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Homework answers / question archive / Suppose that a firm's production function is given by: q = 10K0
Suppose that a firm's production function is given by: q = 10K0.5L0.5 , and the prices of labor and capital are, w= $10, and r=$20 respectively. Derive and plot the MC, AC, AVC curves assuming K is only source of fixed cost.
As given, the cost curves will be derived when the value of K and L are derived. The equilibrium values of K and L are derived when we minimize the cost with the given production function.
MinTC=wL+rK(Given)q=10K0.5L0.5(Lagrange)l=10L+20K+λ{q−10K0.5L0.5}dldK=20−λ[10(0.5)L0.5K0.5]MinTC=wL+rK(Given)q=10K0.5L0.5(Lagrange)l=10L+20K+λ{q−10K0.5L0.5}dldK=20−λ[10(0.5)L0.5K0.5]
dldL=10−λ[10(0.5)K0.5L0.5]dldL,dldK=010[10(0.5)K0.5L0.5]=20[10(0.5)L0.5K0.5]K=12LdldL=10−λ[10(0.5)K0.5L0.5]dldL,dldK=010[10(0.5)K0.5L0.5]=20[10(0.5)L0.5K0.5]K=12L
Putting the values of K and L in the production function,
q=10K0.5L0.5q=10(12L)0.5L0.5q=10√2LL=√210qq=10K0.5L0.5q=10(12L)0.5L0.5q=102LL=210q
Keeping K Fixed
TC=10(√210K)+20¯¯¯¯¯KTC=√2q+20¯¯¯¯¯KMC=dTCdqMC=√2TC=10(210K)+20K¯TC=2q+20K¯MC=dTCdqMC=2
TVC=√2qAVC=qAC=TCqAC=√2+20¯¯¯¯¯Kq
please see the attached file for the complete solution.
