Review: How does RCK model handle government?
- Doesn't enter into utility function (doesn't raise utility)
- But changes budget constraint (increases spending of households)
- The \(\dot k = 0\) line shifts down
- Matters whether its permanent or temporary: if permanent, will lower consumption with no change to capital, if temporary, consumption lowers initially then raises to the new k*.
Diamond Model
Assumptions:
- People live for 2 periods
- \(L_t\) individuals are born at time t
- Thus, \(L_{t + 1} = (1 + n) L_t\)
- Individuals only produce when they're young, consume when they're old
- Utility function: see equation in notes
- production function same as before (Cobb-Douglas) with A growing at g
- Real interest rate and wage per unit of effective worker is determined from the constant-returns-to-scale assumption. This means that the factors of production earn their marginal product:
\(r_t = f'(k_t)\)
\(w_t = f(k_t) - k_t f'(k_t)\)
- Roughly, wages are 70% of income and capital is 30%
- Since the young divide their labor between consumption and savings, their budget constraint splits the income between the first and second period consumption.
This gives us the ability to maximize the equations. The first-order-conditions imply:
- Consumption in first-period is fraction of the income
- Interest rate enters in the denominator, so it will impact consumption and savings
- Savings is 1 - consumption
- When \(\theta =1 \), r does not have an effect on the savings rate: unclear because both substitution and income effect (more attractive to save but also have more income to spend)
Dynamics of the economy:
- investment is at 0 after every new period-- old use it up
- Savings multiplied by new output equals the capital stock
- When deriving the capital per unit of effective labor equation: remember that
\(L_{t + 1} = (1 + n) L_t\)
\(A_{t + 1} = (1 + g) A_t\)
Evolution of k over time
- Utility is logarithmic, meaning \(\theta = 1\). This makes the savings rate (notice no interest rate):
\(\frac{1}{2 + \rho}\)
- \(w_t = (1 - \alpha)\) is the marginal product of labor.
- Steady state means k is not moving, thus \(k_t = k_{t + 1} \)
\(k_{t + 1} = \frac{1}{(1 + n)(1 + g)} \frac{1}{2 + \rho} (1 - \alpha) k_t ^\alpha\)
- Why 45 degree line? It depicts the combos of k such that \(k_t = k_{t + 1} \)
- Stable relationship -- level of k gradually shifts to the steady state... see notes on graph
Properties of steady state:
- k* is globally stable
- All is similar to Solow and RCK
- How does k* respond to a fall in discount rate?
- growth in output per effective worker = 0
- growth in output = n + g
- Why do things shift down when g rises (more technology)? We're looking at output per effective worker. Thus, it's like having more people around since they're more productive.
Dynamic Inefficiency
- There is nothing in the model to bring the economy to the golden-rule level (steady-state level of capital would be too high)
- Possible to over-accumulate capital
- "The action" is in n when you apply k* to both sides:
- \(f'(k) = k(\alpha -1)\) is the derivative in the Solow model
- Hence, it is possible that k* is bigger than k(GR), and this would mean that a central planner could improve this market outcome. The economy is simply investing too much. Why? Essentially, the markets are incomplete since there are transactions that cannot take place in the market economy. Source of incompleteness: We cannot engage in a transaction with those who have not been born. In theory, the central planner has access to those markets by imposing a tax.
- Suppose that k* > k(GR). This means that the central planner can tax the young by k* - k(GR) and give this money to old people. the young will not complain because this level of capital maximizes consumption. Hence the marginal product of capital will improve to n. (formerly was f'(k), which is too small)
- Evidence for dynamic inefficiency: compare marginal product of capital with the growth rate of the economy
- The marginal product of capital, in equilibrium, should be equal to the interest rate
- And then, we see that the actual interest rate is far below the (n + g)
- An economy is dynamically efficient if net capital income exceeds investment
- Finding: national income minus employee's compensation plus income from self-employed and conclusion is that the US and other industrialized economies are Dynamically Efficient
Food for thought:
- Asset Bubbles and Overlapping Generations (famous paper by Tirole): people realize that there is a bubble and keep riding it even though they know the fundamentals aren't there
Introducing Government in the Diamond Model
How does the government change the steady state results in this model?
- It detracts from the income and will lessen consumption and capital
- Does Ricardian Equivalence hold in the Diamond model? No!
Problems with all our models...
Cannot explain difference in income per capita
Growth rate of tech. progress is exogenous
Effect of policy variables on growth (e.g. an increase in savings rate or decrease in rate of time preference) appear to be small
4.1 and 4.2: Human Capital
\(Y_t = K_t^\alpha [A_t H_t] ^{1 - \alpha} \)
Determinants of human capital:
- \(H_t = L_t G_e\), where "e" is education
- This human capital function assumes that the only input to human capital is years of education
- If \(E = 0\), then \(H_t = L_t\) and you're back exactly to the Solow model
- We don't know how education affects labor, so just assume \(G(E) = e^{\phi E}\), which would signify the difference in the wage-gap between skilled and unskilled workers
Dynamics are exactly the same as the Solow model = thus bad because it still leaves g (growth of A) unexplained
Look at growth accounting (slideshow)
Differences in income per capital = fraction of capital invested + differences in skilled and unskilled + "A" catching all else