Given the production function $Q = 100K^{0.5}L^{0.5}$, let's solve for the requested expressions:

i) Marginal Rate of Technical Substitution (MRTS)

The Marginal Rate of Technical Substitution (MRTS) between capital $K$ and labor $L$ is defined as the negative ratio of the marginal products of labor and capital:

$\text{MRTS} = -\frac{\text{Marginal Product of Labor (MPL)}}{\text{Marginal Product of Capital (MPK)}}$

First, let's compute the Marginal Product of Labor (MPL) and the Marginal Product of Capital (MPK):

$\text{MPL} = \frac{\partial Q}{\partial L} = 100 \times 0.5 \times K^{0.5} \times L^{-0.5} = 50 \frac{K^{0.5}}{L^{0.5}}$

$\text{MPK} = \frac{\partial Q}{\partial K} = 100 \times 0.5 \times K^{-0.5} \times L^{0.5} = 50 \frac{L^{0.5}}{K^{0.5}}$

Now, substitute these into the MRTS formula:

$\text{MRTS} = -\frac{50 \frac{K^{0.5}}{L^{0.5}}}{50 \frac{L^{0.5}}{K^{0.5}}} = -\frac{K^{0.5}}{L^{0.5}} \times \frac{K^{0.5}}{L^{0.5}} = -\frac{K}{L}$

So, the expression for MRTS is:

$\text{MRTS} = -\frac{K}{L}$

ii) Elasticity of Output with Respect to Capital ($\varepsilon_K$)

The elasticity of output with respect to capital is defined as:

$\varepsilon_K = \frac{\partial Q}{\partial K} \times \frac{K}{Q}$

We already calculated $\frac{\partial Q}{\partial K} = 50 \frac{L^{0.5}}{K^{0.5}}$, so let's compute $\varepsilon_K$:

$\varepsilon_K = \left(50 \frac{L^{0.5}}{K^{0.5}}\right) \times \frac{K}{100K^{0.5}L^{0.5}} = \frac{50L^{0.5}K^{0.5}}{100K^{0.5}L^{0.5}} = \frac{1}{2}$

Thus, the elasticity of output with respect to capital is:

$\varepsilon_K = \frac{1}{2}$

iii) Elasticity of Output with Respect to Labor ($\varepsilon_L$)

The elasticity of output with respect to labor is defined as:

$\varepsilon_L = \frac{\partial Q}{\partial L} \times \frac{L}{Q}$

We already calculated $\frac{\partial Q}{\partial L} = 50 \frac{K^{0.5}}{L^{0.5}}$, so let's compute $\varepsilon_L$:

$\varepsilon_L = \left(50 \frac{K^{0.5}}{L^{0.5}}\right) \times \frac{L}{100K^{0.5}L^{0.5}} = \frac{50K^{0.5}L^{0.5}}{100K^{0.5}L^{0.5}} = \frac{1}{2}$

Thus, the elasticity of output with respect to labor is:

$\varepsilon_L = \frac{1}{2}$

Summary of Results:

1. MRTS: $\text{MRTS} = -\frac{K}{L}$
2. Elasticity of Output with Respect to Capital: $\varepsilon_K = \frac{1}{2}$
3. Elasticity of Output with Respect to Labor: $\varepsilon_L = \frac{1}{2}$

Would you like any further details or explanations?

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Here are five related questions you might find interesting:

1. How does the MRTS change if the production function has increasing returns to scale?
2. What is the significance of an elasticity of output with respect to capital being equal to 0.5?
3. How would you interpret the elasticity of output with respect to labor in a real-world context?
4. Can you derive the production function if the MRTS is constant?
5. How do changes in capital and labor inputs affect the output in this specific production function?

Tip: Elasticities can help you understand how sensitive output is to changes in inputs, which is crucial for optimizing production in economics.