Economics Assignment 4th year advanced micro university

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2. Producer Theory Practice Set 1
V.Bardis
1. Consider the production functions given below where q and x denote respectively the amount of
output and the amount of the input used to produce it. State the profix maximization problem of the firm
assuming the firm is a price taker in the output and input markets. If the problem can be solved, find
the solution (that is, find the firm's market input demand, supply and maximum profit function). If not,
explain why.
a)
b)
c)
d)
e)
f)
q = min( ln(x+1),c),
where c is a positive constant

3. 3. Consider the functions given below where p and w denote the output price and input price
respectively. Determine whether each function is a (maximum value of) profit function. If it is, find the
firm's supply and market input demand and, if possible, `recover' (i.e., find) the production function.
a)
b)
c)
d)
e)
f)
g)
h)
2. Suppose a firm can produce output (q) using one input (x) as described by the production function
where the parameters A and b are positive and k is non-negative.
a) Identify conditions on k and b such that f is concave on the part of its domain where x>k.
b) Let e(x) denote the elasticity of scale which is defined as
Identify conditions on b and k such that e(x)<1 (decreasing returns to scale),
e(x)=1 (constant returns to scale) and e(x) >1 (increasing returns to scale).
and so in the one-input case, e(x) is simply given by

4. 5. For each of the functions in question 3 that are profit functions, find the real profit function by
dividing the function with the price of output p. Does the real profit function, R, have the same
properties as the profit function ? Other than continuity, what are the properties of the real profit
function ? State and prove each of the properties of R (other than continuity).
6. Find the firm's conditional input demand function and the (mininum value of) cost function for each of
the production functions given in questions 1 and 2. Find and compare MC (marginal cost) with AC
(average total cost) in each case.
7. For each of the proper profit functions given in question 3, find the (minimum value of) cost function of
the firm.
8. TRUE or FALSE: :
a) If a firm's production function exhibits decreasing returns to scale, its average product must exceed its
marginal product and its marginal cost must exceed its average cost.
b) If average cost has a unique minimum value at output qe >0, then it is equal to marginal cost at qe .
c) The production function g(x) = min( ln(x+1),c), where c is a positive constant, is concave.
4. Suppose the profit function of a price taking firm is given by
where p is the price of output and w is the price of the single input used by the firm. Find and
interpret the equation of the tangent plane to the graph of the function at p =2e and w=2, where "e" is
the exponential number.
9. Suppose a single-input firm's (minimum value of) cost function is given by
C(w,q) = w ln(q+1)
where w is the price of the input and q denotes output.
a) Find and compare the firm's marginal cost and average cost. What type of returns to scale does the
firm's production technology exhibit?
b) Find the firm's conditional input demand.
c) Find the firm's production function and calculate the elasticity of scale to confirm your result in part (a).

5. 10. Suppose a firm's (minimum value of) cost function is given by
where w is the price of the input and q denotes output.
a) Find the firm's marginal cost and average cost.
b) Find the minimum value of the firm's average cost and identify the output ranges where the firm's
production exhibits (i) increasing returns and (ii) decreasing returns.
c) Find the firm's conditional input demand.
d) Find the firm's production function and calculate the elasticity of scale to confirm your results
in part (a).
11. In the one-input case and assuming differentiability, show that for input amount x and output
amount q such that q = f(x), we have e(x) = AC(w,q)/MC(w,q) given MC(w,q) =/= 0, i.e., that is, the
elasticity of scale is equal to the ratio of average cost to marginal cost.
12. Find the firm's (maximum value of) profit function from the (minimum value of) cost function giv en in
question 10 without deriving and using the production function of the firm.
13. Determine the allocation of L units of labour among K firms producing the same good such that the total
output produced is maximized and derive the aggregate production function given the production function
of the individual firm i is
(a)
(b)
(c)
(d)
(e)
14. Determine the production possibilities frontier how the opportunity cost of good i in terms of good j
changes as more of good i is produced in an one-input economy with a total of L units of labour in each
of the following mutually exclusive scenarios:
(a) two goods or sectors
(b) three goods or sectors
(c) n goods or sectors
if the production function in sector i is given by
, where b>0
, where b=2
the function in part (a) but with b=1
the function in part (a) but with b=1/2

6. 15. In a two-sector economy with one variable input, labour, find the aggregate demand for labour
assuming the (aggregate) production function of sector i is given by
16. Consider a price-taking firm which, instead of maximizing profit, maximizes output subject to the
constraint that its profit is equal to Z dollars. Assuming the firm sells its output at price p, uses one input
whose price is w, and its production function is
17. Consider a price-taking firm which, instead of maximizing profit, maximizes revenue subject to the
constraint that its profit is equal to Z dollars. Assuming that the firm sells its output at price p, uses one
input whose price is w, and its production function, q=f(x) is twice-differentiable and exhibits decreasing
returns, set up the firm's optimization problem, derive the optimality condition(s), and carry out the
comparative statics with respect to p and w.
a) Derive the choice functions of the firm (i.e. output supply and input demand),
b) the "maximum value of output" function, and
c) carry out the comparative statics with respect to p and w.
18.
, where i = 1,2 and b>1

7. 19.
Tim is a master pianist. If he gives qi performances in city i, i=1,2, he can charge per
Tim's total cost of a total of q performances is
21.
a) How many performances will Tim give in each city?
b) If the total number of performances cannot exceed 20, how many performances will Tim give in
each city?
22. Redo question 21 assuming all else is equal except Tim's total cost is given by
20. Redo question 19 without assuming b = 2.
23. Consider the case of a single-input profit-maximizing firm which is a price taker in all markets.
Assuming the firm's production function is twice-differentiable and strictly concave, determine
whether the behaviour of the firm's will change if the government imposes
(a) a percent tax on profit.
(b) a percent tax on profit and gives the same percent input subsidy.
(c) a percent tax on revenue.
(d) a percent tax on revenue and the same percent input subsidy.
Consider a single-input firm with a strictly concave production function which is a price taker in
the input market. Depict graphically its optimal output and input choice and how these change if
the price of the input increases by a small amount in each of the following mutually exclusive
scenarios:
(a) the firm is a price taker in the output market and its objective is to maximize profit.
(b) the firm is a monopoly in the output market and its objective is to maximize profit.
(c) the firm is a monopoly in the output market and its objective is to maximize revenue subject to
the constraint that its profit is not negative.
(d) the firm is a price taker in the output market and its objective is to maximize input use (the
amount of the input) subject to the constraint that its profit is not negative.
24.
and

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