ECON3102 Tutorial 07 - Week 8
Question 1: Valuation
Charlie’s cheese factory has a very precise business plan for 2019-2028, shown below:
|
Year
|
Profits
|
Investment
|
Dividend
|
|
2019
|
80
|
50
|
30
|
|
2020
|
77
|
49
|
28
|
|
2021
|
86
|
49
|
37
|
|
2022
|
91
|
47
|
44
|
|
2023
|
98
|
47
|
51
|
|
2024
|
98
|
48
|
50
|
|
2025
|
109
|
200
|
-91
|
|
2026
|
120
|
57
|
63
|
|
2027
|
119
|
57
|
62
|
|
2028
|
125
|
61
|
64
|
(in 2025 the main storage facility will need to be replaced, hence the higher investment). From 2029 onwards, dividends will increase at a rate of 2% a year forever. The interest rate is 6%.
(a) Use the Gordon growth formula (See equation 1) to calculate the value that the factory will have in 2028 after paying dividends (i.e. the value not including the value of the dividends it will pay in 2028).
V = r - g/d1 (1)
(b) Compute the present value of the entire infinite stream of dividends that starts in 2019.
Question 2: Credit Constraints
A two-period world exists. A firm’s production function is given by
F(K, L) = At Kα L1-α
where At can have diferent values in each period. In each period, the firm can hire labor in a competitive labor market at the same wage w, which the firm perceives as constant. Unfortunately, there isn’t a market for renting capital: the firm can only use capital it owns. If the firm possesses K units of capital and decides to employ L workers, its earnings will be F(K, L) → wL. Initially, the firm has K1 units of capital in the first period. No depreciation occurs. At the end of period 1, the firm can procure capital for period 2 either by utilizing its earnings or by borrowing. All loans must be repaid in period 2, and the interest rate is r. Earnings can also be used to distribute dividends to shareholders in period 1. The utmost amount lenders are willing to provide to the firm is b.
(a) Formulate the firm’s challenge in deciding the number of workers to employ in each period. Recognize that this issue can be addressed period by period, considering Kt as constant. Derive an expression for the profits of a firm in period t that accepts its capital stock Kt as constant and determines the amount of labor to employ, i.e., for
ω(Kt) = max F(Kt, L) - wL
L
(b) Illustrate the firm’s problem in deciding the dividend amount, borrowing quantity, and investment level.
(c) If b = ∞ , prove that the ideal investment quantity is influenced by A2 but not by A1 . Elaborate.
(d) Assuming b = 0, demonstrate that when A1 is adequately high, then the optimal investment level aligns with part (c). Determine the least level of A1 for this scenario and label it A1(*) . Argue that when A1 < A1(*), the investment is influenced by A1 . Please elucidate.
(e) Suppose A1 < A1(*) . How does the firm respond to a rise in A2 ? How does an increase in b afect the firm’s decisions? Please explain.
Question 3: Aggregate Investment with Risk
The world lasts two periods. The aggregate production technology for period 2 is given by:
F(K, L) = AKα L1-α
K is the aggregate capital stock in period 2 and L is the labor force, which is exogenous and normalized to L = 1. Period 2 is the end of the world, so capital depreciates fully (δ = 1) and the representative household will consume F(K, L) = AKα L1-α . The utility function is u(c) = log(c). A is a random variable, which can take two possible values: AH =1+ ϑ or AL = 1 - ϑ, with equal probability. The interest rate between periods 1
Now consider the following investment project: investing an additional unit in period 1 to obtain an additional unit of capital in period 2.
(a) What is the dividend produced by the marginal unit of capital? Express it as a function of the aggregate capital stock K and realized productivity A.
(b) Suppose ϑ = 0. For what value of K is the net present value of additional investment exactly zero?
(c) Now suppose ϑ > 0. For what value of K is the net present value of additional investment exactly zero? (Hints: Use equation 2 to guide your answer).
(d) How does K depend on ϑ? Explain.