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Tutorial Questions Week 5

ECON8025

Semester 2, 2022.

Exercise 1. Consider a firm that produces an output denoted by y, using a production function f : R2 → R defined by

f (x, z) := min{x, 2z}.

Denote the prices of inputs by wx and wz and the output price by p, and assume that all prices are strictly positive.

(i) Sketch the short-run production function f (x, ) as a function of x, when z is fixed at some arbitrary level . Use appropriate set notation to define the iso- profit lines associated with the corresponding short-run profit maximisation prob- lem, and sketch a family of such iso-profit lines on the same graph as your pro- duction function f (x, ). Make sure to carefully identify all relevant points, lines and slopes. Use your graph to carefully analyse how the solution to the short- run profit maximisation problem varies as a function of the relevant parameter values.

(ii) Now consider the long-run case where both x and z are variable, and sketch an isoquant of f (x, z) corresponding to the output level y = 2. On the same graph, sketch a family of iso-expenditure lines for this firm, assuming that input prices wx and wz are strictly positive. Remember to carefully identify all relevant points, lines and slopes.

(iii) Write down the firm’s long-run cost minimisation problem resulting from an arbitrary output target y > 0, and use your graph from part (ii) to derive the solution to this minimisation problem, including the conditional input demand functions and the resulting cost function. Carefully justify your answers.

Hint: f (x, z) is not a differentiable function, so you cannot just apply the Kuhn-

Tucker conditions!

(iv) Derive the average and marginal cost functions corresponding to the cost func- tion you found in part (iii), and discuss how the firm’s long-run profit-maximising output level varies as a function of the output price p, assuming that the firm is a price-taker in a competitive market. Briefly explain your answers.

Exercise 2. Consider the two-input Cobb-Douglas production function f : R → R defined by

f (x, z) := xazb, where a, b > 0.

Let wx and wz denote the respective input prices, and let p denote the output price.

(i) Calculate the marginal product of input x as a function of (x, z), and analyse whether it is increasing, constant, or decreasing in x, depending on the magni- tude of a.

(ii) Show that f has constant returns to scale if a + b = 1, increasing returns to scale

if a + b > 1, and decreasing returns to scale if a + b < 1.

(iii) For the case where a + b < 1, solve the (long-run) profit maximisation problem

x(m)x{pf (x, z) − wxx − wzz},

and derive the corresponding input demand and output supply functions. You can assume that the profit-maximising input bundle has strictly positive levels of both inputs, and that the necessary conditions discussed in lecture are also sufficient to determine the maximiser.

Hint: You just need to derive the necessary conditions and solve for x and z—this is somewhat complex, but will give your algebraic muscles a work-out.

(iv) Now consider the short-run problem of choosing a profit-maximising level of the first input, x ≥ 0, given a fixed level of the second input, > 0. Discuss how the solution to this problem varies as a function of the magnitude of a. For the case

where a < 1, calculate the corresponding input demand function for input x, as well as the output supply function.

(v) For the case where a < 1, solve the short-run cost minimisation problem

n0{wxx + wz} s.t. f (x, ) ≥ y,

and find the associated (short-run) conditional input demand for good 1, x。(wx, wz, y, ), and the associated (short-run) cost function, c(wx, wz, y, ).