Given a production function of the form q(L, K) = A(min(aL, bK)) where A = 100, a = 1, b = 4, wages = 4, and rent = 25. What is the minimum cost at which the firm can produce 1,000 units when both L and K are variable?Suppose a firm uses labor and capital for its production, the production function is: Q(L, K) = L1/3 K 2/3 Labor wage w=5, capital rental rate r= 2*w; The long-run cost function is: c(q) = a+ Bq Find the value of B.Suppose a firm uses labor and capital for its production, the production function is: Q(L, K) = L1/3 K2/3 Labor wage w=2, capital rental rate r= 2*w; The long-run cost function is: c(q) = a+ BqY Find the value of Y.The long - run cost function can be derived by minimising the cost of producing q units of output . minimize colt ock sulyect to ( L, K ) = q minimize WL trk subject to L K - gg axe Using the lagrange multiplies method , we can write Langragion as : L= COL + eck + 2 Lax - L'3 143 7 Taking the partial derivatives of L cort L . K and d and setting them equal to zero we get, = 0: * - 1 ( 2 ] 1 13 K- 1 3 = 0 9 - LY3 1 43 = 0Solving for L, K and A and setting them egl Solving for L and K in terms of gy,w andre, we get : L = a 3 9 w- K = 9 3 4 w at 0 = m Substituting these values for L and K into the Languagean equation, we get : c ( q ) = WL + uk c ( q ) = ( 20 ] 3/ 4 (3 3 / 4 ccq ) = ~ + B q Here B Z 2 0 3 3 / 4 Putting w = 5 B = 2 ( 5 ) 10 3 3 / 4 30.75 B = 10 H . 38 2 . 28 | B = 4 .38