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.