Answer to Question #228181 in Microeconomics for Emy

Question #228181

If a firm is producing in the long run using capital and labor inputs, how the least cost 

combination of the inputs will be determined? Show graphically and mathematically.


1
Expert's answer
2021-08-30T17:45:13-0400

"\\frac{min}{(L,K)}\n\u200b\t\n wL+rK ....(1.1)"

"Considering;\n\nq=f=(L,K)... (1.2)""The" "Lagrangian" "Function" "Can be" "Define" "as"

"\u2227(L,K,\u03bb)=wL+rK\u2212\u03bb(f(L,K)\u2212q) ....(1.3)""where; \u03bb => Lagrange multiplier"

The initial order conditions for an interior solution when L > 0 and K > 0 include:

"\\frac{\u2202\u2227}{\u2202L} \n\n\u200b\t\n =0\u21d2w=\\frac{\u03bb\u2202f(L,K)}{\u2202L} \n\n\u200b\t\n..... (1.4)"


"\\frac{\u2202\u2227}{\u2202L}\n\u200b\t\n =0\u21d2r= \\frac{\u03bb\u2202f(L,K)}{\u2202L} \n\n\u200b\t...... (1.5)"


"\\frac{\u2202\u2227}{\u2202L}=0\u21d2Q= \\frac{\u03bb\u2202f(L,K)}{\u2202L} \n... (1.6)"


Based on the 6th module


"MPL= \n\\frac{\u2202f(L,K)}{\u2202L}\n\u200b\t\n and MPK=\\frac{\u2202f(L,K)}{\u2202K}\n\u200b"

Substituting 1.4 as well as 1.5 to eliminate Lagrange multiplier yields (expression 1.1):


"\\frac{MPl}{MPk} \n\n\u200b\t\n =\\frac{w}{r} \nr\nw\n\u200b\t\n........ (1.7)"


While 1.6 tends to be the constraint:

"q=f(L,K) ........(1.8)"


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