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| Name | Constrained Optimisation maximize profit function subject to constraint using Lagrange's multiplier |
| Video Uploader | Video From ECON MATHS |
| Upload Date | This Video Uploaded At 20-12-2020 01:26:14 |
| Video Discription | In this video we will solve a problem on constrained Optimisation. maximise the following profit function: π = 50x – 2x2– xy – 3y2 + 95y St: x + y = 25 Where x and y are the outputs of two products produced by the firm. First we form Lagrangian function by first setting the constraint function equal to zero 25 — x — y = 0 The next step in creating the Lagrangian function is to multiply this form of the constraint function by the unknown artificial factor λ and then adding the result to the given original objective function. Thus combining the constraint and the objective function through Lagrangian multplier (λ) we have L= 50x – 2x2 – xy – 3y2 + 95y + λ (25 — x — y) Where L stands for the expression of Lagrangian function. It can be seen that in this function there are three unknowns x, y and λ. Note that the solution that maximises Lagrangian function (L) will also maximise profit (λ) function: For maximising L, we first find partial derivatives of Lwith respect to three unknown x, y and λ and then set them equal to zero. Thus Lx = 50 – 4x – y –λ =0 …..(i) Ly = – x – 6y + 95 — λ =0 …..(ii) Lλ = x + y – 25 = 0 ….(iii) Here Lx, Ly and Lλ are partial derivatives wrt x , y and λ Note that the last equation (iii) is the constraint subject to which the original profit function has to be maximised. In fact, Lagrangian function is so constructed that the partial derivative of Lwith respect to λ (Lagrangian multiplier) always yields the original constraint function. Further, since the partial derivative of L with respect to λ is set equal to zero, it not only ensures that the constraint of the optimisation problem is fulfilled but also converts the Lagrangian function into original constrained profit maximisation problem so that the solution to both of them will yield the same result. now, we solve the above system of three equations with three unknowns to find the optimal values of x and y by substracting the equation (ii) from equation (i) – 45 – 3x + 5y = 0 ….(iv) Xing equation (iii) by 3 and adding it to equation (iv) we have – 45 – 3x + 5y =0 -75 + 3x + 3y =0/ -120 + 8y – =0 Thus, 8y = 120 y= 15 Substituting the value of y = 15 in the constraint function x + y = 25 we get x equal to 10. The value of λ can be obtained by substituting the solved values of x and y in a partial derivative equation containing λ in the Lagrangian function. Thus, in our above example, the value of X can be obtained by substituting x = 10 and y = 15 in equation (ii) above Thus, in the above equation (ii) – x – 6y + 95 + λ =0 λ = x + 6y – 95 λ = 10 + 90 – 95 λ = 5 Here λ can be interpreted as marginal profit at the production level of 25 units. It shows if the firm is required to produce 24 units instead of 25 units, its profits will fall by 5. On the other hand, if firm were to produce 26 instead of 25 units, its profits will increase by about 5. It is to be remembered that Lagrange technique maximises profits under a constraint. It does not solve unconstrained profit maximization. This video will help you to crack any Competitive exam for Economics like UGC NTA NET ECONOMICS, GATE ECONOMICS, UPSC , Delhi School of Economics, MA ENTRANCE ECONOMICS JAM ECONOMICS Check our playlist Algebra in Economics https://youtube.com/playlist?list=PLoJnMTDIbYhvpsJ99b4Nxmu3b5xUN7pfG GATE ECONOMICS https://youtube.com/playlist?list=PLoJnMTDIbYhswtFHC5CaaII3XoORmduMa Mathematical Economics https://youtube.com/playlist?list=PLoJnMTDIbYhtHNOr92jalC0kimJCxqpd5 Integration in ECONOMICS https://youtube.com/playlist?list=PLoJnMTDIbYhubivZ-izIO3jcyD-IYMv0Q Matrix Algebra in Economics https://youtube.com/playlist?list=PLoJnMTDIbYhvpsJ99b4Nxmu3b5xUN7pfG GRAPHING IN ECONOMICS https://youtube.com/playlist?list=PLoJnMTDIbYhvIqWQkTo3Zc2Bbr5-q0m3I Microeconomics https://youtube.com/playlist?list=PLoJnMTDIbYhv2J3KIbUQwEaFAFm-l_cMy Comparative Statics in Economics https://youtube.com/playlist?list=PLoJnMTDIbYhu77zSB02FlWchC9BEwrxtE INPUT OUTPUT MODEL https://youtube.com/playlist?list=PLoJnMTDIbYhuxbcTjmHBt85G6mXUH3yG3 IS-LM MODEL https://youtube.com/playlist?list=PLoJnMTDIbYhtYR3ZMEVfvqa5ve_Te7Ykl You can Join On Facebook https://www.facebook.com/profile.php?id=100028159118237 Facebook page https://www.facebook.com/ECONMATHSS/ On Telegram https://t.me/Hilal885 #MathematicalEconomics #JAMECONOMICS #ImportantQuestionsInEconomics #MAEntranceEconomics #GateEconomics #QuantitativeEconomics #EconMath #GATEEconomics #NETEconomics #DSU #KU #NTA #NetEconomics #JRF #IES #Economics #MathematicalEconomics #Economics #Optimisation #Constrained_Optimisation #Gate_Economics |
| Category | Education |
| Tags | Optimisation in Economics | constrained Optimisation in Economics | constrained Optimisation using Lagrange multiplier method | maximize profit function subject to the constraint | Lagrange's Function | Mathematical Economics | business Economics | constrained optimisation methods | constrained optimisation problem | constrained optimization with equality constraints |
