Formulas and Approaches Consumer theory Lagrangian method for utility maximization Write down the given utility function and budget constraint: Utility function example: 𝑈(𝑍, 𝐵) = 𝑍 0.25 ∗ 𝐵 0.75 Budget constraint example: 𝑌 = 𝑍 + 𝐵 Now set up the lagrangian function: 𝐿(𝑍, 𝐵, 𝜆) = 𝑈(𝑍, 𝐵) − 𝜆(𝑝𝑍 𝑍 + 𝑝𝐵 𝐵 − 𝑌) In the example it would be: 𝐿(𝑍, 𝐵, 𝜆) = 𝑈(𝑍, 𝐵) − 𝜆(𝑍 + 𝐵 − 𝑌) Then you would take the partial derivative to get marginal utility for good Z and B: 𝜕𝐿 = 0.25𝑍 −0.75 𝐵0.75 − 𝜆 𝜕𝑍 𝜕𝐿 𝑀𝑈𝐵 = = 0.75𝑍 0.25 𝐵 −0.25 − 𝜆 𝜕𝐵 𝜕𝐿 =𝑍+𝐵−𝑌 𝜕𝜆 𝑀𝑈𝑍 = Now we isolate lamda in the two equations: 0.25𝑍 −0.75 𝐵0.75 − 𝜆 = 0 → 0.25𝑍 −0.75 𝐵0.75 = 𝜆 0.75𝑍 0.25 𝐵 −0.25 − 𝜆 = 0 → 0.75𝑍 0.25 𝐵 −0.25 = 𝜆 We then equate the two equations: 0.25𝑍 −0.75 𝐵 0.75 = 0.75𝑍 0.25 𝐵 −0.25 Simplify by removing negative exponents and cross multiply: 0.25𝐵 0.75 0.75𝑍 0.25 = 𝑍 0.75 𝐵 0.25 → 0.25𝐵 = 0.75𝑍 Isolate B: 𝐵 = 3𝑍 We substitute this into the lamda partial derivative and solve for Z: 𝑍 + (3𝑍) − 𝑌 = 0 → 4𝑍 = 𝑌 → 𝑍 = 1 𝑌 4 If the income was 800, then Y would equal 800: 𝑍 + (3𝑍) − 800 = 0 → 4𝑍 = 800 → 𝑍 = 800 → 𝑍 = 200 4 Then you would substitute that into the equation of what B is equal (or you could do it with the budget constraint): 𝐵 = 3𝑍 → 𝐵 = 3 ∗ 200 → 𝐵 = 600 This example implies that price of both goods is 1. Producer theory Three Examples of Economic Scale 1. Q = 2K + 3L: To determine the returns to scale, we will begin by increasing both K and L by m. Then we will create a new production function Q’. We will compare Q’ to Q.Q’ = 2(K*m) + 3(L*m) = 2*K*m + 3*L*m = m(2*K + 3*L) = m*Q - After factoring, we can replace (2*K + 3*L) with Q, as we were given that from the start. Since Q’ = m*Q we note that by increasing all of our inputs by the multiplier m we've increased production by exactly m. As a result, we have constant returns to scale. 2. Q=.5KL: Again, we increase both K and L by m and create a new production function. Q’ = .5(K*m)*(L*m) = .5*K*L*m2 = Q * m2 - Since m > 1, then m2 > m. Our new production has increased by more than m, so we have increasing returns to scale. 3. Q=K0.3L0.2: Again, we increase both K and L by m and create a new production function. Q’ = (K*m)0.3(L*m)0.2 = K0.3L0.2m0.5 = Q* m0.5 - Because m > 1, then m0.5 < m, our new production has increased by less than m, so we have decreasing returns to scale.