Solow Model and DMP model

ECONOMICS
Consider the DMP model. Low unemployment is a commonly pursued goal of governments. A subsidy, s, is given to firms to encourage more hiring is a policy option that can be implemented with the intended goal of increasing employment and reducing the unemployment rate
1. What is the firm’s surplus, consumer/worker surplus and total surplus with the introduction of a subsidy?
In a successful firm-worker match, the firm produces an output level, z. The firm’s surplus is given by the profit generated from a certain level of output. Assuming a unit price of 1 and that no capital is used in the production process; the firm generates a profit of
Profit = z – w; where w is the real wage paid to the worker
An employment subsidy, S, increases profits by the amount of the subsidy such that:
Profit/firm’s surplus = z – w + s
A worker will only accept a job if the offered wage exceeds the amount of unemployment insurance benefit (b) received. As such, the consumer surplus is given by:
Consumer/Worker Surplus = w – b
Total surplus = consumer surplus + firm’s surplus
= z – w + s + (w – b)
= z + s – b
2. What is the real wage solution using Nash bargaining
Suppose that the worker and firms share the total surplus in the proportion ? and (1- ? ) respectively; the proportion of the surplus attributed to the worker and firm is given by:
Worker surplus: w – b = ? (z + s - b)
Firm’s surplus: z – w + s = (1- ?) (z + s - b)
The real wage solution to the Nash bargaining problem then becomes:
Max (w – b) ? (z – w + s) 1- ? s.t S = z + s – b; yielding the wage equation:
w = ? (z + s) + (1 – ?) b
3. Equations that determine the supply side of the market V (Q) and demand side of the market em(1/j, 1)
The supply side equation:
V(Q) = b + em(1, j) (w – b); but (w-b) = ? (z + s – b)
Thus: V (Q) = b + em (1, j) ? (z + s – b)
The Demand-Side Equation:
Profits are given by: z – w + s;
Firms will post vacancies until the expected payoff from the same is zero such that the product of the probability of finding a worker and the firm’s surplus is equal to capital investment (k) in posting a vacancy. Denoting the probability of finding a worker as em (1/j, 1), this condition requires that:
em (1/j, 1) (z – w + s) – k = 0
thus; em (1/j, 1) (z – w + s) = k
em (1/j, 1) = but z – w + s = (1- ?) (z + s – b)
Thus: em (1/j, 1) =
The demand-side equation determines (j) given the exogenous variables ?, k, z, and b
The supply-side equation uses the calculated (j) to determine the labor supply Q.
4. Solve for equilibrium and show it graphically
Labor Supply (Q)
V (Q) = b +em (1,j) ?(z +s-b)
(a)
Q*
J* Labor market tightness (j)
(b)
k/[(1-?)(z+s-b)]
em (1/j, 1)
j* Labor market tightness (j)
At equilibrium, the demand and supply equations are equal, as shown in panels (a) and (b) above, yielding Q* level of labor supply and a level of market tightness, j. In panel (b), the ratio of vacancy posting costs to the firm’s surplus from a successful firm-worker match determines the tightness of the labor market, j. The labor market tightness, j, in turn, determines labor supply, Q, in pane (a).
5. What is the impact of the subsidy (s) on labor supply (Q), market tightness (j), real wage (w), and unemployment rate (u)
The subsidy (s) increases the firm’s surplus as well as the total surplus by the amount of the subsidy. This makes it more attractive for firms to post vacancies and increases the chances of workers finding jobs, leading to an increase in labor market tightness (j).
The subsidy pushes the real wage up. The increase in total surplus increases the amount of surplus available for sharing between firms and workers, causing an increase in the real wage.
The higher wages increase the worker surplus since there is no change in unemployment insurance (b). This makes more workers willing to take up jobs at the prevailing wage rate, resulting in an increase in labor supply (Q).
The subsidy increases the firms’ surplus (profits), causing an increase in the level of output (z), and subsequently, opening up more vacancies for employment. At the same time, the increase in wages as a result of the subsidy makes employment more attractive than unemployment, leading more people to seek out jobs. These factors cause an increase in the level of employment and a decrease in the unemployment rate (u).
Consider the Solow growth model. Suppose that with d = 0.1, s= 0.2, n = 0.01, and z = 1; and assuming a period of 1 year
1. Determine capital per worker, income per capita, and consumption per capita in the steady state. Show the theoretical derivation and numerical solution
The Solow growth model is derived from the Cobb-Douglas production function (F = AK ? N(1- ? as it is increasing in both capital and labor, displays decreasing marginal returns, displays constant returns to scale, satisfies the Inada Condition (Berg, 2016).
The stock of capital in discrete time is calculated as: Kt+1 = It + (1- d) Kt
Assuming the Keynesian model and a closed economy; savings and investment are equal and equal a constant fraction, s, of total income (Y); thus
St = It = sYt = sKt?
Suppose the population grows at a constant rate, n, we need an additional investment of nk to ensure that new workers have as much capital to work with as the existing workers (Berg, 2016). Given deprecation, d, an annual investment of n+d would be needed to keep capital steady;
Kt = sKt? + (d + n) Kt ………………………..the Solow Growth Model
The steady state capital stock per-capita is the level of capital (K* ) at which there is no change in capital stock such that:
= sKt? + (d + n) Kt = 0
Assuming ? = 0.33;
Making Kt the subject of the formula: = Kt(? – 1)
= -0.55 = Kt –0.67
= Log 0.55 = - 0.67 log Kt
= 0.3875 = log Kt ; Kt = 2.44
Income per-capita;
y * = k* x (? /(1- ?) = 2.44 x (0.33 / 0.67) = 3.275
Consumption per capita;
Consumption = 1 – savings (s)
Consumption per capita is given by:
c* = k* (1 – s)
c* = 2.44 (1-0.2) = 1.952
2. Suppose that savings increase to s=0.4 in the steady state. Determine capital per worker, output per worker, and consumption per capita in the new steady state
Capital per worker;
Kt(? – 1) =
Kt(-0.67) =
Kt(-0.67) = 0.275 ; Introducing logs;
Log 0.275 = - 0.67 log Kt
0.8368 = log Kt
Kt = 6.868
Income per-capita;
y * = k* x (? /(1- ?) = 6.868 x (0.33 / 0.67) = 12.97
Consumption per capita;
Consumption per capita is given by:
c* = k* (1 – s)
c* = 6.868 (1-0.4) = 4.121
3. Suppose that the depreciation rate, d, increases, what is the effect of this change on the quantity of capital per worker and output per worker from the steady state in a above
The equation of capital per worker is = Kt(? – 1)
The equation indicates a direct relationship between depreciation (d) and capital per-worker. An increase in (d) will, therefore, cause an increase in capital per worker (k*)
Income per-capita;
The equation for income per-capita is;
y * = k* x (? /(1- ?)
The increase in k* due to the increase in (d) will cause a subsequent increase in income per capita (y*) as there is a direct relationship between k* and y* as shown in the equation.
What is the unemployment rate in Canada and the United States currently and how does this compare to the rate of unemployment in these countries in the last decade?
Canada’s unemployment rate in January 2020 stood at 5.5 percent according to Trading Economies, before rising sharply to 10 percent by the end of April 2020 due to the COVID-19 pandemic (Trading Economies, 2020). The US unemployment rate in January 2020 stood at 3.6 percent, and 14.7 percent by the end of April 2020 (Trading Economies, 2020b). The unemployment rate in the US has consistently been lower than that of Canada by between 1 and 3 percentage points over the past decade. The only divergence occurred in the late 1970s, when the US reported a higher unemployment rate than Canada due to the 1974-75 recession, which hit the US harder because of its greater dependence on imported oil. Both countries have, however, reported a steady decrease in unemployment rates since the 1970s. The difference in unemployment rates between the two countries has been on the decline, falling to between 1 and 2 percentage points since 2000.
References
Berg, H. V. (2016). Economic Growth and Development (3rd ed.). Danvers, MA: World Scientific Publishing.
Trading Economies (2020). Canada Unemployment Rate. Trading Economies. Retrieved from https://tradingeconomics.com/canada/unemployment-rate
Trading Economies (2020). United States Unemployment Rate. Trading Economies. Retrieved from https://tradingeconomics.com/united-states/unemployment-rate