3. Given a production function $$Q = \sqrt{(L + K)}$$, answer the following questions:

(a) \((5\%)\) Verify the degree of homogeneity of this production function.

(b) \((5\%)\) Calculate the elasticity of substitution for this production function.

(c) \((5\%)\) Let \(w\) and \(r\) denote the prices for \(L\) and \(K\), respectively. Derive the demand function for \(L\) (i.e., \(L\) as a function of \(Q\), \(w\), and \(r\)).

(d) (3%) Compare your answer in (c) to a demand function for \( L \) derived from a Cobb-Douglas production function (e.g., \( Q = L^{0.5}K^{0.5} \)), which demand for \( L \) is relatively insensitive to \( w \)?

4. Suppose there are 80 identical competitive firms, each faces the same short-run total cost function $$\mathrm{STC}(Q) = 96 + 2Q^2$$, and assume all its fixed costs are sunk. Let the market demand equals $$D(P) = 360 - 4P$$, answer the following questions: (a) \((5\%)\) What is the producer surplus of an individual firm?

(b) \((3\%)\) What do you know about the firm's economic profit?

(c) \((5\%)\) What is the overall producer surplus for the market?

(d) \((5\%)\) What are your answers to (a) and (c) when the market demand becomes \(D(P) = 150 - 5P\)?

5. Suppose Max is a profit-maximizing monopoly firm that sells a well-designed chair. The firm faces the demand curve $$P = 28 - 0.5Q$$. I. If the total cost for producing Q units of output is \(\mathrm{TC} = 24 + 1.5\mathrm{Q}^2\). The fixed cost is sunk, and the marginal cost curve is \(\mathrm{MC} = 3\mathrm{Q}\). (a) (6%) How much extra surplus does the producer capture when it can engage in first-degree price discrimination instead of charging a uniform price?

II. If the marginal cost for each chair is \( \mathrm{MC} = \$ 4 \), and there is no fixed cost. Suppose Max wants to improve its profit by implementing block pricing (assuming just 2 blocks), answer the following questions: (b)(6%) Find the profit-maximizing quantity and price for block 1 \((P{1}, Q{1})\) and for block 2 \((P{2}, Q{2})\).

(c)(3%) What's the firm's producer surplus under the two-block tariff in part b?

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