Handout%203 - Handout #3, M .M ajumdar Department of...

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Unformatted text preview: Handout #3, M .M ajumdar Department of Economics Cornell university 1. Theon of production Firm: one input -->one output labor---> corn [may think of all other inputs as fixed for the moment] ‘ Technology : Production function q=f(L)[“q” quantity of corn; L quantity of labor] f’(L) >0;f”(L) <0 for L >0 Example : q=2\/L . note the inverse function L=q2/4 I shall write L=g(q) where gis the inverse off. ** Digression: f’=1/g’ f’7g” =-m3 ** Verify the example Price of corn “p” per unit; price of labor “w” per unit Example p=1,w=4 T he firm accepts prices as given. Let 1320 heflxed cost ' The cost function C(q) gives the minimum cost of producing q C(q) =Wg(q) + 1* Note that C’(q)=wg’(q) and C’ ’(q) =wg’ ’(q). Since g’ ’(q) >0, the marginal cost is increasing (use handout). Profit «(q)= two-CM) In this case we can also express 11‘ as a function of .L: 1r(L) =pf(L)-[WL + fl] Profit max condition: p=MC (i) pf’=w [VMP=w]. (ii) 1": w/p [MP=REAL WAGE RATE] Verify (i) and (ii) are equivalent In (i) we get p=wg’(q) ' - In (ii) we get p=w/f’(L) ; but f ’(L)=1/g'(q) since g is the inverse function of i ‘ Talk about “average product” and “marginal product” In our example, average and marginal products are falling : for the average to fall, the marginal must be less than the average. Explain the condition in terms of a graph Compute the example A more general case: q =Lu where a is a positive fraction( 0 < a < 1) 11' (L) =pq-WL-fl Set dw/dL =0 ; solve for L*,q* and see: how does the profit maximizing output respond to changes in p how does the profit maximizing demand for L respond to changes in w ? “The role of the fixed cost in the shut down problem; 7 Start with the case q=21/L; pél, w=4 fl=0 Profit 17(L) = ZVI, -4L _ ‘ SettingcrYL) =0, we get (I/w/L) = 4 or, L* = 1/16 q* = 1/2 17* = 1/4 If the fixed cost level were fl = 1/8, you get the same L*,q*; but 17*:‘1/8 : fixed costs affect the profit level, but NOT the quantity of output supplied or the input demanded. Suppose that fl=1. 17*: -3/4. But it is in the interest of the firm to continue in production. A production Function with two inputs (1: F(I(,L) F(K,L)= 41(1/3L1/4: p=1, r=(1/2); W=(1/4) A more general case : q: F (I(,L) ; a widely used form q=KuLflwhere a+f3 < 1 Write profit as a function of K and Bus follows: 1-:- (K,L)= pF(K,L)- (K/Z) - (U4)-fl Set aF/aI( =0; aF/aL =0; solve for the profit maximizing values K *,L* Talk about the average and marginal productivities of each input (say, labor) given the value of the other input. Some assumptions about F Increasing, constant and decreasing returns to scale Isoquants Isoquants cannot intersect Absolute value of the slope of isoquaants is the MRTS Look at the production functions: F(K,L)= min {k,L} F(k,L)= K+L “That is the nature of input substitution ? Given the prices p,r,w we get: 11-(K,L) = pq - rK -wL-fi =pF(K,L)- rK- wL- fl Maximize 11' with respect to ICL. First order conditions : pFK = r pFL =w [VMP of any input = price of that input] Compute an example; ...
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Handout%203 - Handout #3, M .M ajumdar Department of...

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