Quiz 2: Mathematics for Microeconomics
Business
Q 1Q 1
Suppose
a. Calculate
b. Evaluate these partial derivatives at x = 1, y = 2.
c. Write the total differential for U.
d. Calculate dy/dx for dU = 0-that is, what is the implied trade-off between x and y holding U constant?
e. Show U = 16 when x = 1, y = 2.
f. In what ratio must x and y change to hold U constant at 16 for movements away from x = 1, y = 2?
g. More generally, what is the shape of the U = 16 contour line for this function? What is the slope of that line?
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Utility function: It refers to a function use to measure the consumer's level of satisfaction derives from the consumption of goods and services.
The utility function is given by,
a.
Given the utility function, the partial differentiation with respect to a single variable, in a function of several variables is calculated as follows;
b.
Partial differentiation with respect to x is
and with respect to y is
. Compute partial derivatives at x= 1 and y = 2 by substituting the values of x and y in the above partial derivates computed in previous part. It is computed below:
c.
Total differentiation is written with the help of partial differentiation.
d.
Total differentiation is written with the help of partial differentiation.
So,
It means trade-off between x and y is not linear. The negative sign shows that getting more x required forgoing some y and vice-versa.
e.
To prove that value of U is 16, when
, substitute the value of x and y in the utility function and solve as follows:
Hence, the value of utility is 16.
f.
To determine the change in ratio of x and y while holding utility constant at 16, substitute the value of x and y in the
computed in part d. It is computed below:
Hence, (-2/3) ratio of x and y needed to hold U constant at 16.
g.
To plot the utility function, compute the horizontal and the vertical corner points as follows:
Vertical points: When x is 0, then y will be,
Thus, vertical point is (0,2.30)
Horizontal points: When y is 0, then x will be,
Thus, horizontal point is (2,0).
The following figure shows the shape of utility function when utility is 16:
As shown in the above figure, the shape of U is an ellipse centered at the origin.
Slope of this equation is
.
Q 2Q 2
Suppose a firm's total revenues depend on the amount produced (q) according to the function
R = 70q - q 2.
Total costs also depend on q:
C = q 2 + 30q + 100.
a. What level of output should the firm produce to maximize profits (R - C)? What will profits be?
b. Show that the second-order conditions for a maximum are satisfied at the output level found in part (a).
c. Does the solution calculated here obey the ''marginal revenue equals marginal cost'' rule? Explain.
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Total revenue is the amount earned by the firm by its operations.
Total cost is the production cost incurred by the firm to produce the commodity.
Profit is the difference between the total revenue and total cost.
a)
Profit maximizing firm produces the output at that level where difference between total revenue and total cost is maximum.
Necessary condition is as follows:
Sufficient condition is as follows:
Calculate the profit as follows:
Thus, firm produces
of output and earns
profits.
b)
Second-order condition of profit maximization is
. It is also called sufficient condition.
Thus, second order condition is satisfied at 10 unit of output.
c.
Profit maximizes where marginal revenue is equal to marginal cost. Marginal revenue is additional revenue in total revenue while selling one more unit of product. Marginal cost is additional cost in total cost while producing one more unit of product.
Calculate as follows:
Yes, the solution obeys the rule 'marginal revenue equals to marginal cost'.
Q 3Q 3
Suppose that f (x, y) = xy. Find the maximum value for f if x and y are constrained to sum to 1. Solve this problem in two ways: by substitution and by using the Lagrange multiplier method.
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Substitute goods are those good that can be used in place of other good and satisfaction level of both products is almost the same.
Substitution method
Sum of x and y is 1. Therefore, we can write it as:-
Condition of consumer equilibrium (which seeks to solve the maximization and minimization problems subject to some constraints) can be easily understood with the help of Lagrange multiplier.
Lagrange multiplier method:-
Sum of x and y is 1.
Budget constraint can be written as:-
Taking first order condition:-
Q 4Q 4
The dual problem to the one described in Problem 2.3 is minimize x + y subject to xy = 0.25. Solve this problem using the Lagrangian technique. Then compare the value you get for the Lagrange multiplier with the value you got in Problem 2.3. Explain the relationship between the two solutions.
Reference:
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Q 5Q 5
The height of a ball that is thrown straight up with a certain force is a function of the time (t) from which it is released given by
(where g is a constant determined by gravity).
a. How does the value of t at which the height of the ball is at a maximum depend on the parameter g?
b. Use your answer to part (a) to describe how maximum height changes as the parameter g changes.
c. Use the envelope theorem to answer part (b) directly.
d. On the Earth g = 32, but this value varies somewhat around the globe. If two locations had gravitational constants that differed by 0.1, what would be the difference in the maximum height of a ball tossed in the two places?
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Q 6Q 6
A simple way to model the construction of an oil tanker is to start with a large rectangular sheet of steel that is x feet wide and 3 x feet long. Now cut a smaller square that is t feet on a side out of each corner of the larger sheet and fold up and weld the sides of the steel sheet to make a tray like structure with no top.
a. Show that the volume of oil that can be held by this tray is given by
b. How should t be chosen to maximize V for any given value of x?
c. Is there a value of x that maximizes the volume of oil that can be carried?
d. Suppose that a shipbuilder is constrained to use only 1,000,000 square feet of steel sheet to construct an oil tanker. This constraint can be represented by the equation 3x 2 - 4t 2 = 1,000,000 (because the builder can return the cut-out squares for credit). How does the solution to this constrained maximum problem compare with the solutions described in parts (b) and (c)?
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Q 7Q 7
Consider the following constrained maximization problem:
where k is a constant that can be assigned any specific value.
a. Show that if k = 10, this problem can be solved as one involving only equality constraints.
b. Show that solving this problem for k = 4 requires that x 1 = -1.
c. If the x's in this problem must be non-negative, what is the optimal solution when k = 4? (This problem may be solved either intuitively or using the methods outlined in the chapter.)
d. What is the solution for this problem when k = 20? What do you conclude by comparing this solution with the solution for part (a)?
Note: This problem involves what is called a quasi-linear function. Such functions provide important examples of some types of behavior in consumer theory-as we shall see.
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Q 8Q 8
Suppose that a firm has a marginal cost function given by MC ( q ) = q + 1.
a. What is this firm's total cost function? Explain why total costs are known only up to a constant of integration, which represents fixed costs.
b. As you may know from an earlier economics course, if a firm takes price ( p ) as given in its decisions then it will produce that output for which p = MC ( q ). If the firm follows this profit-maximizing rule, how much will it produce when p = 15? Assuming that the firm is just breaking even at this price, what are fixed costs?
c. How much will profits for this firm increase if price increases to 20?
d. Show that, if we continue to assume profit maximization, then this firm's profits can be expressed solely as a function of the price it receives for its output.
e. Show that the increase in profits from p =15 to p = 20 can be calculated in two ways: (i) directly from the equation derived in part (d); and (ii) by integrating the inverse marginal cost function [MC - 1 ( p) = p - 1] from p = 15 to p = 20.
Explain this result intuitively using the envelope theorem.
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Q 9Q 9
Concave and quasi-concave functions
Show that if f ( x 1 , x 2 ) is a concave function then it is also a quasi-concave function. Do this by comparing Equation 2.114 (defining quasi-concavity) with Equation 2.98 (defining concavity). Can you give an intuitive reason for this result? Is the converse of the statement true? Are quasi-concave functions necessarily concave? If not, give a counter example.
Reference: Equation 2.98 and Equation 2.114
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Q 10Q 10
The Cobb-Douglas function
One of the most important functions we will encounter in this book is the Cobb-Douglas function:
where
are positive constants that are each less than 1.
a. Show that this function is quasi-concave using a ''brute force'' method by applying Equation 2.114.
b. Show that the Cobb-Douglas function is quasi-concave by showing that any contour line of the form y = c (where c is any positive constant) is convex and therefore that the set of points for which y c is a convex set.
c. Show that if
then the Cobb-Douglas function is not concave (thereby illustrating again that not all quasiconcave functions are concave).
Note: The Cobb-Douglas function is discussed further in the Extensions to this chapter.
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Q 11Q 11
The power function
Another function we will encounter often in this book is the power function:
where
(at times we will also examine this function for cases where d can be negative, too, in which case we will use the form
to ensure that the derivatives have the proper sign).
a. Show that this function is concave (and therefore also, by the result of Problem 2.9, quasi-concave). Notice that the
is a special case and that the function is ''strictly'' concave only for
.
b. Show that the multivariate form of the power function
is also concave (and quasi-concave). Explain why, in this case, the fact that f 12 = f 21 = 0 makes the determination of concavity especially simple.
c. One way to incorporate ''scale'' effects into the function described in part (b) is to use the monotonic transformation
where
is a positive constant. Does this transformation preserve the concavity of the function? Is g quasi-concave?
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Q 12Q 12
Proof of the envelope theorem in constrained optimization problems
Because we use the envelope theorem in constrained optimization problems often in the text, proving this theorem in a simple case may help develop some intuition. Thus, suppose we wish to maximize a function of two variables and that the value of this function also depends on a parameter, a : f ( x 1 , x 2 , a ). This maximization problem is subject to a constraint that can be written as: g (x 1 , x 2 , a) = 0.
a. Write out the Lagrangian expression and the first-order conditions for this problem.
b. Sum the two first-order conditions involving the x's.
c. Now differentiate the above sum with respect to a-this shows how the x's must change as a changes while requiring that the first-order conditions continue to hold.
d. As we showed in the chapter, both the objective function and the constraint in this problem can be stated as functions of
Differentiate the first of these with respect to a. This shows how the value of the objective changes as a changes while keeping the x's at their optimal values. You should have terms that involve the x's and a single term in
.
e. Now differentiate the constraint as formulated in part (d) with respect to a. You should have terms in the x's and a single term in
f. Multiply the results from part (e) by l (the Lagrange multiplier), and use this together with the first-order conditions from part (c) to substitute into the derivative from part (d). You should be able to show that
which is just the partial derivative of the Lagrangian expression when all the x's are at their optimal values. This proves the envelope theorem. Explain intuitively how the various parts of this proof impose the condition that the x's are constantly being adjusted to be at their optimal values.
g. Return to Example 2.8 and explain how the envelope theorem can be applied to changes in the fence perimeter P-that is, how do changes in P affect the size of the area that can be fenced? Show that in this case the envelope theorem illustrates how the Lagrange multiplier puts a value on the constraint
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Q 13Q 13
Taylor approximations
Taylor's theorem shows that any function can be approximated in the vicinity of any convenient point by a series of terms involving the function and its derivatives. Here we look at some applications of the theorem for functions of one and two variables.
a. Any continuous and differentiable function of a single variable, f (x), can be approximated near the point a by the formula
Using only the first three of these terms results in a quadratic Taylor approximation. Use this approximation together with the definition of concavity given in Equation 2.85 to show that any concave function must lie on or below the tangent to the function at point a.
b. The quadratic Taylor approximation for any function of two variables, f (x, y), near the point (a, b) is given by
Use this approximation to show that any concave function (as defined by Equation 2.98) must lie on or below its tangent plane at (a, b).
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Q 14Q 14
More on expected value
Because the expected value concept plays an important role in many economic theories, it may be useful to summarize a few more properties of this statistical measure. Throughout this problem, x is assumed to be a continuous random variable with PDF f (x).
a. ( Jensen's inequality) Suppose that g (x) is a concave function. Show that
. Hint: Construct the tangent to g (x) at the point E(x). This tangent will have the form c + dx ? g (x) for all values of x and c + dE(x) = g[E(x)] where c and d are constants.
b. Use the procedure from part (a) to show that if g (x) is a convex function then
c. Suppose x takes on only non-negative values-that is,
. Use integration by parts to show that
where F(x) is the cumulative distribution function for x [that is
d. (Markov's inequality) Show that if x takes on only positive values then the following inequality holds:
e. Consider the PDF
1. Show that this is a proper PDF.
2. Calculate F(x) for this PDF.
3. Use the results of part (c) to calculate E(x) for this PDF.
4. Show that Markov's inequality holds for this function.
f. The concept of conditional expected value is useful in some economic problems. We denote the expected value of x conditional on the occurrence of some event, A, as E ( x | A ). To compute this value we need to know the PDF for x given that A has occurred [denoted by f ( x | A )]. With this notation,
. Perhaps the easiest way to understand these relationships is with an example. Let
1. Show that this is a proper PDF.
2. Calculate E(x).
3. Calculate the probability that
.
4. Consider the event
, and call this event A. What is f (x|A)?
5. Calculate E(x|A).
6. Explain your results intuitively.
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Q 15Q 15
The definition of the variance of a random variable can be used to show a number of additional results.
a. Show that Var(x) = E(x 2 ) - [E(x)] 2.
b. Use Markov's inequality (Problem 2.14d) to show that if x can take on only non-negative values,
This result shows that there are limits on how often a random variable can be far from its expected value. If
this result also says that
Therefore, for example, the probability that a random variable can be more than two standard deviations from its expected value is always less than 0.25. The theoretical result is called Chebyshev's inequality.
c. Equation 2.197 showed that if two (or more) random variables are independent, the variance of their sum is equal to the sum of their variances. Use this result to show that the sum of n independent random variables, each of which has expected value
, has expected value nm and variance
. Show also that the average of these n random variables (which is also a random variable) will have expected value
This is sometimes called the law of large numbers-that is, the variance of an average shrinks down as more independent variables are included.
d. Use the result from part (c) to show that if x 1 and x 2 are independent random variables each with the same expected value and variance, the variance of a weighted average of the two
is minimized when k= 0.5.
How much is the variance of this sum reduced by setting k properly relative to other possible values of k?
e. How would the result from part (d) change if the two variables had unequal variances?
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Q 16Q 16
More on covariances
Here are a few useful relationships related to the covariance of two random variables, x l and x 2.
a. Show that
An important implication of this is that if
That is, the expected value of a product of two random variables is the product of these variables' expected values.
b. Show that
c. In Problem 2.15d we looked at the variance
Is the conclusion that this variance is minimized for k = 0.5 changed by considering cases where
d. The correlation coefficient between two random variables is defined as
Explain why
and provide some intuition for this result.
e. Suppose that the random variable y is related to the random variable x by the linear equation
. Show that
Here ? is sometimes called the (theoretical) regression coefficient of y on x. With actual data, the sample analog of this expression is the ordinary least squares (OLS) regression coefficient.
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