1. True or false?
In the IS-LM model, if animal spirits make the demand for investment less responsive to changes in the interest rate, then
(a) fiscal policy will become more effective;
(b) monetary policy will become more effective.
Assume standard slopes of IS and LM. Provide graphical solution, analytical solution and explain intuitively.
Solution.
(a) Effectiveness of the policy is measured by the absolute value of the change in equilibrium output.
Consider fiscal expansion (bond financed increase in government purchases).
Analytical solution.
Initial equilibrium: 
Effect of fiscal expansion: 
.
Solving this system we get the fiscal policy multiplier 
.
If interest responsiveness of investment, i.e. 
goes down, then 
falls and as a result fiscal policy multiplier goes up, implying that fiscal policy becomes more effective.
Intuitive explanation.
If government purchases go up, then planned spending increase results in excess demand for goods at any interest rate. To meet excess demand firms expand output, which results in an increase in income and rise in money demand. Resulting excess demand for money leads to fall in interest rate and fall in investment (partial crowding out takes place), which decreases efficiency of fiscal policy.
If investment becomes less responsive to interest rate, then crowding out effect is smaller, and fiscal policy becomes more effective.
Note that if investment is less sensitive to interest rate, then IS curve is steeper as with the same increase in the interest rate we observe smaller change in investment, smaller change in planned spending and smaller fall in output. Under linearity assumption the two IS curves intersect at zero interest rate.
Conclusion: the statement is true with recpect to fiscal policy.
(b) Consider monetary expansion.
Analytical solution.
Effect of monetary expansion: 
.
Solving this system we get the monetary policy multiplier 
. If interest responsiveness of investment, i.e. goes down, then 
goes up as 
and as a result monetary policy multiplier goes down, implying that policy becomes less effective.
Intuitive explanation.
If money supply goes up, resulting in excess supply of money and excess demand for bonds, then bonds prices goes up and interest rate falls at every output level. Due to fall in interest rate investment goes up, which increases planned spending and leads to excess demand for goods. To meet excess demand firms expand output.
If investment becomes less responsive to interest rate, then planned spending increases less, excess demand is smaller and so it can be satisfied with smaller output expansion. Thus if investment is less interest sensitive, then the same increase in money supply results in smaller increase in output, indicating that monetary policy becomes less effective.
Conclusion: the statement is false as monetary policy becomes less effective.
2. Assume a small open economy with fixed prices and wages and perfect capital mobility can be described by the following equations: 
, 
, 
, 
. Money demand is given by 
, where 
is the nominal interest rate, 
- world’s income, 
- domestic income, 
- nominal exchange rate (price of a unit of foreign currency). Let 
, 
. Suppose production function is given by 
, where 
stays for labour. Nominal wages are sticky at 
.
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(a) If the exchange rate is fixed at 4, specify the conditions for equilibrium in the foreign exchange market and calculate equilibrium output and nominal money supply. Illustrate equilibrium graphically.
(b) Suppose we want to achieve 
. Is it possible to do this via the change of government purchases? If yes, then find by how much government purchases have to be increased or decreased and illustrate adjustment graphically. If no, then prove that it is impossible.
(c) Is it possible to achieve via exchange rate policy? If yes, specify the policy, find the corresponding parameters, and illustrate adjustment graphically. If no, then prove that it is impossible.
(c) Suppose now capital control takes place so that there is no more capital mobility. Specify the conditions for equilibrium in the foreign exchange market and calculate equilibrium output and nominal money supply. Illustrate equilibrium graphically.
(d) Suppose we want to achieve 
under capital control. Is it possible to do this via the change of government purchases? If yes, then find by how much government purchases have to be increased or decreased and illustrate adjustment graphically. If no, then prove that it is impossible.
(e) Assuming capital control, is it possible to achieve via exchange rate policy? If yes, specify the policy, find the corresponding parameters, and illustrate adjustment graphically. If no, then prove that it is impossible.
Solution.
(a) Due to perfect capital mobility foreign exchange market is in equilibrium if 
. Under fixed exchange rate (if this policy is credible) there is no exchange rate risk and expected depreciation is 0. Assuming absence of country risk, equilibrium appears at 
.
IS curve equation: 
or 
or 
, then 
.
LM curve equation: 
.
SRAS in sticky wage model is determined by labour demand. As marginal product og labour is constant, then labour demand is horizontal which results in horizontal SRAS at 
. 
.
(b) Yes, it is possible. As required output level is above the existing one we need expansionary policy that could be implemented via an increase in government purchases.
Rightward shift of IS (caused by increase in autonomous spending) increases domestic interest rate and results in huge capital inflow. Resulting balance of payments surplus leads to excess supply of foreign currency.
In case of fixed exchange rate excess supply of foreign currency creates a need for Central Bank intervention. To prevent domestic currency from appreciation Central Bank has to buy foreign currency, which increases money supply and shifts LM to the right and output expand even more that initial: new equilibrium is achieved at 
.
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.
(c) Thus could be done via devaluation of national currency (policy has to be unanticipated as otherwise BP curve shifts up).
Devaluation of national currency brings an improvement in trade balance and leads to the rightward shift of IS. As a result domestic interest rate goes up provoking huge capital inflow. Resulting balance of payments surplus leads to excess supply of foreign currency.
Under fixed exchange rate the position of LM curve becomes endogenous as CB has to intervene in response to external disbalance by selling/buying foreign currency. Central Bank buys foreign currency, which increases money supply and shifts LM to the right up to point .
, 
or 
.
Thus the required output expansion could be achieved via 40% devaluation of national currency.
(c) As CF=0, then BP=NX+CF=0 implies 
or 
. Thus at initial equilibrium 
, 
.
(d) It’s impossible as BP curve is not affected by fiscal policy and to keep equilibrium at foreign exchange market output has to stay constant.
Fiscal expansion brings a rightward shift of IS which results in an increase in domestic interest rate. But due to absence of capital mobility there is no capital inflow. As domestic income goes up demand for imported goods increases and balance of pauments goes into deficit. Thus we observe excess demand for foreign currency.
Central Bank intervenes by selling foreign currency, money supply goes down and as a result LM shifts up. New equilibrium is achieved at 
with the same level of output but increased interest rate.
(e) Thus could be done via devaluation of national currency Devaluation of national currency (it does not matter whether policy is anticipated or not as there is no capital mobility) brings an improvement in trade balsnce. As a result both IS and BP cirves shift to the right (and BP shifts nore than IS).
Intersection of shifted IS curve with initial LM appears at poit that lies to the left from the new BP curve, indicating that there sis balance surplus. Resulting excess supply of foreign currency requires CB intervention: CB buys foreign currency, which increases the money suplly and shifts LM curve down. New equilibrium appears at point 
with increased output and reduced interest rate.
, 
or 
.
Thus the required output expansion could be achieved via 15% devaluation of national currency.
3. Consider AD-AS model with aggregate demand curve given by 
and short run aggregate supply curve represented by equation 
.
(a) Suppose that initially economy is in the long run equilibrium. Find the equilibrium output and price level and illustrate on the graph (denote it by A).
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(b) Assume static expectations. What happens with output and price level in the short run if nominal money supply increases by 3? Illustrate the new equilibrium on the same graph (denote it by B).
(c) How would your answer to part (b) change if agents have rational expectations and monetary expansion was fully expected. Illustrate the new equilibrium on the same graph (denote it by C).
(d) How would your answer to (b) change if expectations are rational and monetary expansion was unexpected. Illustrate the new equilibrium on the same graph (denot it by D).
Solution.
(a) Long run equilibrium is given by intersection of aggregate demand curve with long-run aggregate supply curve (point A). 
and 
, 
.
(b) SR equilibrium: 
and as 
we get 
or 
and 
. Graphically new equilibrium is achieved at the intersection of initial SRAS curve and the shifted AD curve. (point B)
(c) If expansion was fully expected, then there will be no mistake and and there will be no deviation from full employment even in SR as according to policy ineffectiveness proposition anticipated economic policy does not affect output: 
and from aggregate demand curve we get 
.
Graphically it can be illustrated by simultaneous upward shifts of AD and SRAS curves. (point C)
(d) If expansion was unanticipated, then there will be deviation from full employment output in the SR and the equilibrium will coincide with the one found under static expectation in (b) so that points D and B coincide.
4. Consider an economy represented by AD-AS model. Suppose that there is a one time positive productivity shock due to technological improvements.
(a) Represent the short- and long- run impact of such policy on output and prices.
(b) Policy authorities wish to use demand-side policies in order to counteract the effect of the supply-side shock. How could the Central Bank intervene, and what would be the effect of the intervention?
(c) How could the Government intervene, and what would be the effect of the intervention?
Solution.
(a) Positive productivity shock increases MPL and shifts up the labour demand curve. As a result full employment output increases from 
to 
.
Using sticky wages model for SRAS we can treat this change as a rightward shift of SRAS for the same nominal wage due to increased demand for labour. Note: SRAS shifts more than LRAS.
As a result price falls and output increase from to 
in the short run. In the long-run new contracts are signed with adjusted nominal wage (
) and economy shifts to .
Note: depending on AD slope point E’ may lie either to the left or to the right from LRAS and so SRAS may either shift further to the right or partially back to the left.
(b) To move faster to the new full employment level of output CB can use expansionary monetary policy (by purchasing government bonds), which shifts AD up (or contractionary policy if we have flatter AD and point E’ is to the right from ). Then if SRAS and AD shift simultaneously, economy would jump to the new full employment level of output.
(c) To move faster to the new full employment level of output Government could use expansionary policy (or contractionary if we have flatter AD and point E’ is to the right from ) fiscal policy (for example by increasing government purchases), which shifts AD to the right. Then if SRAS and AD shift simultaneously, economy would jump to the new full employment level of output.
5. Consider the sticky-wage model of the labour market. Suppose that instead of being fixed, money wages are partially indexed to inflation, that is, if the price level rises by 1% then money wages rise by x%, with 0<x<1. How does this indexation affect the SRAS curve?
Solution.
As indexation is only partial, then following the price increase even with this adjustment real wage is still below than equilibrium one.
Thus labour is cheaper and firms are willing to increase employment, which results in an increase in output. But definitely this time real wage is higher that in absence of indexation and thus we observe smaller rise in employment (economy moves from 
to 
and 
), which results in smaller output expansion: output increases from to 
, which is less that in absence of indexation as 
. As a result short-run aggregate supply curve is steaper.
The slope of SRAS with indexation could be derived formally.
and 
.
Differentiating we get
Slope of SRAS: 
.
6. Consider a small open economy with flexible wages and prices and no capital mobility. How investment is affected by a balanced budget increase in government purchases in this economy?
Solution.
Absence of capital mobility implies that in equilibrium NX=0.
Under flexible prices and wages economy operates at potential (full employment output), thus output stays the same and there are no changes in income taxes, which implies that government purchases are financed by an increase in the lump-sum taxes: 
1 point. As a result disposable income falls (
) and consumption decreases but consumption fall less than YD as we assume that marginal propensity to consume is less than one: 
but 
or 
.
So 
, i.e. investment has to fall.
7. Assume the monetary authority has the following loss function 
, where 
is target inflation rate and 
is the log of target level of output, 
. Also assume Lucas type aggregate supply function of the form 
, where 
is the log of full employment level of output. Assume that 
.
(a) Find equilibrium inflation rate under discretionary policy. Comment on the parameters affecting equilibrium inflation rate.
(b) Is there a time inconsistency problem? Illustrate your answer by the graph.
(c) Show how inflation bias could be eliminated by
(i) appointment an independent central banker with different loss function;
(ii) setting for the central banker inflation target that is different from the socially optimal target ;
(iii) offering contract to the central banker.
Solution.
(a) Let’s find subgame-perfect Nash equilibrium using the method of backward induction. Thus we consider the last step, i.e. given the private sector expectations we will find the best rate of inflation from the solution to the following problem:
We substitute 
from constraint into the objective function:
.
First-order condition gives: 
, which gives
.
As expectations are rational, then 
.
Solving equation with respect to 
we get 
.
As a result 
, and from supply curve we find the level of employment 
.
It should be noticed that the equilibrium inflation rate is above the target level. It increases in 
as with higher the problem of inflation becomes less important, which results in larger deviation of equilibrium inflation rate from the socially optimal one.
Equilibrium rate of inflation also increases in 
as with larger the aggregagte supply curve becomes flatter. As a result the same change in rate of inflation allows sharp change in output, so the incentive for increase in inflation rate is large.
Equilibrium corresponds to tangency point of loss function level lines and the SRAS curve that corresponds to 
(point A).
(b) If monetary authorities announces inflation rate and population believe in this announcement so that economy opereates along SRAS curve that corresponds to 
, then central banker would find optimal to deviate from as allowing higher inflation rate reduces social loss: 
. As a result economy moves to point B. It implies that announcement of target rate of inflation is not credible and population would never believe in it.
Therefore, the government faces a time-inconsistency problem: target inflation rate is desirable in terms of loss minimisation but once the time to choose the economic policy comes the monetary authorities find it optimal to deviate from .
(c) (i) 
iff 
.
(ii) 
iff 
.
(iii) Optimal contract. Consider transfer function of the form 
. Then transfer inclusive loss function is given by 
.
FOC 
, which results in equilibrium inflation rate 
iff 
.
8. Read paper of Yellen J., Efficiency wage models of unemployment, American Economic Review, 74, May, 1984 and answer the following questions:
(a) State all explicit and implicit assumptions of the efficiency wage model considered by Yellen.
(b) Write down the representative firm problem and derive the first order conditions.
(c) Explain the statements based on the model:
(i) “A profit maximizing firm which can hire all the labour it wants at the wage it chooses to offer will offer a real wage 
, which satisfies the condition that the elasticity of effort with respect to wage is unity”.
(ii) “… this wage choice [ ] minimizes labour costs per efficiency unit”.
(d) Explain how unemployment may arise in this model.
(e) Analyze the effect of positive productivity shock in the framework of efficiency wage model.
(f) List and briefly comment on different explanations for efficiency wages mentioned by Yellen.
Solution.
(a) Assumptions of the efficiency wage model.
Ø Identical perfectly competitive(on output market) firms, represented by one aggregate firm that act as price maker (different from monopsonistic one) on labour market,
Ø Firm’s output depends on the number of workers employed 
and on their effort 
, labour and effort enter the production function multiplicatively 
,
Ø MPL is assumed to positive and diminishing: 
and 
,
Ø Effort level is positively related to real wage paid: 
and 
,
Ø If 
exceeds reservation wage and resulting Ld falls short of LS, employment is determined by Ld.
In fact firms choose effort rather than wage rate but as wage is the only determinant of effort effort-setting is equivalent to wage setting.
(b) The representative firm problem takes the form:
First order conditions: 
(1), 
(2).
(c) (i) From (1) 
and from 
(2). Thus 
(3).
(ii) Labour costs per efficient unit are given by 
. If we minimize this cost w.r.t. , the FOC gives
. (4)
Thus the real wage rate for which 
satisfies (4) and so minimizes labour costs per efficiency unit.
(d) Thus real wage rate is given by solution of (3) and is determined completely by effort function. The corresponding Ld can be found from equality of MPL and effective costs of labour as it follows from (2):
(5).
If 
and 
defined by (5) is less than the corresponding labour supply, then firms are unconstrained in choice of and will set 
. As a result there will be unemployment of 
. Thus some workers that are willing to work at given(and even lower) wage rate are not hired as then MPL will be below efficiency labour cost. Firm could set lower wage rate but with lower wage rate effort level of all workers goes down and productivity of those who were employed falls.
If exceeds 
, then firms are constrained by , wage rate goes up and there is no unemployment.
(e) Let 
reflects the state of technology, then the firm’s problem takes the form
FOCs: 
and 
. Rearranging we get
(6) and 
(7).
If productivity increases (s goes up), then according to (6) wage rate is not affected and so effort and efficiency costs are not affected as well.
Only L demand responds: it follows from (7) that L will increase under assumption of diminishing MP to keep LHS constant.
(f) Different explanations for efficiency wages.
Ø Nutrition explanation for LDC is mentioned: higher real wage allow workers to get better food and as a result workers’ health improves, which improves average productivity by reducing illness.
Ø Shirking: under unobservable efforts wage rate above the market clearing provides incentive to work rather than to shirk. Unemployment that results from efficiency wage makes job loss more costly and “serves as a worker discipline device” in moral hazard problem.
Ø To reduce costly labour turnover: under high unemployment (that results from wage above market clearing) workers have lower incentive to move from one job to the other as the probability of getting the new one is lower;
Ø Adverse selection: if workers differ in their productivity characteristics and productivity is unobservable but is positively correlated with reservation wages, then firms with higher wage will attract workers with higher productivity.
Ø Sociological models: if individual worker’s effort depends on the work norms, the firm can raise the group work norms and as a result the average effort by paying worker in excess of his opportunity costs “in return for his effort above the minimum required”.
9. Consider Cagan money demand function of the form 
, where 
. Assume that current growth rate of money supply is zero. Suppose that at 
Central Bank announces that starting from 
money supply would be increased from 
to 
. Assuming that agents have rational expectations and announcement ics credible find the change in current price level. Draw the time path of 
.
Solution.
In equilibrium money demand equals money supply: 
. Solving for the price level we get
, where 
.
Similarly for the next period 
. Thus we get expected price level by taking expectations from both sides
.
Plugging it back into the formula for equilibrium current price level we obtain 
. Itereating forward and assuming 
finally we get
.
As initially money supply was constant and equals then initial price level was constant at 
.
The price level would be affected starting from the moment of announcement : 
Thus we have 
.
So price level is constant at 
up to the moment of unnouncement, then it jumps up due to revision of expectations concerning future monetary policy and gows continuosly up to the level of 
which is achieved at the point when announced policy is implemented and then it is stabilized at this level.
10. Consider the following version of the Solow model with government sector (without technical progress). Assume that an income tax at the rate t is levied and that, accordingly, private saving per capita is equal to 
. The government purchases per capita are constant and equal 
and there are no government transfers.
(a) Derive the capital accumulation equation. Explain each step carefully.
(b) Define the steady state and illustrate it graphically. How many steady states are there? Discuss stability of steady states.
(c) Analyze the impact of an increase in the tax rate on the steady-state capital per capita and output per capita. Draw a chart of the time paths of the growth rate of capital per capita.
(d) Analyze the impact of an increase in government purchases on the steady-state per capita
(e) In view of your answers to (c) and (d) comment on the following statement: “To increase the growth rate of per capita output government should run budget surplus to free resources for investment.”
Solution.
(а) In equilibrium in case of closed economy aggregate savings (i.e. sum of private and government) has to be equal to investment: 
or 
.
Taking into account that gross investment is equal to the total of capital stock increase and depreciation expenses (
), the equilibrium condition transforms to the following:
We divide both sides of the equation by Lt and taking into account homogeneity of degree one of production function get:
.
Proceeding to per capita terms by denoting capital per capita by k and output per capita by f(k), taking into account that 
we obtain the capital accumulation equation:
This equation shows that any excess of per capita aggregate savings over required investment is used for capital accumulation.
(b) We define the steady state in the model under consideration as a situation where capital per worker remains unchanged: 
. Steady state per capita capital stock k* is given by:
The point of intersection of the aggregate savings curve and required investment line determines the steady state per capita capital k*.
There are two steady states (
and 
) but only one is stable- . If economy operates at some point 
, then total per capita savings exceed required investment and capital per worker goes up. As a result in the long run such an economy operates at . (c) With an increase in the tax rate disposable income fall, resulting in reduction of private saving but government saving goes up and the last effect dominates: 
increases in 
.
As a result aggregate saving function shifts upwards and steady state capital per worker increases from 
to 
and per capita output increases as well.
Immediately after an increase in the tax k goes up (excess of savings over required investment is used for increase in k). Initially economy was in steady state with growth rate of k, equal zero. So when t goes up, growth rate becomes positive. As capital accumulates, the gap between aggregate savings and required investment becomes smaller and as a result growth rate start to fall back until it becomes again equal to zero at the new steady state.
(d) With an increase in government purchases government saving falls while private saving are unaffected, thus aggregate saving goes down and steady state capital per worker falls from to and per capita output falls as well.
Immediately after an increase in g k goes down as total savings falls short of required investment. Initially economy was in steady state with growth rate of k, equal zero. So when g goes up, growth rate becomes negative. As capital falls, the gap b/w required investment and per capita aggregate saving accumulates becomes smaller and as a result growth rate approaches zero.
(e) Conclusion: tax rate and government purchases do not affect the LR growth rate but if tax rate goes up and/or government purchases falls, BS goes up and as a result during transition to the new steady state the growth rate increases.
11. Consider the Hall’s version of intertemporal consumption model. Suppose that agents have rational expectations and maximize expected lifetime utility 
subject to the budget constraint 
. Assume that 
.
(a) Show that the first order conditions imply 
.
(b) Assume quadratic utility function: 
with 
and 
. Show that 
for all 
.
(c) Suppose that 
and your life has four periods (
). The one when you study and have no income 
, when you work but at a low wage 
, one when you work at a high wage 
and finally one when you are retired. Let 
, 
, 
, 
, 
, 
. Solve for the optimal consumption paths.
(d) At time 
the government announces to tax you one single time at period with an amount 
. How does your consumption change (assuming that announcement is credible)?
(e) At time the government announces to tax for all periods 
with an amount . How does your consumption change (assuming that announcement is credible)?
Solution.
(a) At period consumer faces with the following intertemporal expected utility maximimisation problem:
Denoting by 
the Lagrangean multiplier accosiated with the budget constraint we get the following first order conditions:
.
As 
this implies 
for every . If 
then 
, which gives the required equality 
, which suggests that current marginal utility of cunsumptio is a best predictore of the future marginal utility.
(b) If then 
and 
. Thus 
which is equivalent to 
, i.e. consumption today is a best predictor of consumption tomorrow.
(c) If , then budget constraint together with suggests that 
.
At period 
we get new information concerning income in period and slo reivise expectations concerning future incomes, which changes the consumption profile:
At period consumption goes up due to upward revision of income expectations. We can see that 
At period 
the realized income appears to be less than the expected one which brings a reduction in consumption relative to the level planned in the previous period:
so that 
.
As there are no news in the last period consumption stays the same: 
.
(d) As announcement is made at period there are no changes in period t consumption level 
. But at consumer revises his expectations about disposal income of period and smooth this reduction over the remaining three periods:
(e) Again there would be no changes in period t consumption level as the change in taxes was unexpected. But starting from consumer would revise his expectations about disposal income of each period. Now he faces with permanent fall in disposal income (rather than temporary that was observed in part (d)) and as a result there would be no change in saving but consumption is reduced by the fall in permanent income: 