The golden rule rate of saving

      (a) In the Solow model, the steady-state capital stock associated with the golden rule rate of saving is a decreasing function of the depreciation rate of capital. (b) In the Diamond-Mortensen-Pissarides model of the labour market, the fundamental theorems of welfare do not apply. (c) The equity premium puzzle would be less severe if, in the data, consumption growth and excess returns of stocks over bonds were less strongly correlated, everything else equal. (d) The Hodrick-Prescott filter can be used to take a trend with constant growth out of a data series. (e) Consider a labour market matching function m = µuγ v 1−γ , where m is the number of new matches, u is the unemployment rate, v is the number of job vacancies, and γ ∈ (0, 1) and µ > 0 are parameters. Given this matching function, the job finding rate is negatively related to the vacancy filling rate. (f) The Kaldor growth facts imply that, on average, the aggregate capital stock grows at the same rate as aggregate output. 1 Question 2. Labour supply (25 points) Consider a static model of a household with preferences given by: U(c, h) = c 1−σ − 1 1 − σ − κ 1 + ψ h 1+ψ , where σ, κ, ψ > 0 are preference parameters. The household chooses consumption (c) and hours worked (h) to maximize U(c, h), subject to the following budget constraint: c = wh, where w is the wage rate per hour worked. (a) Derive the first-order conditions for consumption and hours worked (labour supply). (b) Derive the Frisch elasticity of labour supply in this model. (c) Show that when σ = 1 (so that U(c, h) = ln c − κ 1+ψ h 1+ψ ), the number of hours worked chosen by the household does not react to a change in the wage rate w. (d) Suppose now that ψ = 1 and σ = 2. If the wage increases by 1 percent, by how much does labour supply change? Decompose your answer into the underlying effects. (e) Derive an analytical expression for elasticity of consumption with respect to the wage (for general σ, κ, ψ > 0). Show that this elasticity strictly exceeds 1 if and only if σ < 1 and explain intuitively why this is the case. 2 Question 3. Consumption of durables and non-durables (15 points) Consider an infinitely-lived household which consumes both durables and non-durables. Expected discounted utility at time zero is given by: E0 X∞ t=0 β tu (ct , dt) where ct denotes consumption of non-durables in period t, dt is the stock of durables owned by the household in period t, β ∈ (0, 1) is the household’s subjective discount factor, and Et is the expectations operator conditional upon information available in period t. The price of both durables and non-durables is one. The budget constraint of the household in period t is given by: ct + dt = yt + (1 − δ) dt−1, t = 0, 1, 2, 3.. where δ ∈ (0, 1) is the depreciation rate of durables and yt > 0 is an exogenous income variable, which follows a stochastic but stationary process. In each period t, the household chooses ct and dt such as to maximize the expected present value of lifetime utility. (a) Derive the first-order optimality conditions associated with the household’s decision problem. Suppose now that the utility function is given by u (ct , dt) = ct + γ ln dt . 2 (b) Derive an analytical expression for dt in terms of the model parameters and show that dt remains constant over time (given the parameters). (c) Show that dt is increasing in β and explain intuitively why this is the case. 2For simplicity, ignore any non-negativity constraint on ct. 3 Question 4. Pricing inflation risk (20 points) Consider an infinitely-lived, representative household with preferences given by U0 = E0 X∞ t=0 β t c 1−ρ t − 1 1 − ρ , where ct denotes consumption at time t, β ∈ (0, 1) is the discount factor and ρ > 0 is the coefficient of risk aversion. The household can invest in two types of one-period bonds: (i) nominal bonds, denoted Bn t , which offer a nominal interest rate r n t , (ii) inflation-linked bonds, denoted Bi t , which offer a nominal interest rate r i t (1 + πt+1), where πt+1 = Pt+1 Pt − 1 is the inflation rate in period t+1, and where Pt is the nominal price level in period t. Inflation is uncertain, i.e. it evolves stochastically over time. The budget constraint of the household in period t, in nominal terms, is given by: Ptct + B n t + B i t = Ptyt + 1 + r n t−1 B n t−1 + 1 + r i t−1 (1 + πt) B i t−1 , where yt is an exogenous and stochastic income flow. In each period, the household chooses ct ,Bn t and Bi t in order to maximize the utility objective given above, subject to the budget constraint. Let b n t ≡ Bn t /Pt and b i t ≡ Bi t /Pt denote the real values of nominal and inflationlinked bonds, respectively. (a) Re-write the budget constraint in real terms (as opposed to nominal terms). (b) Derive the Euler equations for the two types of bonds. Which of the two bonds would the household consider to be risk-free? Suppose now that in equilibrium it holds that yt = ct and that the central bank sets a monetary policy according to a rule which targets inflation as a function of output growth: πt = γ yt yt−1 − 1 , where γ > 0 is a policy parameter. (c) Consider the risk premium formula derived in class for the excess return of equity, but apply it instead to the two bonds considered above. Which of the two bonds earns a higher ex-ante expected return? (d) Discuss the intuition for your answer under (c).