Questions: The output of an economic system Q, subject to two inputs, such as labor L and capital K, is often modeled by the Cobb-Douglas production function Q=c L^a K^b. When a+b=1, the case is called constant returns to scale. Suppose Q=1451, a=8/9, b=1/9, and c=72. Complete parts (a) and (b). a. Find the rate of change of capital with respect to labor, dK/dL. First substitute the given values of Q, a, b, and c into the production function.

The output of an economic system Q, subject to two inputs, such as labor L and capital K, is often modeled by the Cobb-Douglas production function Q=c L^a K^b. When a+b=1, the case is called constant returns to scale. Suppose Q=1451, a=8/9, b=1/9, and c=72. Complete parts (a) and (b).
a. Find the rate of change of capital with respect to labor, dK/dL. First substitute the given values of Q, a, b, and c into the production function.

Transcript text: The output of an economic system $Q$, subject to two inputs, such as labor $L$ and capital $K$, is often modeled by the Cobb-Douglas production function $Q=c L^{a} K^{b}$. When $a+b=1$, the case is called constant returns to scale. Suppose $Q=1451, a=\frac{8}{9}, b=\frac{1}{9}$, and $c=72$. Complete parts (a) and (b). a. Find the rate of change of capital with respect to labor, $\frac{d K}{d L}$. First substitute the given values of $Q, a, b$, and $c$ into the production function.

Solution

Solution Steps

To find the rate of change of capital with respect to labor, $\frac{dK}{dL}$, we need to use implicit differentiation on the Cobb-Douglas production function $Q = c L^a K^b$. Given the values $Q = 1451$, $a = \frac{8}{9}$, $b = \frac{1}{9}$, and $c = 72$, we can substitute these into the function and then differentiate implicitly with respect to $L$.

Solution Approach

Substitute the given values into the Cobb-Douglas production function.
Differentiate implicitly with respect to $L$.
Solve for $\frac{dK}{dL}$.

Step 1: Substitute Given Values into the Production Function

The Cobb-Douglas production function is given by: \[ Q = c L^a K^b \]

We are given: \[ Q = 1451 \] \[ a = \frac{8}{9} \] \[ b = \frac{1}{9} \] \[ c = 72 \]

Substituting these values into the production function, we get: \[ 1451 = 72 L^{\frac{8}{9}} K^{\frac{1}{9}} \]

Step 2: Solve for $ K $ in Terms of $ L $

To find the rate of change of capital with respect to labor, we first need to express $ K $ in terms of $ L $. Rearrange the equation to solve for $ K $: \[ 1451 = 72 L^{\frac{8}{9}} K^{\frac{1}{9}} \] \[ \frac{1451}{72} = L^{\frac{8}{9}} K^{\frac{1}{9}} \] \[ \left( \frac{1451}{72} \right)^9 = \left( L^{\frac{8}{9}} K^{\frac{1}{9}} \right)^9 \] \[ \left( \frac{1451}{72} \right)^9 = L^8 K \] \[ K = \frac{\left( \frac{1451}{72} \right)^9}{L^8} \]

Step 3: Differentiate $ K $ with Respect to $ L $

Now, we differentiate $ K $ with respect to $ L $: \[ K = \frac{\left( \frac{1451}{72} \right)^9}{L^8} \] Let $ C = \left( \frac{1451}{72} \right)^9 $, then: \[ K = \frac{C}{L^8} \]

Differentiate both sides with respect to $ L $: \[ \frac{dK}{dL} = \frac{d}{dL} \left( \frac{C}{L^8} \right) \] \[ \frac{dK}{dL} = C \cdot \frac{d}{dL} \left( L^{-8} \right) \] \[ \frac{dK}{dL} = C \cdot (-8) L^{-9} \] \[ \frac{dK}{dL} = -8C L^{-9} \]

Substitute back $ C = \left( \frac{1451}{72} \right)^9 $: \[ \frac{dK}{dL} = -8 \left( \frac{1451}{72} \right)^9 L^{-9} \]