Derive the cost-minimizing demands for K and L as a function of output (q), wage rates (w) and rental rates (r?

Suppose the process of producing lightweight parkas by Polly's Parkas is described by the function





q = 10(K^0.8)((L - 40)^0.2)





where q is the number of parkas produced, K the number of computerized stitching-machine hours and L the number of person-hours of labor. Also, $10 worth of raw materials is used in production of each parka.Derive the cost-minimizing demands for K and L as a function of output (q), wage rates (w) and rental rates (r?
Doesn't seems like all assumptions are given.


Usually factor demand is given by FactorCost=w


MFC=w


MPxMR=MRP


MRP=MFC


MRP=w


But we have only Marginal product but not marginal revenue.


We can assume what each factor earns it's marginal product, thus


MPK=r


MPL=w


MPK = 鈭俀/鈭侹 = 8x((L-40)/K)^(1/5)


MPL = 鈭俀/鈭侺 = 2x(K/(L-40))^(4/5)





Demand for factor is sometimes marginal cost curve,thus MFC=鈭俆C[F]/鈭侳


TC=wL+rK+10Q


Q[K;L] = 10x(K^(4/5))x((L-40)^(1/5))


Equilibrium is given by:


MPK/r=MPL/w


MPK = 鈭俀/鈭侹 = 8x((L-40)/K)^(1/5)


MPL = 鈭俀/鈭侺 = 2x(K/(L-40))^(4/5)





Now from MPK/r = MPL/w derive K and L:


K[L] = 4w(L-40)/r


L[K] = (rK/4w)+40





Now substitute L[K] and K[L] into Q[K;L] to get Q[K] and Q[L]


Q[L] = 20 x 2^(3/5) x (L-40)^(1/5) x ((L-40)w/r)^(4/5)


(from this we can derive L[Q;w;r] but formula is too long)


Q[K] = 5 x 2^(3/5) x K^(4/5) x (Kr/w)^(1/5)


(from this we can derive K[Q;w;r] but formula is too long)





To derive cost-minimizing we canput all values into TC formula:


TC[L;K;Q]=wL+rK+10Q


TC[K]=wL[K]+rK+10Q[K]


TC[L]=wL+rK[L]+10Q[L]


鈭俆C[K]/鈭侹 = MC[K] = Factor cost = r


鈭俆C[L]/鈭侺 = MC[L] = Factor cost = w





P.S. I'm quite sure what I've messed up something here, so may be this solution is totally wrong.