Chapter The Firm Exercise. Suppose that a unit of output can be produced by any of the following combinations of inputs z = 0: 0:5 ; z 0:3 = 0:. Construct the isouant for =. ; z 3 0:5 = 0:. Assuming constant returns to scale, construct the isouant for =. 3. If the techniue z 4 = [0:5; 0:5] were also available would it be included in the isouant for =? Outline Answer z 0.5 z z 4 0. 0. z z 3 = 0 0. 0.3 0.5 z Figure.: Isouant simple case 3
Microeconomics CHAPTER. THE FIRM z 0.5 z z 4 0. z 0. z 3 = 0 0. 0.3 0.5 z Figure.: Isouant alternative case z 0.5 z 0. z 0. z 3 = = 0 0. 0.3 0.5 z Figure.3: Isouants under CRTS cfrank Cowell 006 4
Microeconomics. See Figure. for the simplest case. However, if other basic techniues are also available then an isouant such as that in Figure. is consistent with the data in the uestion.. See Figure.3. Draw the rays through the origin that pass through each of the corners of the isouant for =. Each corner of the isouant for =. lies twice as far out along the ray as the corner for the case =. 3. Clearly z 4 should not be included in the isouant since z 4 reuires strictly more of either input to produce one unit of output than does z so that it cannot be e cient. This is true whatever the exact shape of the isouant in see Figures. and. cfrank Cowell 006 5
Microeconomics CHAPTER. THE FIRM Exercise. A rm uses two inputs in the production of a single good. The input reuirements per unit of output for a number of alternative techniues are given by the following table: Process 3 4 5 6 Input 9 5 7 3 4 Input 4 6 0 9 7 The rm has exactly 40 units of input and 40 units of input at its disposal.. Discuss the concepts of technological and economic e ciency with reference to this example.. Describe the optimal production plan for the rm. 3. Would the rm prefer 0 extra units of input or 0 extra units of input? Outline Answer. As illustrated in gure.4 only processes,,4 and 6 are technically e - cient.. Given the resource constraint (see shaded area), the economically e cient input combination is a mixture of processes 4 and 6. z 4 5 Economically Efficient Point. Attainable Set 6 3 0 z Figure.4: Economically e cient point 3. Note that in the neighbourhood of this e cient point MRTS=. So, as illustrated in the enlarged diagram in Figure.5, 0 extra units of input clearly enable more output to be produced than 0 extra units of input. cfrank Cowell 006 6
Microeconomics Original Isouant 0 0 Figure.5: E ect of increase in input cfrank Cowell 006 7
Microeconomics CHAPTER. THE FIRM Exercise.3 Consider the following structure of the cost function: C(w; 0) = 0, C (w; ) = int() where int(x) is the smallest integer greater than or eual to x. Sketch total, average and marginal cost curves. Outline Answer From the uestion the cost function is given by so that average cost is see Figure.6. C(w; ) = k k + ; k < k; k = ; ; 3::: k + k ; k < k; k = ; ; 3::: C(w,) C (w,) C(w,)/ 0 3 Figure.6: Stepwise marginal cost cfrank Cowell 006 8
Microeconomics Exercise.4 Suppose a rm s production function has the Cobb-Douglas form = z z where z and z are inputs, is output and, are positive parameters.. Draw the isouants. Do they touch the axes?. What is the elasticity of substitution in this case? 3. Using the Lagrangean method nd the cost-minimising values of the inputs and the cost function. 4. Under what circumstances will the production function exhibit (a) decreasing (b) constant (c) increasing returns to scale? Explain this using rst the production function and then the cost function. 5. Find the conditional demand curve for input. z z Figure.7: Isouants: Cobb-Douglas Outline Answer. The isouants are illustrated in Figure.7. They do not touch the axes.. The elasticity of substitution is de ned as ij := @ log (z j =z i ) @ log j (z)= i (z) which, in the two input case, becomes = cfrank Cowell 006 9 @ log z @ log z (z) (z) (.)
Microeconomics CHAPTER. THE FIRM In case we have (z) = z z and so, by di erentiation, we nd: Taking logarithms we have z log = log or z (z) (z) = = z z u = log log v (z) (z) where u := log (z =z ) and v := log ( = ). Di erentiating u with respect to v we have @u = : (.) @v So, using the de nitions of u and v in euation (.) we have = @u @v = : 3. This is a Cobb-Douglas production function. This will yield a uniue interior solution; the Lagrangean is: L(z; ) = w z + w z + [ and the rst-order conditions are: z z ] ; (.3) @L(z; ) @z = w z z = 0 ; (.4) @L(z; ) @z = w z z = 0 ; (.5) @L(z; ) = z @ z = 0 : (.6) Using these conditions and rearranging we can get an expression for minimized cost in terms of and : w z + w z = z z + z z = [ + ] : We can then eliminate : which implies w w z = w z = w z = 0 z = 0 : (.7) Substituting the values of z and z back in the production function we have = w w cfrank Cowell 006 0
Microeconomics which implies = w w + (.8) So, using (.7) and (.8), the corresponding cost function is C(w; ) = w z + w z = [ + ] w w 4. Using the production functions we have, for any t > 0: (tz) = [tz ] [tz ] = t + (z): + : Therefore we have DRTS/CRTS/IRTS according as + S. If we look at average cost as a function of we nd that AC is increasing/constant/decreasing in according as + S. 5. Using (.7) and (.8) conditional demand functions are H (w; ) = H (w; ) = w w w w and are smooth with respect to input prices. + + cfrank Cowell 006
Microeconomics CHAPTER. THE FIRM Exercise.5 Suppose a rm s production function has the Leontief form z = min ; z where the notation is the same as in Exercise.4.. Draw the isouants.. For a given level of output identify the cost-minimising input combination(s) on the diagram. 3. Hence write down the cost function in this case. Why would the Lagrangean method of Exercise.4 be inappropriate here? 4. What is the conditional input demand curve for input? 5. Repeat parts -4 for each of the two production functions = z + z = z + z Explain carefully how the solution to the cost-minimisation problem di ers in these two cases. z A B z Figure.8: Isouants: Leontief Outline Answer. The Isouants are illustrated in Figure.8 the so-called Leontief case,. If all prices are positive, we have a uniue cost-minimising solution at A: to see this, draw any straight line with positive nite slope through A and take this as an isocost line; if we considered any other point B on the isouant through A then an isocost line through B (same slope as the one through A) must lie above the one you have just drawn. cfrank Cowell 006
Microeconomics z z Figure.9: Isouants: linear z z Figure.0: Isouants: non-convex to origin cfrank Cowell 006 3
Microeconomics CHAPTER. THE FIRM 3. The coordinates of the corner A are ( ; ) and, given w, this immediately yields the minimised cost. C(w; ) = w + w : The methods in Exercise.4 since the Lagrangean is not di erentiable at the corner. 4. Conditional demand is constant if all prices are positive 5. Given the linear case Isouants are as in Figure.9. H (w; ) = H (w; ) = : = z + z It is obvious that the solution will be either at the corner (= ; 0) if w =w < = or at the corner (0; = ) if w =w > =, or otherwise anywhere on the isouant This immediately shows us that minimised cost must be. w C(w; ) = min ; w So conditional demand can be multivalued: 8 >: >< i H (w; ) = z h0; 8 >< H (w; ) = z >: if w w < if w w = 0 if w w > 0 if w w < i h0; if w w = if w w > Case 3 is a test to see if you are awake: the isouants are not convex to the origin: an experiment with a straight-edge to simulate an isocost line will show that it is almost like case the solution will be either at the corner ( p = ; 0) if w =w < p = or at the corner (0; p = ) if w =w > p = (but nowhere else). So the cost function is : r p C(w; ) = min w ; w = : cfrank Cowell 006 4
Microeconomics The conditional demand function is similar to, but slightly di erent from, the previous case: 8 if w w < >< o H (w; ) = z n0; if w w = >: 8 >< H (w; ) = 0 if w w > 0 if w w < o z n0; if w w = >: if w w > Note the discontinuity exactly at w =w = p = cfrank Cowell 006 5
Microeconomics CHAPTER. THE FIRM Exercise.6 Assume the production function (z) = h i z + z where z i is the uantity of input i and i 0, < are parameters. This is an example of the CES (Constant Elasticity of Substitution) production function.. Show that the elasticity of substitution is.. Explain what happens to the form of the production function and the elasticity of substitution in each of the following three cases:!,! 0,!. 3. Relate your answer to the answers to Exercises.4 and.5. Outline Answer. Writing the production function as (z) := h i z + z it is clear that the marginal product of input i is. Therefore the MRTS is which implies log z z i (z) := h i z + z i z i (.9) (z) (z) = z z (.0) = log log (z) : (z) Therefore = @ log @ log z z (z) (z) =. Clearly! yields = 0 ((z) = min f z ; z g),! 0 yields = ((z) = z z ),! yields = ((z) = z + z ). 3. The case! corresponds to that in part of Exercise.5;! 0. corresponds to that in Exercise.4;!. corresponds to that in part 5 of Exercise.5. cfrank Cowell 006 6
Microeconomics Exercise.7 For the CES function in Exercise.6 nd H (w; ), the conditional demand for good, for the case where 6= 0;. Verify that it is decreasing in w and homogeneous of degree 0 in (w,w ). Outline Answer From the minimization of the following Lagrangean we obtain mx L(z; ; w; ) := w i z i + [ i= (z)] [z ] = w (.) [z ] = w (.) On rearranging: Using the production function we get w = [z] w = [z] w + w = Rearranging we nd = [w ] + [w ] Substituting this into (.) we get: w = [z ] [w ] + [w ] Rearranging this we have: z = " # w + w Clearly z is decreasing in w if <. Furthermore, rescaling w and w by some positive constant will leave z unchanged. cfrank Cowell 006 7
Microeconomics CHAPTER. THE FIRM Exercise.8 For any homothetic production function show that the cost function must be expressible in the form C (w; ) = a (w) b () : z 0 z Figure.: Homotheticity: expansion path Outline Answer From the de nition of homotheticity, the isouants must look like Figure.; interpreting the tangents as isocost lines it is clear from the gure that the expansion paths are rays through the origin. So, if H i (w; ) is the demand for input i conditional on output, the optimal input ratio H i (w; ) H j (w; ) must be independent of and so we must have H i (w; ) H i (w; 0 ) = Hj (w; ) H j (w; 0 ) for any ; 0. For this to true it is clear that the ratio H i (w; )=H i (w; 0 ) must be independent of w. Setting 0 = we therefore have H (w; ) H (w; ) = H (w; ) H (w; ) = ::: = Hm (w; ) H m (w; ) = b() and so H i (w; ) = b()h i (w; ): cfrank Cowell 006 8
Microeconomics Therefore the minimized cost is given by C(w; ) = = where a(w) = P m i= w ih i (w; ): mx w i H i (w; ) i= mx w i b()h i (w; ) i= = b() mx w i H i (w; ) i= = a(w)b() cfrank Cowell 006 9
Microeconomics CHAPTER. THE FIRM Exercise.9 Consider the production function = z + z + 3 z3. Find the long-run cost function and sketch the long-run and short-run marginal and average cost curves and comment on their form.. Suppose input 3 is xed in the short run. Repeat the analysis for the short-run case. 3. What is the elasticity of supply in the short and the long run? Outline Answer. The production function is clearly homogeneous of degree in all inputs i.e. in the long run we have constant returns to scale. But CRTS implies constant average cost. So LRMC = LRAC = constant Their graphs will be an identical straight line. z z Figure.: Isouants do not touch the axes. In the short run z 3 = z 3 so we can write the problem as the following Lagrangean ^L(z; ^) = w z + w z + ^ h z + z + 3 z 3 i ; (.3) or, using a transformation of the constraint to make the manipulation easier, we can use the Lagrangean L(z; ) = w z + w z + z + z k (.4) where is the Lagrange multiplier for the transformed constraint and cfrank Cowell 006 0 k := 3 z 3 : (.5)
Microeconomics Note that the isouant is z = k z From the Figure. it is clear that the isouants do not touch the axes and so we will have an interior solution. The rst-order conditions are which imply : w i i z i = 0; i = ; (.6) z i = r i w i ; i = ; (.7) To nd the conditional demand function we need to solve for. Using the production function and euations (.5), (.7) we get k = X j= = j j (.8) w j from which we nd where p = b k b := p w + p w : Substituting from (.9) into (.7) we get minimised cost as (.9) Marginal cost is and average cost is ~C (w; ; z 3 ) = X w i zi + w 3 z 3 (.0) i= = b = k + w 3z 3 (.) b 3 z 3 3z 3 : (.) b 3 z 3 (.3) b 3 z 3 + w 3z 3 : (.4) Let be the value of for which MC=AC in (.3) and (.4) at the minimum of AC in Figure.3 and let P be the corresponding minimum value of AC. Then, using p =MC in (.3) for p p the short-run supply 8 0 if p <p >< curve is given by = S(w; p) = 0 or if p =p h i >: = z3 b 3 p p if p >p cfrank Cowell 006
Microeconomics CHAPTER. THE FIRM 3. Di erentiating the last line in the previous formula we get d ln d ln p = p d dp = p > 0 p=b Note that the elasticity decreases with b. In the long run the supply curve coincides with the MC,AC curves and so has in nite elasticity. marginal cost average cost Figure.3: Short-run marginal and average cost cfrank Cowell 006
Microeconomics Exercise.0 A competitive rm s output is determined by = z z :::zm m where z i is its usage of input i and i > 0 is a parameter i = ; ; :::; m. Assume that in the short run only k of the m inputs are variable.. Find the long-run average and marginal cost functions for this rm. Under what conditions will marginal cost rise with output?. Find the short-run marginal cost function. 3. Find the rm s short-run elasticity of supply. What would happen to this elasticity if k were reduced? Outline Answer Write the production function in the euivalent form: log = mx i log z i (.5) i= The isouant for the case m = would take the form which does not touch the axis for nite (z ; z ). z = z (.6). The cost-minimisation problem can be represented as minimising the Lagrangean " # mx mx w i z i + log i log z i (.7) i= where w i is the given price of input i, and is the Lagrange multiplier for the modi ed production constraint. Given that the isouant does not touch the axis we must have an interior solution: rst-order conditions are which imply i= w i i z i = 0; i = ; ; ::; m (.8) Now solve for. Using (.5) and (.9) we get z i i = = z i = i w i ; i = ; ; ::; m (.9) i i ; i = ; ; ::; m (.30) my i= w i z i i = cfrank Cowell 006 3 A m Y i= w i i (.3)
Microeconomics CHAPTER. THE FIRM where := P m j= j and A := [ Q m we nd i= i i ] = A Q m i= w i i = are constants, from which = = A [w w :::wm m ] = : (.3) Substituting from (.3) into (.9) we get the conditional demand function: H i (w; ) = zi = i A [w w w :::wm m ] = (.33) i and minimised cost is C (w; ) = mx i= w i z i = A [w w :::wm m ] = (.34) = B = (.35) where B := A [w w :::wm m ] =. It is clear from (.35) that cost is increasing in and increasing in w i if i > 0 (it is always nondecreasing in w i ). Di erentiating (.35) with respect to marginal cost is C (w; ) = B (.36) Clearly marginal cost falls/stays constant/rises with as T.. In the short run inputs ; :::; k (k m) remain variable and the remaining inputs are xed. In the short-run the production function can be written as kx log = i log z i + log k (.37) where i= k := exp mx i=k+ i log z i! (.38) and z i is the arbitrary value at which input i is xed; note that B is xed in the short run. The general form of the Lagrangean (.7) remains unchanged, but with replaced by = k and m replaced by k. So the rst-order conditions and their corollaries (.8)-(.3) are essentially as before, but and A are replaced by and A k := h Qk i= i i k := kx j (.39) j= i =k. Hence short-run conditional demand is ~H i (w; ; z k+ ; :::; z m ) = =k i A k w w i w :::w k k (.40) k cfrank Cowell 006 4
Microeconomics and minimised cost in the short run is ~C (w; ; z k+ ; :::; z m ) = kx w i zi + c k i= =k = k A k w w :::w k k + c k (.4) k = k B k = k + ck (.4) where mx c k := w i z i (.43) i=k+ is the xed-cost component in the short run and B k := A k [w w :::w k k = k] = k. Di erentiating (.4) we nd that short-run marginal cost is ~C (w; ; z k+ ; :::; z m ) = B k k k 3. Using the Marginal cost=price condition we nd B k k k = p (.44) where p is the price of output so that, rearranging (.44) the supply function is p k k = S (w; p; z k+ ; :::; z m ) = (.45) wherever MCAC. The elasticity of (.45) is given by @ log S (w; p; z k+ ; :::; z m ) @ log p B k = k k > 0 (.46) It is clear from (.39) that k k k ::: and so the positive supply elasticity in (.46) must fall as k falls. cfrank Cowell 006 5
Microeconomics CHAPTER. THE FIRM Exercise. A rm produces goods and using goods 3,...,5 as inputs. The production of one unit of good i (i = ; ) reuires at least a ij units of good j, ( j = 3; 4; 5).. Assuming constant returns to scale, how much of resource j will be needed to produce units of commodity?. For given values of 3 ; 4 ; 5 sketch the set of technologically feasible outputs of goods and. Outline Answer. To produce units of commodity a j units of resource j will be needed. a i + a i R i :. The feasibility constraint for resource j is therefore going to be a j + a j R j : Taking into account all three resources, the feasible set is given as in Figure.4 points satisfying a 3 3 + a 3 3 R 3 points satisfying a 4 4 + a 4 R 4 Feasible Set points satisfying a 5 5 + a 5 5 R 5 Figure.4: Feasible set Exercise. [see Exercise.4] cfrank Cowell 006 6
Microeconomics Exercise.3 An agricultural producer raises sheep to produce wool (good ) and meat (good ). There is a choice of four breeds (A, B, C, D) that can be used to stock the farm; each breed can be considered as a separate input to the production process. The yield of wool and of meat per 000 sheep (in arbitrary units) for each breed is given in Table.. A B C D wool 0 65 85 90 meat 70 50 0 0 Table.: Yield per 000 sheep for breeds A,...,D. On a diagram show the production possibilities if the producer stocks exactly 000 sheep using just one breed from the set {A,B,C,D}.. Using this diagram show the production possibilities if the producer s 000 sheep are a mixture of breeds A and B. Do the same for a mixture of breeds B and C; and again for a mixture of breeds C and D. Hence draw the (wool, meat) transformation curve for 000 sheep. What would be the transformation curve for 000 sheep? 3. What is the MRT of meat into wool if a combination of breeds A and B are used? What is the MRT if a combination of breeds B and C are used?and if breeds C and D are used? 4. Why will the producer not nd it necessary to use more than two breeds? 5. A new breed E becomes available that has a (wool, meat) yield per 000 sheep of (50,50). Explain why the producer would never be interested in stocking breed E if breeds A,...,D are still available and why the transformation curve remains una ected. 6. Another new breed F becomes available that has a (wool, meat) yield per 000 sheep of (50,50). Explain how this will alter the transformation curve. Outline Answer. See Figure.5.. See Figure.5. 3. The MRT if A and B are used is is going to be 0 50 85 65 = 3 : 70 50 0 65 = 4 9. If B and C are used it 4. In general for m inputs and n outputs if m > n then m n inputs are redundant. 5. As we can observe in Figure.5, by using breed E the producer cannot move the frontier (the transformation curve) outwards. cfrank Cowell 006 7
Microeconomics CHAPTER. THE FIRM 80 60 meat 40 0 0 0 0 40 60 80 00 wool Figure.5: The wool and meat tradeo 6. As we can observe in Figure.6 now the technological frontier has moved outwards: one of the former techniues is no longer on the frontier. 80 60 meat 40 0 0 0 0 40 60 80 00 wool Figure.6: E ect of a new breed cfrank Cowell 006 8
Microeconomics Exercise.4 A rm produces goods and uses labour (good 3) as input subject to the production constraint [ ] + [ ] + A 3 0 where i is net output of good i and A is a positive constant. Draw the transformation curve for goods and. What would happen to this transformation curve if the constant A had a larger value? Outline Answer. From the production function it is clear that, for any given value 3, the transformation curve is the boundary of the the set of points ( ; ) satisfying [ ] + [ ] A 3 ; 0 where the right-hand side of the rst expression is positive because 3 is negative. This is therefore going to be a uarter circle as in Figure.7.. See Figure.7. Large A Small A Figure.7: Transformation curves cfrank Cowell 006 9