I also have the class materials if needed.
Econ 4373-1: Economics of Financial Crises Homework assignment 1 To receive credit, answers must be prepared in the form of a PDF report of at most 5 pages, 10-12 point font. Please attach the spreadsheet file or the code that you used to make your calculations, but treat it as reference. Do not assume I will be going over it, the report should be self-contained. Think of this assignment as practice for creating professional reports in your future job. Put some effort into writing and editing it nicely. In particular, grammar and spelling errors will be penalized. Make sure your graphs are legible and clearly labeled. Consider a population of entrepreneurs whose skill at running a business is measured by the probability of causing the firm to close down. Suppose there are 3 types of skill: high, medium, and low and assume that the medium type has a probability of failing equal to the population-wide average, while this probability for the high and low types are 50% lower and higher, respectively. At birth, individuals draw one of the skill types from a uniform distribution (i.e. 1/3, 1/3, 1/3), but they do not observe it and they cannot change it. Instead, they learn about it over time by observing their own path of business successes and failures. Assume that entrepreneurs are “born” at the age of 25, they “die” at the age of 65, and after every bankruptcy they “restart” with another firm. i. Let us restrict our attention to businesses in the US that are small in size (up to 500 employees) and mature (6 years and older). Find empirical evidence to approximate the average probability that a firm goes out of business in any given year. Briefly describe your data source and the reasoning behind your calculation (hint: you can use the survey of the US Small Business Administration). Summarize the obtained probabilities that describe the three types of entrepreneurs. ii. Simulate a path of business failures for a large population of entrepreneurs during their entire 40-year lifespan. At age 65, ask each one of them which skill group they believe they belong to with the highest probability. Summarize the distribution of perceived skill types and explain carefully if and why it differs from the true distribution. iii. Now suppose each entrepreneur can only bankrupt one firm during a career. Following a second failure he/she must become a worker and can no longer learn about their skill. Summarize the distribution of perceived skill types among 65-year-olds now, and discuss how and why it differs from the original one. Then consider a sub-population consisting of those individuals, who are no longer entrepreneurs at age 65. Summarize and discuss the distribution of perceived skill among them. iv. Now expand your population to include an age structure: suppose you can interview waves of entrepreneurs in 10-year intervals, i.e. at age 25, 35, 45, 55, 65. Maintain the assumption that each entrepreneur can only fail and restart a business once in their lifetime. Summarize and discuss the resulting distribution of perceived skill types. 1 v. Is this a good approach to think about overconfidence? List some upsides and down- sides. What elements could be added to make it more realistic? vi. [EXTRA CREDIT] Now, propose your own parameters for the true distribution of skills in the society. In particular, modify the assumption that the high and low types’ probability of failure is spread evenly around the population-wide average and/or ad- just the proportion of true types in the society. Just make sure that the new population average is still equal to what you found in the data. Argue why your proposed pa- rameters make sense. Can you provide some evidence supporting it? Show how the answers to previous questions change under your preferred parametrization. Note: this is an exercise for creativity. There are no correct or incorrect answers, but instead an answer can be more or less convincing. Hint: for clarity of exposition, consider summarizing your distributions by plotting identical bar charts for each bullet point above. Make sure you are precise about the assumptions behind your calculations, such as the assumed population size, etc. Submit your report as a PDF, rather than a doc file, if you want to be sure that your document will look the same on my computer as on yours. Econ 4389-1: Economics of Financial Crises Math and intermediate economics review notes Exponential functions Of great importance in economics are the functions where x appears as an exponent. Definition 1. An exponential function f : R→ R+ is given by f(x) = ax, where a > 0. Definition 2. The Euler’s number e is defined as e ≡ lim n→∞ ( 1 + 1 n )n ≈ 2.718218 Function f(x) = ex is called THE exponential function and frequently referred to as exp(x). For some argument r the value of exp(r) can then naturally be interpreted as limn→∞ ( 1+ r n )n . Consider a general exponential function y = ax with base a > 1. The inverse function of y cannot be computed explicitly, but it is called a logarithm with base a. We write y = loga(z) ⇐⇒ ay = z There are some general properties of exponential functions, mirrored by the corresponding properties of logarithmic function: ar · as = ar+s a−r = 1/ar ar/as = ar−s (ar)s = ar·s a0 = 1 log(r · s) = log r + log s log(1/s) = −log s log(r/s) = log r − log s log rs = s · log r log 1 = 0 The change of base formula for logarithms is logax = logbx logba . A special case of the logarithmic function is that with number e as base. It is called the natural logarithm and formally denoted as y = ln x ⇐⇒ ey = x Derivatives Definition 3. Derivative of function f : R→ R is defined as df dx = f ′(x) = lim h→0 f(x+ h)− f(x) h (1) Notice that the derivative of a function f(x) at a point x0 is the slope of that function at that point. Definition 4. If f ′(x) ≥ 0 we say that f(x) is increasing and if f ′(x) ≤ 0 we say that it is decreasing. 1 Example 1. Suppose f(x) = ax. Then f ′(x) = lim h→0 a(x+ h)− ax h = lim h→0 ax+ ah− ax h = lim h→0 a h h = a From this simple calculation we get the following differentiation rule: if f(x) = ax then f ′(x) = a. Following the same procedure (although the calculations may become more involved), we can obtain the derivative of any function. Here’s a list of some commonly used derivatives: f(x) f ′(x) a 0 ax+ b a xk kxk−1 ax ax ln a log x 1 xln a Rules for computing derivatives: 1. The Sum Rule (SR): d dx [f(x) + g(x)] = f ′(x) + g′(x) 2. The Product Rule (PR): d dx [f(x) g(x)] = f ′(x)g(x) + f(x)g′(x) 3. The Quotient Rule (QR): d dx [ f(x) g(x) ] = f ′(x)g(x)− f(x)g′(x) [g(x)]2 4. The Chain Rule (CR): d dx [f(g(x))] = f ′(g(x))g′(x) Here are some examples: 1. Using the Sum Rule: d dx [ax+ ln(x)] (SR) = d dx [ax] + d dx [ln(x)] = a+ 1 x 2. Using the Product Rule: d dx [axex] (PR) = d dx [ax]ex + ax d dx [ex] = aex + axex 2 3. Using the Quotient Rule: d dx [ ax ln(x) ] (QR) = d dx [ax]ln(x)− ax d dx [ln(x)] [ln(x)]2 = a ln(x)− ax x [ln(x)]2 4. Using the Chain Rule: d dx [eax] (CR) = d d(ax) [eax] d dx [ax] = aeax 5. Combining two rules: d dx [(ax− b)c] (CR)= d d(ax− b) [(ax− b)c] d dx [ax− b] (SR)= c(ax− b)c−1 ( d dx [ax] + d dx [−b] ) = c(ax− b)c−1a Definition 5. Higher-order derivatives are defined in the similar manner. For example, the second derivative of f is f ′′(x) = lim h→0 f ′(x+ h)− f ′(x) h (2) If the first derivative is the slope of the function, then the second derivative is the slope of the slope. That is, how much the slope of a function varies if x is changed. Example 2. Let f(x) = x3 + 2x2, then f ′(x) = 3x2 + 4x and f ′′(x) = 6x+ 4 Concavity Definition 6. Function f is concave (convex) if for all x and y and all t such that 0 ≤ t ≤ 1 we have f(tx+ (1− t)y) ≥ (≤) tf(x) + (1− t)f(y) Lemma 1. If f ′′(x) ≥ 0 then f(x) is convex, while if f ′′(x) ≤ 0 then it is concave. Definition 7. Partial derivatives are defined as derivatives of a function of multiple variables when all but the variable of interest are held fixed during the differentiation. For example, let f(x, y) be a two-variable function, then fx(x, y) ≡ ∂ ∂x [f(x, y)] ≡ limh→0 f(x+ h, y)− f(x, y) h Example 3. Let f(x, y) = x2y3. When we differentiate with respect to x, we treat y as constant: fx(x, y) = 2xy 3 3 Equivalently, when we differentiate with respect to y, we treat x as constant: fy(x, y) = 3x 2y2 Optimization Definition 8. Let f : [a, b]→ R be a function. We say that f achieves a maximum at x∗ if f(x∗ ≥ f(x)) for all x in [a, b]. If f(x∗) > f(x) for all x 6= x∗ then we say that x∗ is a strict maximum. Definition 9. Analogously, f achieves a minimum at x∗ if f(x∗ ≤ f(x)) for all x. If f(x∗) < f(x) for all x 6= x∗ then we say that x∗ is a strict minimum. notice that the problem of maximizing f(x) with respect to x is the same as the problem of minimizing −f(x). theorem 1. (the extreme value theorem) a continuous function f defined on a closed interval [a, b] always has a maximum and a minimum. definition 10. a critical point of a one-variable function f : r → r is a number c in its domain for which f ′(x) = 0. example 4. to find the critical points of f(x) = (x− 5)3, we calculate its first derivative: f ′(x) = 3(x− 5)2 and then solve f ′(x) = 0 for x: 3(x− 5)2 = 0 =⇒ x = 5 hence, 5 is the only critical point of f(x). theorem 2. if f is continuous on a closed interval [a, b], then any maximizer or minimizer of f must be either an endpoint of the interval (i.e. a or b), or at a critical point in (a, b). notice that the opposite statement is not true. that is, if a point is an endpoint or a f(x)="" for="" all="" x="" 6="x∗" then="" we="" say="" that="" x∗="" is="" a="" strict="" minimum.="" notice="" that="" the="" problem="" of="" maximizing="" f(x)="" with="" respect="" to="" x="" is="" the="" same="" as="" the="" problem="" of="" minimizing="" −f(x).="" theorem="" 1.="" (the="" extreme="" value="" theorem)="" a="" continuous="" function="" f="" defined="" on="" a="" closed="" interval="" [a,="" b]="" always="" has="" a="" maximum="" and="" a="" minimum.="" definition="" 10.="" a="" critical="" point="" of="" a="" one-variable="" function="" f="" :="" r="" →="" r="" is="" a="" number="" c="" in="" its="" domain="" for="" which="" f="" ′(x)="0." example="" 4.="" to="" find="" the="" critical="" points="" of="" f(x)="(x−" 5)3,="" we="" calculate="" its="" first="" derivative:="" f="" ′(x)="3(x−" 5)2="" and="" then="" solve="" f="" ′(x)="0" for="" x:="" 3(x−="" 5)2="0" =⇒="" x="5" hence,="" 5="" is="" the="" only="" critical="" point="" of="" f(x).="" theorem="" 2.="" if="" f="" is="" continuous="" on="" a="" closed="" interval="" [a,="" b],="" then="" any="" maximizer="" or="" minimizer="" of="" f="" must="" be="" either="" an="" endpoint="" of="" the="" interval="" (i.e.="" a="" or="" b),="" or="" at="" a="" critical="" point="" in="" (a,="" b).="" notice="" that="" the="" opposite="" statement="" is="" not="" true.="" that="" is,="" if="" a="" point="" is="" an="" endpoint="" or="">