So The last equality follows because fixed costs do not change as output changes. by definition. avc The marginal cost curve cuts the average cost and the average variable cost curve at their minimum points. Why? The average variable cost curve and the marginal cost curve take the same value for the first unit of output. Why? The area underneath the marginal cost curve is equal to variable costs. Why iction -Frans Long Run Cost Curve When all factors are variable long run average costs must be no larger than short run average costs. The long run average cost curve is the lower envelope of all the short run average cost curves SRAC SRMC LRAn LrAc y Notice the minima of the SRAC curves do not intercept the LRAC curve except at the minimum. Why? The second graph shows the connections between long run marginal cost and the other cost curves is simply the collection of points from the" optimal"SRMC curves at each output level y
Production — Firms 11 Marginal Cost • Marginal cost measures the change in cost given a small change in output. So: MC(y) = ∆c(y) ∆y = c(y + ∆y) − c(y) ∆y = ∆cv(y) ∆y • The last equality follows because fixed costs do not change as output changes, by definition. .. .................. .. .... ............................ .. ................................................................................................................................................................................................................................................................................ ................................................................................................................................................................................................................................................................................... 0 Costs y 0 MC y MC AC AV C MC Variable cost • The marginal cost curve cuts the average cost and the average variable cost curve at their minimum points. Why? • The average variable cost curve and the marginal cost curve take the same value for the first unit of output. Why? • The area underneath the marginal cost curve is equal to variable costs. Why? Production — Firms 12 Long Run Cost Curves • When all factors are variable long run average costs must be no larger than short run average costs. • The long run average cost curve is the lower envelope of all the short run average cost curves. ..................................................................................................................................................................................... ............................ .......................... ............................ . ......................... ............................ . ......................... ..................................................................................................................................................................................... ............................ . ......................... . ............................................................................... ................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................................................................................................................. . 0 Costs y 0 Costs y LRAC SRAC y ∗ y ∗ LRAC SRMC LRMC . . . . . . . . . . • Notice the minima of the SRAC curves do not intercept the LRAC curve except at the minimum. Why? • The second graph shows the connections between long run marginal cost and the other cost curves. • The LRMC curve is simply the collection of points from the “optimal” SRMC curves at each output level y ∗
