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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)= twoCM)
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) =pqWLﬂ 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 wLﬁ =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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 '06
 MASSON
 Microeconomics

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