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Topic 3: Theory of the Firm

1. A profit-maximising firm has a production function given by:

q = ALαKβ

where α > 0, β > 0, α + β < 1, , L denotes labour and K denotes capital. The firm’s total costs are given by: WL + rK where W denotes wages, r denotes the unit price of capital. The firm sells its output at a given market price, p.

(a) Explain the difference between decreasing, constant and increasing returns to scale and indicate which describes this firm’s production function.          [20%]

(b) Derive expressions for the firm’s optimal demand for labour and capital and comment on their properties.                [20%]

(c) Derive the supply function of the firm and comment on its properties. Explain what happens to the supply function if α + β = 1            [20%]

(d) The firm now aims to maximise output subject to incurring costs, C*. Derive expressions for the optimal demand for labour and capital as a function of output, comment on their properties and compare the expressions to those derived in part (b).  [40%]

2. A firm has a production function given by:

q = ALαKβ

where q denotes output, L denotes labour, K denotes capital, A>0 and A>0. The firm’s total costs are given by:

C = WL + rK

where w denotes wages and r denotes the unit price of capital.            

(a) The producer aims to maximise output subject to total costs being equal to C*. Derive the input demand functions for L and K and comment on how realistic their properties are.   [40%]

(b) Derive the cost function associated with part (a) and comment on its properties. [20%]

(c) Present a general proof of Shephard’s Lemma in the context of producer theory and comment on its role in the theory of the firm. [40%]

3. A firm has a cost function, derived via output constrained cost minimisation, given by:

where denotes the target output level, w denotes wages, r denotes the unit price of capital, α + β = 1, α > 0 and β > 0.

(a) Give an example of an industry where output constrained cost minimisation might apply and analyse four properties of the above cost function.          [20%]

(b) Use the above cost function to derive the conditional input demand functions for capital and labour. [40%]

(c) Analyse four properties of the conditional input demand functions derived in part (b) and comment on the realism of these properties. [20%]

(d) Comment on the significance of Shephard’s Lemma for producer theory.   [20%]

4. A firm has a production function given by:

q = ALαKβ

where q denotes output, L denotes labour, K denotes capital, A>0 and α + β = 1. The firm’s total costs are given by:

C = WL + rK

where w denotes wages and r denotes the unit price of capital.

(a) Explain whether the production function has increasing, decreasing or constant returns to scale. [10%]

(c)  The producer aims to minimize total costs subject to producing a target output level of . Derive the conditional input demand functions for K and L and state four properties of the conditional input demand functions. [50%]

(c) Derive the cost function associated with part (b) and state four of its properties. [20%]

(d) Use Shephard’s Lemma to derive the conditional input demand functions from the cost function. [20%]