Derive the law of motion for capital stock and the steady-state level of capital stock?

 Derive the law of motion for capital stock and the steady state level of capital stock?

The law of motion tells you how capital per unit of effective labor evolves over time. In particular, it says that when actual investment (sk(t)α) is higher that required investment ((n + g + δ)k(t)), capital per unit of effective labor would be rising.
(a)The law of motion tells you how capital per unit of effective labor evolves over time. In particular, it says that when actual investment (sk(t) α ) is higher that required investment ((n + g + δ) k(t)), capital per unit of effective labor would be rising. Actual investment is the new investment in the economy. Required investment (or break even) is the investment that has to be made to keep k at its existing level. This investment has to be made because part of the capital stock wears out (δ) and because AL is growing at a rate n + g.
 (b) First you have to solve for the steady state level of k, which occurs when skα
 = (n + g + δ)k, k ∗ =  s n + g + δ  1 1−α
Then you solve for the golden rule level of k, which is the k that maximize
 c = k α − (n + g + δ)k, kGR =  α n + g + δ  1 1−α Comparing these two solutions it is clear that k ∗ = kGR when α = s.

(c) The Solow model predicts conditional convergence, i.e., that after controlling for country characteristics, there should be convergence. The reason is that according to the Solow model countries converge to their own balanced growth paths, and not necessarily a common one. Divergence is not consistent with the Solow model. In the long run, once countries have converged to their steady states income levels will be different if they have different characteristics. But these differences will persist, 1 they will not grow larger. Further in the short run, countries further away from their steady states grow faster than countries closer to their steady states.
(d) At time t0 after the permanent one time increase in the level of efficiency, k = K AL will fall below the BGP value k ∗ . Note however that the BGP will not change, since no characteristic of the economy changed.
             Suppose, we call the value of k after the rise in A, k0. Then at k0, we have that actual investment is higher than required investment: sf(k0) > (n + g + δ)k0. By the law of motion for capital this implies that ?k > 0, and thus k will rise until it returns to the original steady state.
            (a) Substitute the firm’s first order condition with respect to capital into the household’s Euler to get the economy’s first key dynamic equation,

 c?(t) c(t) = f 0 (k(t)) − δ − ρ − θg θ

The Euler equation tells you how this economy is going to choose consumption/saving over time. On the margin the decision of giving up consumption today for more consumption tomorrow, depends on the economy’s net return to capital (market return) and on preference parameters. The second key dynamic equation of the model is the law of motion for capital per unit of effective ?
k(t) = f(k(t)) − c(t) − (n + g + δ) k(t)
 This is different from the one in class because now we have depreciation. Required investment is higher, i.e., the level of investment required just to keep k constant is now higher. The reason is that now you have to account for the replacement of worn out capital.
(b). Suppose the economy starts off from a level of capital per unit of effective labor below the steady state: k(0) < k∗ . At time 0, k(0) is pre determined and cannot be adjusted. Consumption per unit of effective labor at time 0, c(0), is a “control” variable, i.e., it is to be chosen. There is a whole continuum of possible choices for c(0). Each one of these places the economy on a given trajectory. However, not all of these trajectories will take the economy to the steady state. In fact there is only one trajectory that takes the economy to the steady state: the saddle path. This is the only trajectory that satisfies, household optimization, law of motion for capital, the household intertemporal budget constraint and the No Ponzi Game condition. Thus if the economy is at k(0) it chooses a level of 2 consumption c(0) that places it on the saddle path. Once the economy is on the saddle path it will go to the steady state. For any point above this level of c(0) the economy will diverge up and to the left, while for any point below it, it will diverge down and to the right.
 (c) The increase in δ means that break even investment increases. From the law of motion the amount of consumption possible is now lower than before. Further, the level of extra investment required is higher at higher levels of k. Thus the ?k = 0 locus shifts down but not in a parallel fashion, i.e., it shifts down more at higher levels of k.
                The ?c = 0 locus will shift to the left. To see this note that when ?c = 0, we have that f 0 (k ∗ ) = ρ+δ +θg. Then δnew > δ implies that ρ+δnew +θg > ρ+δ +θg, and thus f 0 (k ∗ new) > f0 (k ∗ ). Since we have a diminishing marginal product of capital this implies k ∗ new < k∗ . Both    (c ∗ , k∗ ) will be lower in the new steady state. At the time of the change in δ k is predetermined and cannot change. Thus c must jump to place the economy on the new saddle path. Whether c will jump up or down depends on whether the new saddle path crosses the original cc? = 0
(a) This type of government expenditures constitute a resource cost and affect the law of motion for capital: the more the government takes away from the economy the fewer resources are available to the private sector. When government expenditures are high at level G0, before time t0, the law of motion for k is, ?k(t) = f(k(t)) − c(t) − G0 − (n + g)k(t) After time t0, when government expenditures are low, the law of motion becomes, ?k(t) = f(k(t)) − c(t) − G1 − (n + g)k(t) The Euler equation is the same in both cases, c?(t) c(t) = f 0 (k) − ρ − θg θ At time t0 the ?k = 0 locus shifts up by G1 − G0. At that point k is pre determined at k ∗ the original BGP. Thus, c must jump up by G1 − G0 to place the economy on the new saddle path at point E’ (With a permanent decrease in government expenditures and thus taxes, lifetime wealth increases for households. Because this change in permanent the households will not alter the time pattern of their consumption but instead will take a permanent rise in how much they consume. 3 Thus we go directly from E to E’ and k is not affected (remains at k ∗ throughout), while c ∗ rises.
 (b) Before time t0 and after time t1 the law of motion for k is, ?k(t) = f(k(t)) − c(t) − G0 − (n + g)k(t) Between times (t0, t1) the law of motion is, ?k(t) = f(k(t)) − c(t) − G1 − (n + g)k(t) Again the Euler in both cases is, c?(t) c(t) = f 0 (k) − ρ − θg θ In this case at time t0, c will not rise by the full amount G1 − G0. If it did, then consumption would have to drop discontinuously at t1, and this marginal utility would increase discontinuously. But since the return to G0 is anticipated, so will the discrete jumps in consumption and marginal utility. This however is not optimal for rational individuals who like smooth consumption patterns. So at time t0, c will increase less than G1 − G0, such that the dynamics implied by the Euler and the law of motion with G(t) = G1 will push the economy to the original saddle path as time t1arrives. After that the economy rides the original saddle path to the original BGP. So k first increases and then decreases.
(a) The capital accumulation equation is, ?k(t) = Ak(t) − c(t) − δk(t) ?k(t) = (A − δ) k(t) − c(t)
 (b) Real interest rate: r(t) = (1 − τ ) A − δ Growth of consumption per worker: c?(t) c(t) = (1 − τ ) A − (δ + ρ) θ Consider the capital accumulation equation from part (a). Divide both sides by k(t), and use the fact that k? (t) k(t) will be constant. Let k? (t) k(t) ≡ b. Then we have, b = (A − δ) − c(t) k(t) This implies that k? (t) k(t) = c?(t) c(t) , i.e., the growth rate of capital per worker is the same as that of consumption per worker. From the production function y(t) = Ak(t), you can see that also y?(t) y(t) = k? (t) k(t) . The savings rate s = y−c y , can be shown to be, s = Ak − c Ak = 1 − 1 A c k = 1 − 1 A  A − δ − (1 − τ ) A − (δ + ρ) θ  = 1 −  (θ − 1 + τ ) A + ρ + δ (1 − θ) θA
 (c) You can see that the interest rate, the savings rate, and the growth rate are all decreasing in the tax, by calculating the derivatives: ∂r ∂τ = −A < 0, ∂s ∂τ = − 1 θ < 0, ∂( y? y ) ∂τ = − A θ < 0.
 (d) Since the economies have the same tax rates their growth rates will be the same.

Graphical Representation

Problems

Consider the Diamond model. Assume that utility is logarithmic and production is Cobb Douglas, f(k) = Akα , α ∈ (0, 1). 1.

Describe how the following changes in parameters affect the capital accumulation schedule kt+1(kt):

Answer: Each young consumer solves the problem
                                   max c1,t,c2,t+1 U = ln(c1,t) + 1 1 + ρ ln(c2,t+1), ρ > −1
                                   st. c1,t = wt − st , c2,t+1 = (1 + rt+1)st , c1,t ≥ 0, c2,t+1 ≥ 0
The problem is rewritten as max
                                                st U = ln(wt − st) + 1 1 + ρ ln(1 + rt+1)st  st. st ∈ [0, wt ]
Taking the first order condition yields
                          0 = − 1 wt − st + 1 1 + ρ (1 + rt+1) 1 (1 + rt+1)st ⇔ (1 + ρ)st = wt − st ⇔ st = wt 2 + ρ
The problem of the representative firm is given by max
                                  Kt≥0,Lt≥0 AKα t L 1−α t − wtLt − rtKt , α ∈ (0, 1)
Note that f(k) = Akα ⇔ F(K, L) = AKαL 1−α . Using the first order conditions, rt and wt can be expressed as functions of kt .
                                               ∂πt(·) ∂Lt = (1 − α)AKα t L −α t − wt = 0 ⇔
                                                                       wt = (1 − α)Akα t
                                                    ∂πt(·) ∂Kt = αAKα−1 t L 1−α t − rt = 0 ⇔ rt = αAkα−1 t
The aggregate capital level at time
                                                          t + 1 is given by Kt+1 = stLt = st Lt+1 1 + n ⇔
                                                                             kt+1 = st 1 + n

In sum we thus have
                                        kt+1 = st 1 + n = wt (2 + ρ)(1 + n) = (1 − α)A (2 + ρ)(1 + n) k α t

(a) An increase in the rate of population growth, n.
 Answer:
                                  ∂kt+1/∂n = − 1 (1 + n) 2 (1 − α)A 2 + ρ k α t < 0
(b) A fall in A.
Answer:
                             ∂kt+1/ ∂A = 1 − α (1 + n)(1 − ρ) k α t > 0
(c) An increase in the share of capital in production, α.
 Answer:
                                   ∂kt+1/ ∂α = ∂ ∂α (1 − α)A (1 + n)(2 + ρ) k α t ! = B ∂ ∂α  (1 − α)k α t
                                     = B  −k α t + (1 − α) ln(kt)k α t  = Bkα t  (1 − α) ln(kt) − 1
 where B ≡ A (1+n)(1+ρ) .
                                                          Clearly, B > 0 and k α t > 0.
Consequently, the sign of ∂kt+1 ∂α is determined by the expression: (1 − α) ln(kt) − 1.
                                (1 − α) ln(kt) > 1 ⇔ ln(kt) > 1 1 − α ⇔ kt > e 1 1−α
Hence
∂kt+1 ∂α ? ??? ??? > 0 if kt > e 1 1−α = 0 if kt = e 1 1−α < 0 if kt < e 1 1−α
 Introduce depreciation in the Diamond model such that δ is the rate at which capital depreciates.
(B)
WHAT IS ECONOMY?:

Economy is the collective resource base and production capability and services capacity in all of a country. A system of production, Distribution and consuming wealth                                                                                        Economic parameter

Target
The first spreadsheet brings the socioeconomic data (Gross Domestic Product (GDP) population) introduced by the user into the Venin model. The GDP per capita is calculated in the base year as well as over time.
              The view also includes some unit variables needed in the model and information on the initial and final times, as entered by the user into the For FITS Excel file.
 Structure
 The socioeconomic parameters (GDP and population) are located on the top of the spreadsheet.

Inputs
 The GDP and the population both are specified by service and area. Both are exogenous inputs entered by the user (Socioeconomic data sheet of the For FITS Excel file). Two other user inputs are the base year values and their evolution over time. The latter are entered by evolution indexes that define the growth of the relevant parameter under consideration.
The exogenous inputs INITIAL TIME and FINAL TIME are the initial and final projection times set by the user in the excel data file (Modeling switches sheet of the For FITS Excel file). The default values are from 2010 (initial time, i.e. base year) up to 2040 (final time).
Outputs GDP and population are used to obtain GDP per capita which is one of the main drivers of the For FITS model: they have a major influence on the generation of transport demand. Several factors that determine transport activity over time are expressed as functions of a country’s GDP per capita:
 · People per active bike (view demand (passenger, not));
· Ownership of personal passenger road motor vehicles (view demand (pass. personal motor road));
 · Ownership of personal passenger LDVS (view demand (pass. personal motor road));
 · Ownership of personal passenger VESSELS (view demand (pass. personal vessels));
· Share that public transport represents in the total pkm of public transport and personal passenger vehicles (view demand (passenger, public));
 · Share of air transport pkm in the total pkm including personal vehicles, public and air transport (view demand (passenger, air));
· Tones lifted in large freight service (view demand (large freight, GDP and structure));
· Share of light vehicles in total road freight vehicles (view demand (light road freight vehicle shares)).

Parameters Restrictions

In spite of the strong theoretial case that can be made for free international trade, every country in the world has ereted at least some barriers to trade. Trade restrictions are typicaly under taken in an effort to protect companes and workers in the home economy from competition by foreign firms. A protectionist policy is one in which a country restricts the importation of goods and services produced in foreign countries. The slowdown in the U.S. economy late in 2007 and in 2008 has produced a new round of protectionist sentiment—one that became a factor in the 2008 U.S. presidential campaign.
The United States, for example, uses protectionist policies to limit the quantity of foreign produced sugar coming into the United States. The effect of this poliy is to reduce the supply of sugar in the U.S. market and increase the price of sugar in the United States. The 2008 U.S. Farm Bill sweetened things for sugar growers even more. It raised the price they are guaranteed to receive and limited imports of foreign sugar so that American growers will always have at least 85% of the domestic market. The bill for the first time set an income limit—only growers whose incomes fall below $1.5 million per year (for couples) or $750,000 for individuals will receive direct subsidies (The Wall Street Journal, 2008).

Tariffs
A  Tariff is a duty or Tax imposed by the government of a country on the Traded  community  as it cross the national boundaries  For example, the U.S. tariff on imported frozen orange juice is 35 cents per gallon (which amounts to about 40% of value). The tariff on imported canned tuna is 35%, and the tariff on imported shoes ranges between 2% and 48%.

Antidumping Proceedings
The United States Department of Justice, which is the U.S. agency in charge of determining whether a foreign firm has charged an unfair price, has sometimes defined normal profit rates as exceeding production cost by well over 50%, a rate far higher than exists in most U.S. industry.
Voluntry Export Restrictions
They became prominent in the United States in the 1980s, when the U.S. government persuaded foreign exporters of automobiles and steel to agree to limit their exports to the United States.
                              Although such restrictions are called voluntary, they typically are agreed to only after pressure is applied by the country whose industries they protect. The United States, for example, has succeeded in pressuring many other countries to accept quotas limiting their exports of goods ranging from sweaters to steel.
                                   A voluntary export restriction works precisely like an ordinary quota. It raises prices for the domestic product and reduces the quantity consumed of the good or service affected by the quota. It can also increase the profits of the firms that agree to the quota because it raises the price they receive for their products.

Other Barriers
Justifications for Trade Restriction: An Evaluation

There is no disputing the logic of the argument that free trade increases global production, worldwide consumption, and international efficiency. But critics stress that the argument is a theoretical one. In the real world, they say, there are several arguments that can be made to justify protectionist measures.
                                                Infant Industries

One argument for trade barriers is that they serve as a kind of buffer to protect fledgling domestic industries. Initially, firms in a new industry may be too small to achieve significant economies of scale and could be clobbered by established firms in other countries. A new domestic industry with potential economies of scale is called an infant industry.

(C) Suppose a government introduces a pay as you go social security system where young individuals pay a lump sum tax T and old individuals receive (1 + n)T. Suppose that if we set T = 0, the equilibrium allocation corresponds to the one computed in part (a), and furthermore, suppose that the latter satisfies the restrictions of part.
A lump sum payment is an often large sum that is paid in one single payment instead of broken up into installments. It is also known as a bullet repayment when dealing with a loan. They are sometimes associated with pension plans and other retirement vehicles, such as 401k accounts, where retirees accept a smaller upfront lump sum payment rather than a larger sum paid out over time. These are often paid out in the event of debentures.
Lump sum payments are also used to describe a bulk payment to acquire a group of items, such as a company paying one sum for the inventory of another business. Lottery winners will also typically have the option to take a lump sum payout versus yearly payments.
 Lump sums Annuity payments
if you choose the annuity option, the payments could come to you over several decades. For example, instead of $10 million of income in one year, your annuity payment might be $300,000 a year. Although the $300,000 would be subject to income tax, it would likely keep you out of the highest state tax brackets. You would also avoid the highest federal income tax bracket of 37% (as of 2020) for single people with incomes greater than $518,400 or $622,050 for married couples filing jointly.

The Welfare Theorems
We now turn to a more formal statement of the theorem suggested above – that every Walrasian equilibrium allocation is a Pareto optimal allocation. We then prove a converse result that if an initial allocation is Pareto optimal, there is a Walrasian equilibrium at which no trade occurs.
 Theorem 1 (First Welfare Theorem) (Arrow, 1951; Debreu, 1951) Let (p,(xi )i∈I) be a Walrasian equilibrium for the economy E. Then if (A2) holds, the allocation (xi )i∈I is Pareto optimal. Proof. By way of contradiction, suppose there is a feasible allocation (ˆxi )i∈I such that ui (ˆxi ) ≥ ui (xi ) for all i ∈ I with strict inequality for some i 0 . By revealed preference and (A2), p · xˆi ≥ p · xi for all i ∈ I, and also p · xˆi0 > p · xi0 (this is Walras’ law). Therefore, because prices are non negative, there must be at least one good l for which P i xˆi l > P i xi l = P i ei l.
Therefore xˆ is not feasible. Q.E.D. This result provides formal support for Adam Smith’s claim that individuals acting in their own interests end up behaving in a way that is efficient from a societal standpoint. It is a powerful statement about the efficiency properties of decentralized markets: despite the fact that there is no explicit social coordination and agents simply maximize their utilities given prices, the resulting equilibrium outcome is efficient from a social perspective.
 Note that in a sense, the assumptions are quite weak. Given our model of the exchange economy, the only assumption on preferences that we require is monotonicity (and local nonsatiation would suffice). Of course, it should be emphasized that the model itself contains a large number of heroic implicit assumptions that seem highly unlikely to be satisfied in any real economy. Among these are:
(1) all agents face precisely the same prices;
 (2) all agents are price takers – i.e. they take prices as given and don’t believe that their purchasing decisions will move prices;
 (3) markets exist for all goods and agents can freely participate. Moreover, we have said nothing so far about how a group of agents might arrive at equilibium prices. So you’ll probably want to withhold judgment on the efficiency of decentralized markets. The first welfare theorem states that equilibrium outcomes are efficient. Our next result states that efficient outcomes are Walrasian equilibria given the correct prices and endowments.
Theorem 2 (Second Welfare Theorem)
(Arrow, 1951; Debreu, 1951) Let E be an economy that satisfies (A1)—(A4). If (ei )i∈I is Pareto optimal then there exists a price vector p ∈ RL + such that (p,(ei )i∈I) is a Walrasian equilibrium for E
Proof. To prove this we need a version of the separating hyperplane theorem. Suppose you have an open convex set A ⊂ Rn and a point x /∈ A. Then there exists a p 6= 0 such that p · a ≥ p · x for all a ∈ cl(A). To prove the theorem, let’s define
:                                           Ai = {a ∈ RL : ei + a ≥ 0 and ui (ei + a) > ui (ei )}.
 Because ui is concave, Ai is a convex set. Therefore the set
                                            A = X i∈I Ai = {a ∈ RL : ∃a1 ∈ A1, ..., aI ∈ AI with a = X i∈I ai}
 is a convex set. Moreover, 0 ∈/ A, because if it were there would exist some (ai )i∈I with P i∈I ai = 0 and ui (ei +ai ) > ui (ei ) for all i ∈ I, contradicting the assumption that e is Pareto optimal. The separating hyper plane theorem now implies that there is some price vector p 6= 0 such that p · a ≥ 0 for all a ∈ cl(A). Furthermore, p ≥ 0 because if a À 0 then a ∈ A by monotonicity and if some pl < 0 we could take al arbitrarily big and all the other al0 small but positive and get a contradiction. Because p 6= 0 and p ≥ 0, this means that p > 0. The claim is that this p will support the allocation e as a Walrasian equilibrium. Obviously e satisfies the market clearing part of the definition of equilibrium. Moreover, fixing prices p, consider a given agent i. Suppose xi ∈ RL + and ui (xi ) > ui (ei ). We will show that xi is not in i’s budget set, thus proving the individual optimization part of equilibrium. First, by definition of Ai and p, p · xi ≥ p · ei . Moreover, by continuity, the fact that ui (xi ) > ui (ei ) implies that for λ just less than 1, ui (λxi ) > ui (ei ). Therefore p · λxi ≥ p · ei . This can’t be the case if p · xi = p · ei , so therefore p · xi > p · ei . Q.E.D. Note that the second welfare theorem does not say that starting from a given endowment, every Pareto optimal allocation is a Walrasian equilibrium. Rather it says that if we were to start from a given endowment then for any Pareto optimal allocation there is a way to re distribute resources and a set of prices that makes the allocation a Walrasian equilibrium outcome.
Arrow Debreu Equilibrium
An Arrow Debreu equilibrium are prices fpˆt(s t )g ∞ t=0,s t2S t and allocations (fcˆ i t (s t )g ∞ t=0,s t2S t)i=1,..,I such that: 1 Given fpˆt(s t )g ∞ t=0,s t2S t , for i = 1, .., I, fcˆ i t (s t )g ∞ t=0,s t2S t solves: max fc i t (s t)g ∞ t=0,s t 2S t ∞ ∑ t=0 ∑ s t2S t β tπ(s t )u(c i t (s t )) s.t. ∞ ∑ t=0 ∑ s t2S t pbt(s t )c i t (s t )  ∞ ∑ t=0 ∑ s t2S t pbt(s t )e i t (s t ) c i t (s t )  0 for all t 2 Markets clear: I ∑ i=1 cˆ i t (s t ) = I ∑ i=1 e i t (s t ) for all t, all s t 2 S t
(D) Show how a benevolent government can attain a better outcome in terms of welfare than the market allocation of part (a), given that the market allocation satisfies the restrictions of part

This chapter deals with the most simple kind of macroeconomic model, which abstracts from all issues of heterogeneity and distribution among economic agents. Here, we study an economy consisting of a representative ¯rm and a representative consumer. As we will show, this is equivalent, under some circumstances, to studying an economy with many identical ¯rms and many identical consumers. Here, as in all the models we will study, economic agents optimize, i.e. they maximize some objective subject to the constraints they face. The preferences of consumers, the technology available to ¯rms, and the endowments of resources available to consumers and ¯rms, combined with optimizing behavior and some notion of equilibrium, allow us to use the model to make predictions. Here, the equilibrium concept we will use is competitive equilibrium, i.e. all economic agents are assumed to be price takers.

A Static Model
Walras law states that the value of excess demand across markets is always zero, and this then implies that, if there are M markets and M ¡ 1 of those markets are in equilibrium, then the additional market is also in equilibrium. We can therefore drop one market clearing condition in determining competitive equilibrium prices and quantities. Here, we eliminate
            These are ¯ve equations in the ¯ve unknowns `; n, k; w; and r; and we can solve for c using the consumers budget constraint. It should be apparent here that the number of consumers, N; is virtually irrelevant to the equilibrium solution, so for convenience we can set N = 1, and simply analyze an economy with a single representative consumer. Competitive equilibrium might seem inappropriate when there is one consumer and one ¯rm, but as we have shown, in this context our results would not be any di®erent if there were many ¯rms and many consumers. We can substitute in equation (1.6) to obtain an equation which solves for equilibrium `
                                         zf2(k0; 1 ¡ `)u1(zf(k0; 1 ¡ `); `) ¡ u2(zf(k0; 1 ¡ `); `) = 0
Given the solution for l we then substitute in the following equations to obtain solutions for r; w; n; k, and c:
                                                                f1(k0; 1 ¡ i) = r
                                                                zf2(k0; 1 ¡ l`) = w
                                                                   n = 1 ¡ l
                                                                   k = k0
                                                                     c = zf(k0; 1 ¡ `)
Equilibrium Effects

The discussion of monopoly welfare loss in the previous section is an example of partial equilibrium analysis. It considered the monopolist in isolation and did not consider any potential spillovers into related markets nor the consequences of rent seeking for the economy as a whole. This section will go some way towards remedying these omissions.
Consider an economy that produces two goods and has a fixed supply of labor. The production possibility frontier depicting the possible combinations of output of the two goods is denoted by G (y1, y2)=0 in Figure 5.2. The competitive equilibrium prices ratio pc = p1 p2 determines the gradient of the line tangent to G (y1, y2)=0 at point a. This will be the equilibrium for the economy in the absence of lobbying.
 The lobbying that we consider is for the monopolization of industry 1. If this is successful it will have two effects. The first effect will be to change the relative prices in the economy. The second will be to use some labor in the lobbying process which could usefully be used elsewhere. The consequences of these effects will now be traced on the production possibility diagram. Let the monopoly price for good 1 be given by pm 1 and the monopoly price ratio by pm = pm 1 p2 . Since pm > pc the monopoly price line will be steeper than the competitive price line. The price effect alone move the economy from point a to point b around the initial production possibility frontier see Figure 5.2. Evaluated at the competitive prices, the value of output can be seen to have reduced. The effect of introducing lobbying can be seen by realizing that the labor of lobbyists does not produce either good 1 or good 2 but is effectively lost to the economy. With labor used in lobbying the potential output of the economy must fall. Hence, the production possibility frontier with lobbying must lie inside that without lobbying. This is shown in Figure 5.3 where the production possibility frontier with lobbying is denoted GL (y1, y2)=0. When faced with the monopoly price line the equilibrium with monopoly and lobbying will be at point c in Figure 5.3. The outcome in Figure 5.3 is what might have been expected. The move to monopoly pricing shifts the equilibrium around the frontier and lobbying shifts the frontier inward. The value at competitive prices of output at a is higher than at b, and value at b is higher than at c. Hence the lobbying has resulted in a diminution of the value of output. At the aggregate level, this is damaging for the economy. At the micro level there will be income transfers towards the owners of the monopoly and the lobbyists, and away from the consumers so the outcome is not necessarily bad for all individuals.

PROBLEMS

Let the island economy in this chapter have a productivity shock that takes on two possible values, {θL, θH} with 0 < θL < θH. An island’s productivity remains constant from one period to another with probability π ∈ (.5, 1), and its productivity changes to the other possible value with probability 1 − π. These symmetric transition probabilities imply a stationary distribution where half of the islands experience a given θ at any point in time. Let xˆ be the economy’s labor supply (as an average per market).
 a. If there exists a stationary equilibrium with labor movements, argue that an island’s labor force has two possible values, {x1, x2} with 0 < x1 < x2.
Solutions:
(a)We know from the text that labor movements are characterized by two increasing functions X−(θ) ≤ X+(θ). Assume that the current shock is θ. If, at the beginning of period, the island labor force is x < X −(θ), then outside workers move in the island so that next period labor force is X −(θ). If, at the beginning of a period, the island labor force is x > X +(θ), then workers move out of the island so that next period labor force is X +(θ). Agents who move out cannot work this period. Otherwise, that is if X −(θ) ≤ x ≤ X+(θ), all workers stay and no outside workers move in. We discuss two cases.
                                                Case 1, no movements : X−(θH) ≤ X+(θL)
This implies that, since X− and X+ are increasing in θ:
X − (θL) ≤ X − (θH) ≤ X + (θL) ≤ X + (θH).
 If the island labor force is x ∈ [X−(θH), X+(θL)], then for all s = L, H, it is true that X−(θs) ≤ x ≤ X+(θs). Thus, x is within the “moving boundaries” for all possible θ. It implies that the island labor force never change in equilibrium.
If the initial island labor force is x < X −(θH), then, at the first θ = θH the island labor force becomes X−(θH) and stay constant afterward. Similarly, if the initial labor force is x > X+(θL), then, at the first θ = θL, the island labor force becomes x = X+(θL) and stay constant afterward.
Case 2, movements:                           X+(θL) < X−(θH)
First observe that if the initial island labor force is x ≤ X +(θL), then at the first θ = θH, the island labor force is X−(θH). Similarly, if the initial island labor force is x ≥ X−(θL), then, at the first θ = θL, the island labor force is X+(θL). Lastly, assume that the initial island labor force is X +(θL) ≤ x ≤ X−(θH). If θ = θL, workers move out and next period labor force is X +(θL). If θ = θH, workers move in and next period labor force is X −(θH).
The above discussion shows that, for any initial labor force, the island labor force lies eventually in the set {X+(θL), X−(θH)}. Furthermore, once in the set, the labor force switch back and forth between X +(θL) and X−(θH). From X+(θL) to X−(θH) after a “positive shock” θH, and back to X−(θL) after a “negative shock” θL. Therefore, in a stationary equilibrium with movement, an island labor force take only two possible values :
                                                       x1 ≡ X + (θL) < X − (θH) ≡ x2.