Question

The output \(Q\) of an economic system subject to two inputs, such as labor \(L\) and capital \(K,\) is often modeled by the Cobb-Douglas production function \(Q(L, K)=c L^{a} K^{b} .\) Suppose \(a=\frac{1}{3}, b=\frac{2}{3},\) and \(c=1\) a. Evaluate the partial derivatives \(Q_{L}\) and \(Q_{K}\) b. Suppose \(L=10\) is fixed and \(K\) increases from \(K=20\) to \(K=20.5 .\) Use linear approximation to estimate the change in \(Q\) c. Suppose \(K=20\) is fixed and \(L\) decreases from \(L=10\) to \(L=9.5 .\) Use linear approximation to estimate the change in \(Q\) d. Graph the level curves of the production function in the first quadrant of the \(L K\) -plane for \(Q=1,2,\) and 3 e. Use the graph of part (d). If you move along the vertical line \(L=2\) in the positive \(K\) -direction, how does \(Q\) change? Is this consistent with \(Q_{K}\) computed in part (a)? f. Use the graph of part (d). If you move along the horizontal line \(K=2\) in the positive \(L\) -direction, how does \(Q\) change? Is this consistent with \(Q_{L}\) computed in part (a)?

Short answer

Answer: When moving in the positive K direction along the vertical line L = 2, the output Q increases. This is consistent with the computed \(Q_{K}\) in part (a) since \(Q_K(2, K)\) is positive.

Step-by-step solution

Calculate \(Q_L\)

Differentiate the function \(Q(L,K) = L^{\frac{1}{3}}K^{\frac{2}{3}}\) with respect to L to get \(Q_{L}= \frac{1}{3}L^{-\frac{2}{3}}K^{\frac{2}{3}}\).

Calculate \(Q_K\)

Similarly, differentiate the function \(Q(L,K)\) with respect to K to get \(Q_{K}= \frac{2}{3}L^{\frac{1}{3}}K^{-\frac{1}{3}}\). b. Estimate the change in Q using linear approximation for L = 10, K = 20 to 20.5

Calculate \(\Delta Q\) for L = 10

For the change of Q when \(L=10\) and \(K\) increases from 20 to 20.5, we have: \(\Delta Q \approx Q_{K}(10, 20) \cdot \Delta K\), where \(\Delta K = 0.5\). Plug in the values for \(Q_K\) and do the calculation: \(\Delta Q \approx \frac{2}{3}(10)^{\frac{1}{3}}(20)^{-\frac{1}{3}}(0.5) \approx 0.1453\). c. Estimate the change in Q using linear approximation for K = 20, L = 10 to 9.5

Calculate \(\Delta Q\) for K = 20

For the change of Q when \(K=20\) and \(L\) decreases from 10 to 9.5, we have: \(\Delta Q \approx Q_{L}(20, 10) \cdot \Delta L\), where \(\Delta L = -0.5\). Plug in the values for \(Q_L\) and do the calculation: \(\Delta Q \approx \frac{1}{3}(20)^{-\frac{2}{3}}(10)^{\frac{2}{3}}(-0.5) \approx -0.09625\). d. Graphing level curves for Q = 1, 2, and 3

Solve for K

For graphing level curves, we need a function of K in terms of L: \(Q = L^{\frac{1}{3}}K^{\frac{2}{3}} \Rightarrow K = (\frac{Q}{L^{\frac{1}{3}}})^{\frac{3}{2}}\). The level curves of this function for Q = 1, 2, and 3 will be plotted. e. Analyzing Q change along the vertical line L = 2.

Evaluate \(Q_K\) at L=2

Calculate \(Q_K\) for the point L=2: \(Q_K(2, K) = \frac{2}{3}(2)^{\frac{1}{3}}(K)^{-\frac{1}{3}}\). Since this value is positive, Q increases as we move in the positive K direction along L = 2. f. Analyzing Q change along the horizontal line K = 2.

Evaluate \(Q_L\) at K=2

Calculate \(Q_L\) for the point K=2: \(Q_L(L, 2) = \frac{1}{3}(L)^{-\frac{2}{3}}(2)^{\frac{2}{3}}\). Since this value is positive, Q increases as we move in the positive L direction along K = 2.

Key concepts

Partial Derivatives

In the context of the Cobb-Douglas production function, partial derivatives help us understand how changes in individual inputs, like labor (\(L\)) or capital (\(K\)), affect production output (\(Q\)). The formula for the Cobb-Douglas function is \(Q(L, K) = cL^aK^b\). To find the partial derivative with respect to \(L\), we fix \(K\) and differentiate with respect to \(L\), resulting in \(Q_L = \frac{1}{3}L^{-\frac{2}{3}}K^{\frac{2}{3}}\). This shows how output changes with a small increase in labor, keeping capital constant.
Similarly, for the partial derivative with respect to \(K\), we differentiate keeping \(L\) fixed, leading to \(Q_K = \frac{2}{3}L^{\frac{1}{3}}K^{-\frac{1}{3}}\). This tells us how output responds to changes in capital while labor remains the same.
Understanding these derivatives helps businesses and economists predict the effects of varying inputs in production, which is crucial for optimizing resource allocation.

Linear Approximation

Linear approximation provides a simple method to estimate changes in a function based on small changes in the variables. It is particularly useful in economics when evaluating the changes in production output (\(Q\)) due to changes in inputs, without recalculating the entire function.
If \(L\) or \(K\) changes slightly, we can estimate the change in \(Q\) using the partial derivatives. For example, if \(L = 10\) and \(K\) increases from 20 to 20.5, the change \(\Delta Q\) is approximately \(Q_K(10, 20) \cdot \Delta K\). This calculation simplifies to \(0.1453\), showing a small increase in output.
Similarly, for a decrease in \(L\) from 10 to 9.5 with \(K = 20\), use \(Q_L(20,10) \cdot \Delta L\), yielding \(\Delta Q \approx -0.09625\). This reflects a slight decrease in output with less labor. Thus, linear approximation is a handy tool to predict shifts in production efficiently.

Level Curves

Level curves are a graphical representation of all input combinations that produce the same output level in the Cobb-Douglas production function. Given by \(Q = L^{\frac{1}{3}}K^{\frac{2}{3}}\), level curves help visualize the relationship between inputs and output.
By fixing \(Q\) at a constant value, like 1, 2, or 3, and varying \(L\) and \(K\), we obtain curves that graphically display where these output levels occur. The formula \(K = \left(\frac{Q}{L^{\frac{1}{3}}}\right)^{\frac{3}{2}}\) describes these curves.
Each curve indicates sets of \(L\) and \(K\) that provide the same \(Q\). They are crucial in economic analysis as they show how changes in one resource can be compensated by another while maintaining the same output level, assisting in making informed production and cost decisions.

Economic Modeling

Economic modeling, using functions like the Cobb-Douglas, is fundamental in understanding and predicting economic behaviors. By modeling outputs based on inputs, these functions can simulate how changing conditions affect production.
The Cobb-Douglas model helps economists and businesses examine the elasticity of output in relation to labor and capital. It shows how responsive output is when inputs change, helping forecast potential increases or decreases in production.
These models become significant in resource distribution and policy creation. By analyzing the results, decision-makers can allocate resources more efficiently, plan for changes in economic conditions, and anticipate future needs. Models like the Cobb-Douglas effectively connect theory with real-world economic dynamics, providing a framework for strategic planning and comparative analysis.