ECON8026 Advanced Macroeconomic Analysis Assignment 8 Semester 2, 2022
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ECON8026 Advanced Macroeconomic Analysis
Assignment 8
Semester 2, 2022
Question 1
The Laffer Curve. Consider the model of optimal labor income taxation we studied in Chapter 19. All the notation and ideas are as developed there. For each of the following utility functions, first derive the function n(t) (i.e., the function relating the equilibrium quantity of labor supply to the labor tax rate), then express total tax revenue as a function of the labor tax rate TR(t), then determine whether a Laffer Curve arises.
a. u(c, 1 - n) = 2^c + (1 - n)
b. u(c, 1 - n) = ln c + ln(1 - n)
Question 2
A Consumption Tax. In the one-period model from Chapter 19, suppose that instead of having access to labor income taxes, the government can levy only a consumption tax (i.e., a sales tax) in order to pay for its expenditures. The household budget constraint is thus (1 + tC )c = w . n, where tC > 0 is the consumption tax rate (e.g., in Washington DC, tC = 0.0575). The government budget constraint is thus tC . c = govt. For each of the following utility functions, first derive a function c(tC ) (which is a function relating the equilibrium quantity of consumption to the consumption tax rate), then express total tax revenue as a function of the consumption tax rate TR(tC ) , then determine whether a Laffer Curve for consumption tax revenue arises. You should proceed in as close an analogy as possible with the analysis we conducted in Chapter 19. Furthermore, continue to assume that firm-optimization leads to the condition w = 1 and the resource constraint of the economy is c + govt = n.
Now let’s proceed to the specific utility functions under consideration.
a. u(c, 1 - n) = 2^c + (1 - n)
b. u(c, 1 - n) = ln c + ln(1 - n)
Question 3
Stock, Bonds, “Bills,” and the Financial Accelerator. In this problem, you will study an enriched version of the accelerator framework we studied in class. As in our basic analysis, we continue to use the two-period theory of firm profit maximization as our vehicle for studying the effects of financial-market developments on macroeconomic activity. However, rather than supposing it is just “stock” that is the financial asset at firms’ disposal for facilitating physical capital purchases, we will now suppose that both “stock” and “bonds” are at firms’ disposal for facilitating physical capital purchases.
Before describing more precisely the analysis you are to conduct, a deeper understanding of “bond markets” is required. In “normal economic conditions,” (i.e, in or near a “steady state,” in the sense we first discussed in Chapter 8), it is usually sufficient to think of all bonds of various maturity lengths in a highly simplified way: by supposing that they are all simply one-period face-value = 1 bonds with the same nominal interest rate. Recall, in fact, that this is how our basic discussion of monetary policy proceeded. In “unusual” (i.e., far away from steady state) financial market conditions, however, it can become important to distinguish between different types of bonds and hence different types of nominal interest rates on those bonds.
You may have seen discussion in the press about central banks, such as the U.S. Fed- eral Reserve, considering whether or not to “begin buying bonds” as a way of conducting policy. Viewed through the standard lens of how to understand open-market operations, this discussion makes no sense because in the standard view, central banks already do buy (and sell) “bonds” as the mechanism by which they conduct open-market operations!
A difference that becomes important to understand during unusual financial market con- ditions is that open-market operations are conducted using the shortest-maturity “bonds” that the Treasury sells, of duration one month or shorter. In the lingo of finance, this type of “bond” is called a “Treasury bill.” The term “Treasury bond” is usually used to refer to longer-maturity Treasury securities – those that have maturities of one, two, five, or more years. These longer-maturity Treasury “bonds” have typically not been assets that the Federal Reserve buys and sells as regular practice; buying such longer-maturity bonds is/has not been the usual way of conducting monetary policy.
In the ensuing analysis, part of the goal will be to understand/explain why policy-makers are currently considering this option. Before beginning this analysis, though, there is more to understand.
In private-market borrower/lender relationships, longer-maturity Treasury bonds (“bonds”) are typically allowed to be used just like stocks in financing firms’ physical capital purchases. We can capture this idea by enriching the financing constraint in our financial accelerator framework to read:
P1 . (k2 - k1 ) = RS . S1 . a1 + RB . P1(b) . B1
The left hand side of this richer financing constraint is the same as the left hand side of the financing constraint we considered in our basic theory (and the notation is identical, as well – refer to your notes for the notational definitions).
The right hand side of the financing constraint is richer than in our basic theory, however. The market value of “stock,” S1a1 , still affects how much physical investment firms can do, scaled by the government regulation RS . In addition, now the market value of a firm’s “bond holdings” (which, again, means long-maturity government bonds) also affects how much physical investment firms can do, scaled by the government regulation RB . The notation here is that B1 is a firm’s holdings of nominal bonds (“long-maturity”) at the end of period 1, and P1(b) is the nominal price of that bond during period 1. Note that RB and RS need not be equal to each other.
In the context of the two-period framework, the firm’s two-period discounted profit function now reads:
P1f(k1 , n1 ) + P1k1 + (S1 + D1)a0 + B0 - P1w1n1 - P1k2 - S1a1 - P1(b)B1
+ [P2f(k2 , n2 ) + P2k2 + (S2 + D2)a1 + B1 - P2w2n2 - P2k3 - S2a2 - P2(b)B2]
The new notation compared to our study of the basic accelerator mechanism is the following: B0 is the firm’s holdings of nominal bonds (which have face value = 1) at the start of period one, B1 is the firm’s holdings of nominal bonds (which have face value = 1) at the end of period one, and B2 is the firm’s holdings of nominal bonds (which have face value = 1) at the end of period two.
Note that period-2 profits are being discounted by the nominal interest rate i: in this problem, we will consider i to be the “Treasury bill” interest rate (as opposed to the “Treasury bond” interest rate). The Treasury-bill interest rate is the one the Federal Reserve usually (i.e., in “normal times”) controls.
We can define the nominal interest rate on Treasury bonds as
iBOND = - 1
Thus, note that iBOND and i need not equal each other.
The rest of the notation above is just as in our study of the basic financial accelerator framework. Finally, because the economy ends at the end of period 2, we can conclude (as usual) that k3 = 0, a2 = 0, and B2 = 0.
With this background in place, you are to analyze a number of issues.
a. Using λ as your notation for the Lagrange multiplier on the financing constraint, con- struct the Lagrangian for the representative firm’s (two-period) profit-maximization prob- lem.
b. Based on this Lagrangian, compute the first-order condition with respect to nominal bond holdings at the end of period 1 (i.e., compute the FOC with respect to B1). (Note: This FOC is critical for much of the analysis that follows, so you should make sure that your work here is absolutely correct.)
c. Recall that in this enriched version of the accelerator framework, the nominal interest rate on “Treasury bills”, i, and the nominal interest rate on “Treasury bonds”, iBOND , are potentially different from each other. If financing constraints do NOT at all affect firms’ investment in physical capital, how does iBOND compare to i? Specifically, is iBOND equal to i, is iBOND smaller than i, is iBOND larger than i, or is it impossible to determine? Be as thorough in your analysis and conclusions as possible (i.e., tell us as much about this issue as you can!). Your analysis here should be based on the FOC on B1 computed in part b above. (Hint: if financing constraints “don’t matter,” what is the value of the Lagrange multiplier λ?)
d. If financing constraints DO affect firms’ investment in physical capital, how does iBOND compare to i? Specifically, is iBOND equal to i, is iBOND smaller than i, is iBOND larger than i, or is it impossible to determine? Furthermore, if possible, use your solution here as a basis for justifying whether or not it is appropriate in “normal economic condi- tions” to consider both “Treasury bills” and “Treasury bonds” as the “same” asset. Be as thorough in your analysis and conclusions as possible. Once again, your analysis here should be based on the FOC on B1 computed in part b above. (Note: the government regulatory variables RS and RB are both strictly positive – that is, neither can be zero or less than zero).
The above analysis was framed in terms of nominal interest rates; the remainder of the analysis is framed in terms of real interest rates.
e. By computing the first-order condition on firms’ stock-holdings at the end of period 1, a1 , and following exactly the same algebra as presented in class, we can express the
Lagrange multiplier λ as
λ = ┌ ┐ . (P.1)
Use the first-order condition on B1 you computed in part b above to derive an analogous expression for λ except in terms of the real interest rate on bonds (i.e., TBOND ) and RB (rather than RS). (Hint: Use the FOC on B1 you computed in part b above and follow a very similar set of algebraic manipulations as we followed in class.)
f. Compare the expression you just derived in part e with expression (P.1). Suppose T = TSTOCK . If this is the case, is TBOND equal to T , is TBOND smaller than T , is TBOND larger than T , or is it impossible to determine? Furthermore, in this case, does the financing constraint affect firms’ physical investment decisions? Briefly justify your conclusions and provide brief explanation.
g. Through late 2008, suppose that T = TSTOCK was a reasonable description of the U.S. economy for the preceding 20+ years. In late 2008, TSTOCK fell dramatically below T , which, as we studied in class, would cause the financial accelerator effect to begin. Suppose government policy-makers, both fiscal policy-makers and monetary policymakers, decide that they need to intervene in order to try to choke off the accelerator effect. Furthermore, suppose that there is no way to change either RS or RB (because of coordination delays amongst various government agencies, perhaps). Using all of your preceding analysis as well as drawing on what we studied in class, explain why “buying bonds” (which, again, means long-maturity bonds in the sense described above) might be a sound strategy to pursue. (Note: The analysis here is largely not mathematical. Rather, what is required is an careful logical progression of thought that explains why buying bonds might be a good idea.)
Question 4
Financing Constraints and Housing Markets. Consider an enriched version of the two-period consumption-savings framework from Chapters 3 and 4, in which the represen- tative individual not only makes decisions about consumption and savings, but also housing purchases. For this particular application, it is useful to interpret “period 1” as the “young period” of the individual’s life, and interpret “period 2” as the “old period” of the individ- ual’s life.
In the young period of an individual’s life, utility depends only on period-1 consumption c1 . In the old period of an individual’s life, utility depends both on period-2 consumption c2 , as well as his/her “quantity” of housing (denoted h). From the perspective of the beginning of period 1, the individual’s lifetime utility function is
ln c1 + ln c2 + ln h
in which ln(.) stands for the natural log function; the term lnh indicates that people directly obtain happiness from their housing.
Due to the “time to build” nature of housing (that is, it takes time to build a housing unit), the representative individual has to incur expenses in his/her young period to pur- chase housing for his/her old period. The real price in period 1 (i.e., measured in terms of period-1 consumption) of a “unit” of housing (again, think of a unit of housing as square footage) is p1(H) , and the real price in period 2 (i.e., measured in terms of period-2 consump- tion) of a unit of housing is p2(H) .
In addition to housing decisions, the representative individual also makes stock purchase decisions. The individual begins period 1 with zero stock holdings (a0 = 0), and ends period 2 with zero stock holdings (a2 = 0). How many shares of stock the individual ends period 1 with, and hence begins period 2 with, is to be optimally chosen. The real price in period 1 (i.e., measured in terms of period-1 consumption) of each share of stock is s1 , and the real price in period 2 (i.e., measured in terms of period-2 consumption) of each share of stock is s2 . For simplicity, suppose that stock never pays any dividends (that is, dividends = 0 always).
Because housing is a big-ticket item, the representative individual has to accumulate financial assets (stock) while young to overcome the informational asymmetry problem and be able to purchase housing. Suppose the financing constraint that governs the purchase of
housing is
p1(H)h
(technically an inequality constraint, but we will assume it always holds with strict equality). In the financing constraint, RH > 0 is a government-controlled “leverage ratio” for housing. Note well the subscripts on variables that appear in the financing constraint.
Finally, the real quantities of income in the young period and the old period are y1 and y2 , over which the individual has no choice.
The sequential Lagrangian for the representative individual’s problem lifetime utility maximization problem is:
L = ln c1 + ln c2 + ln h + λ1 ┌y1 - c1 - s1 a1 -p1(H)h┐ + λ2 ┌y2 + s2 a1 +p2(H)h - c2 ┐ + µ ┌ s2 a1 - ┐
in which µ is the Lagrange multiplier on the financing constraint, and λ 1 and λ2 are, respectively, the Lagrange multipliers on the period-1 and period-2 budget constraints.
a. In no more than two brief sentences/phrases, qualitatively describe what an informa- tional asymmetry is, and why it can be a serious problem in financial transactions.
b. In no more than three brief sentences/phrases, qualitatively describe the role that the leverage ratio RH plays in the “housing finance” market. In particular, briefly de- scribe/discuss what higher leverage ratios imply for the individual’s ability to finance a house purchase (i.e., “obtain a mortgage”).
c. Based on the sequential Lagrangian presented above, compute the two first-order conditions: with respect to a1 and h. (You can safely ignore any other first-order conditions.)
d. Based on the first-order condition with respect to h computed in part c, solve for the period-1 real price of housing p1(H) (that is, your final expression should be of the form p1(H) = ... where the term on the right hand side is for you to determine). (Note: you do NOT have to eliminate Lagrange multipliers from the final expression.)
e. Based on the expression for p1(H) computed in part d, and assuming that the Lagrange multiplier µ > 0 (recall, furthermore, that RH > 0), answer the following: is the period-1 price of housing larger than or smaller than what it would be if financing constraints for housing were not at all an issue? Or is it impossible to determine? Carefully explain the logic of your argument/analysis, and provide brief economic interpretation of your conclu- sion.
For the remainder of this problem (i.e., for parts f, g, and h), suppose that λ 1 = λ2 = 1.
f. Consider the period-1 housing market, with the quantity h of housing drawn on the horizontal axis and the period-1 price, p1(H) , of housing drawn on the vertical axis. Using the house-price expression computed in part d, qualitatively sketch the relationship between h and p1(H) that it implies. Your sketch should make clear whether the relationship is upward- sloping, downward-sloping, perfectly horizontal, or perfectly vertical. Clearly present the algebraic/logical steps that lead to your sketch, and clearly label your sketch.
g. In the same sketch in part f, clearly show and label what happens if p2(H) rises. (Examples of what could “happen” are that the relationship you sketched rotates, or shifts, or both rotates and shifts, etc.) Explain the logic behind your conclusion, and provide brief economic interpretation of your conclusion.
h. In the same sketch in part f, clearly show and label what happens if RH rises. (Examples of what could “happen” are that the relationship you sketched rotates, or shifts, or both rotates and shifts, etc.) Explain the logic behind your conclusion, and provide brief economic interpretation of your conclusion.
2023-02-28