Ekkehart Schlicht: Economic Surplus and Derived Demand

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1 Ekkehart Schlicht: Ecoomic Surplus ad Derived Demad Muich Discussio Paper No Departmet of Ecoomics Uiversity of Muich Volkswirtschaftliche Fakultät Ludwig-Maximilias-Uiversität Müche Olie at

2 Ecoomic Surplus ad Derived Demad Ekkehart Schlicht 1 Itroductio ALFRED MARSHALL (1920, v.vi.4) has poited out that most demad especially labor demad is derived from the demad for some other product. He wrote: To take aother illustratio, the direct demad for houses gives rise to a joit demad for the labour of all the various buildig trades, ad for bricks, stoe, wood, etc. which are factors of productio of buildig work of all kids, or as we may say for shortess, of ew houses. The demad for ay oe of these, as for istace the labour of plasterers, is oly a idirect or derived demad. This ote demostrates that the usual aalysis of ecoomic ret, as typically explaied for the case of cosumers surplus, carries over to the case of derived demad. The assertio comes as o surprise; rather such is typically presupposed i all kids of cost-beefit cosideratios ivolvig derived demad, yet there is, to the best of my kowledge, o demostratio to the be foud i the texbooks or the literature. This ote is iteded to fill the gap. Istitutioal Ecoomics Group, Departmet of Ecoomics, Uiversity of Muich, Schackstr. 4, Muich, Germay, web: I thak Robert Schlicht for a simplifyig suggestio. 1

3 2 Derived Demad Cosider a market for a cosumer good with the fallig demad curve (iverse demad fuctio) p = p (x), p < 0 (1) where x deotes the quatity demaded at price p. Assume that the commodity is produced by may firms by meas of some iput (labor, for istace) accordig to a productio fuctio x i = f i ( i ), f i > 0, f i < 0. where the idex i refers to firm umber i. 1 Profits of firm i are p i f i ( i ) w i, ad the profit maximizig level of productio is characterized by the margial productivity coditio pf i ( i ) = w. (2) Deote by = i i the aggregate labor iput ad defie the aggregate productio fuctio f as { } f () := max f i ( i ) i =. (3) Assumig a uique iterior maximum, we obtai from (2) ad (3) i i pf () = w. (4) For ay give product price p, this equatio gives the ordiary (iverse) demad curve for the iput (e.g. labor). Its slope is equal to pf < 0. The sum of profits accruig to all firms together is i (pf i ( i ) w i ) ad hece π = pf () w. (5) Ay level of productio x is, accordig to (1), uiquely related to the product price p that is ecessary to clear the market. These iterrelatios ca be icorporated withi the aalysis by isertig (1) ad (3) ito (4), yieldig a relatioship betwee 1 I the simplest case, oly oe iput (such as labor) is eeded. For reasos of simplicity this is assumed i the followig expositio. The more geeral case of may iputs ca be covered by re-iterpretig f i ( i ) as the maximum value added obtaiable for firm i if factor iput i is costlessly provided, etc. 2

4 w derived demad curve ordiary demad curves at alterative product prices Figure 1: The derived iput demad curve is steeper tha the ordiary iput demad curves. factor iput ad iput price: p (f ()) f () = w. This gives rise to the idirect iput demad curve w () := p (f ()) f (). (6) Its slope is w = pf + p (f ) 2 < pf. It is, therefore, steeper tha the ordiary demad curve (Figure 1). 1 Cosider a wage reductio from w 0 to w 1 that goes alog with a employmet icrease from to 1, a output icrease from to x 1, ad a price reductio from p 0 to p 1. These quatities relate as follows: = f ( ), p 0 = p ( ) x 1 = f ( 1 ), p 1 = p (x 1 ). Cosider the area uder the demad curve (1) betwee ad output level x = f () belogig to some employmet level : P () = f() Differetiatio of (8) with respect to yields (7) p (x) dx. (8) P = p (f ()) f () = w () 1 See PINDYCK ad RUBINFELD (2005, 521). Note that some authors use the cocept i a differet sese; see VARIAN (1996, 338), for istace. 3

5 p w p(x) w() x 1 x 1 (a) (b) Figure 2: The area uder the demad curve for output (a) ad the area uder the derived demad curve for iput (b) betwee correspodig values of iput ad output are idetical. which implies P ( 1 ) = 1 w () d. This is the just the area uder the derived iput demad curve. Hece the correspodig areas uder the product demad curve ad the idirect iput demad curve are idetical i size (Figure 2). 3 Derived Surplus Cosider ow the chage i cosumer s surplus that results from a price chage from p 0 to p 1. It is cs = p 0 p 1 x 1 + x1 p (x) dx (9) ad is depicted i Figure 3. Accordig to the previous argumet, the itegral over the demad curve equals the correspodig itegral over the idirect demad curve ad we have x 1 p (x) dx = 1 w () d. The profits associated with the differet levels of productio are π 0 = p 0 w 0 π 1 = p 1 x 1 w 1 1 4

6 p w p 0 p(x) w 0 w() cs cs+ π p 1 w 1 x 1 x 1 (a) (b) Figure 3: The chage i cosumers surplus cs is give by the shaded area left of the fial demad curve (a). The aalogous area left of the derived demad curve (b) gives the chage i ecoomic ret as the sum of the chages i cosumers surplus cs ad profits π. ad the chage i profits is π = π 1 π 0. Hece (9) ca be writte as cs + π = w 0 w w () d. (10) The left-had side gives the chage i ecoomic ret as the sum of the chages i cosumers surplus cs ad profits π. The right-had side gives the surplus area i the derived demad diagram, which is, thus, a measure for the icrease i ecoomic ret iduced by a price decrease of iput from w 0 to w 1 ad a etailed price decrease of the fial product from p 0 to p 1. (Figure 3). Refereces MARSHALL, A. 1920, Priciples of Ecoomics, 8th ed., Lodo: Macmilla, olie at reprit 1949, first editio

7 PINDYCK, R. S. ad D. L. RUBINFELD 2005, Microecoomics, 6th ed., Pearso, Upper Saddle River, New Jersey, first editio VARIAN, H. R. 1996, Itermediate Microecoomics, fourth ed., Norto, New York, first editio

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