<p>Don't get stuck with math.</p>

<p>Look again at the production function. Perhaps draw out isoquant curves (for the same output, how could you vary <span class="math-container">$K$</span> and <span class="math-container">$L$</span>). How are the isoquants for this production function different from those of standard Cobb-Douglas production function?</p>

<p>Hint: perfect substitute.</p>

CC BY-SA 4.0
2020-03-04T05:19:07.037Z
Thank you for this! Now I feel stupid. It was such a simple solution. Thank you so much!
Magic
CC BY-SA 4.0
2020-03-04T02:56:21.053Z
2020-03-04T02:56:21.053Z
Art
<p>Suppose that I have a production function <span class="math-container">$(aK + bL)^3$</span> in a perfect competition where a and b are constants. I am confused on how to obtain the long-run cost function from this production function without using Lagrange. I have obviously tried <span class="math-container">$MRTS=w/r$</span>, where w is wage and r is rent for capital, to express L in terms of K or vice versa. This does not yield to anything as the variables K and L just get cancelled in the MRTS. I am utterly lost. How do I go about this?</p>

CC BY-SA 4.0
2020-03-03T23:33:49.443Z
Can you show what you have tried so far?
Brennan
CC BY-SA 4.0
2020-03-04T04:50:51.977Z
Well, the usual way to proceed is to express L in term of K and plug that into the production function. The way to do this is by definition that MRTS = w/r. However, MRTS in this case is (3b(aK + bL)^2)/(3a(aK + bL)^2). So (aK + bL)^2 would just cancel and I cannot find a definition to utilize.
Magic
<microeconomics><production-function><cost><long-run>
Deriving long-run cost functions from production function
Magic
# Deriving long-run cost functions from production function

(Asked by: username_0 on 2020-03-03)

Suppose that I have a production function $(aK + bL)^3$ in a perfect competition where a and b are constants. I am confused on how to obtain the long-run cost function from this production function without using Lagrange. I have obviously tried $MRTS=w/r$, where w is wage and r is rent for capital, to express L in terms of K or vice versa. This does not yield to anything as the variables K and L just get cancelled in the MRTS. I am utterly lost. How do I go about this?

**Tags:** microeconomics, production-function, cost, long-run

> **Comments:**
> **username_1**: Can you show what you have tried so far?  
> **username_0**: Well, the usual way to proceed is to express L in term of K and plug that into the production function. The way to do this is by definition that MRTS = w/r. However, MRTS in this case is (3b(aK + bL)^2)/(3a(aK + bL)^2). So (aK + bL)^2 would just cancel and I cannot find a definition to utilize.  

---

## 1 answers

### Answer by username_2 on 2020-03-04. Score: 3 | Accepted answer

Don't get stuck with math.

Look again at the production function. Perhaps draw out isoquant curves (for the same output, how could you vary $K$ and $L$). How are the isoquants for this production function different from those of standard Cobb-Douglas production function?

Hint: perfect substitute.

> **Comments:**
> **username_0**: Thank you for this! Now I feel stupid. It was such a simple solution. Thank you so much!  

---

