3 Firm Borrowing Problem
Please read this first: Course Webpage Optimal Borrowing Choice Firm Maximization.
3.1 Write down the problem
1. Write down a firm’s profit maximization problem. Assume that the firm has fixed free labor,
but can choose capital input. At the start of a period, a firm rents capital inputs and combines
capital with labor to produce. At the end of the period, the firm sells its output and pays
interest rates based on how much capital it rented (no wage costs). Profit is denoted by π,
period interest rate is r, the price of output is p, the firm makes y units of output, and the
production function is Cobb-Douglas: A · Kα · L
0.5
2. Today, you and two friends open a promising business based on a promising idea, annual
interest rate is r = 1.05, how much do you want to borrow from the bank to buy equipments
etc? The bank has no limit on borrowing, but you have to pay back by the end of the year.
Do NOT compute anything, this is a survey question.
3.2 Finding optimal choice using three methods at a specific r
Use the following matlab code to draw a random number for coefficient α, and for A:
1 % d e f i n e max and min
2 max_alpha = 0 . 3 5 ;
3 min_alpha = 0 . 1 5 ;
4 max_A = 1 . 5 ;
5 min_A = 0 . 5 ;
6 % Draw random number between min and max
7 ( max_alpha−min_alpha ) ∗ rand ( ) + min_alpha
8 (max_A−min_A) ∗ rand ( ) + min_A
1. Modify the matlab code from the last homework. Replace the utility function by the profit
function. For your random draws of A and α, based on the grid-brute-force method, draw the
profit function values along a grid of capital points, and find the optimal capital borrowing
choice. Interest r = 1.05, output price p = 1, and fixed labor L = 2.
2. Analytically solve the firm maximization problem by taking derivative of the profit function
with respect to the capital borrowing choice.
3.3 Demand curve of credit
1. Given the parameters you drew earlier, solve your problem now at 21 interest rate points,
evenly spaced between r = 1.0 and r = 1.20. Write down your firm’s optimal choice at each
rate point. Draw a demand curve for the firm. Write a few sentences interpreting this curve.
You can solve for this computationally. You can also derive this curve using the analytical
equations from earlier. Solve your analytical problem with the r inside the equation, what is
the optimal borrowing choice as a function of r?
2. Given the parameters you drew earlier, at what r level would the demand for capital would
be K = 25. And at what r level would the demand for capital would be K = 0.05.4. Type 1 has high A and high α: A = 1.2 and α = 0.30, type 2 has high A and low α: A = 1.2
and α = 0.20, type 3 low A and high α: A = 0.8 and low α = 0.30, and type 4 low A and low
α: A = 0.8 and α = 0.20. What are the demand curves for each of the four types of firms?
What are the different effects of the two parameters?
5. Given the fractions of firms you have from each of these groups, solve for the aggregate demand
curve for credit in your economy.
6. Plot your aggregate supply curve for credit from last homework, over-lay it on top of the
aggregate demand curve here.
3.4 Elasticity
First, based on the problem you just solved, what is the elasticity of capital demand with respect
to interest rates? You should solve this without plugging in any numerical values.
1. Write down the elasticity formula
2. Solve for the demand elasticity with respect to price given the solution from the previous
problem
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3. For what price ranges is demand elasticity, unit-elastic, or inelastic? Or is the elasticity fixed?
Second, what is the elasticity of output with respect to capital input changes?
1. Solve for the elasticity of output with respect to input
2. For what price ranges is demand elasticity, unit-elastic, or inelastic? Or is the elasticity fixed?
3. Log-linearize the production function, how does the log-linearized equation relate to elasticity?
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