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Instructions. This consists of four questions in total. There are 100 points to be earned. Please limit your answer to maximally 500 words per subquestion (i.e. max 500 words for question 1(a), max 500 words for question 1(b), and so forth). This word limit is provided as guidance only. You will not be penalised if you exceed the word limit. Question 1. Agree or disagree? (40 points) Consider four out of the six statements below.1 Do you agree or disagree? Explain your answers. (a) In the Solow model without population growth, the steady-state capital-to-output ratio exceeds one if the saving rate exceeds the depreciation rate. (b) In the Diamond-Mortensen-Pissarides model of the labour market, the wage can never fall below the unemployment benefit. (c) In OverLapping Generations models, a pay-as-you-go pension can improve social welfare only if the model features productive capital. (d) In the Real Business Cycle model, consumption growth and the marginal product of capital are positively correlated. (e) If output increases but consumption and labour supply remain unchanged, then frictions in the labour market may have become less severe. (f ) In the money-in-the-utility model, the Friedman rule is implemented via a sustained contraction of the money supply. 1 Question 2. Labour taxation (30 points) Consider a model with a representative, infinitely-lived household and a representative firm. Discounted utility at time t = 0 is given by: U0 = 1 X t=0 t ? ln ct ? h1+ t 1+ ? , where ?, > 0 and 2 (0, 1) are preference parameters. Households rent out capital and supply labour to firms. For every period, the household chooses consumption (ct ), capital (kt+1 ) and hours worked (ht ) to maximize utility, subject to the following budget constraint: ct + kt+1 = (1 ? ) wt ht + (1 + rk,t )kt , where wt is the wage rate per hour worked, and rk,t is the rental rate of capital, both of which are taken as given by the household. Moreover, 2 (0, 1) is the depreciation rate of capital and ? is a labour income tax rate. (a) Derive the first-order optimality conditions for the household’s choice variables. (b) Derive an expression for the overall elasticity of labour supply with respect to a change in the wage rate. (c) Decompose your answer under (b) into a substitution and an income e↵ect. The representative firm behaves competitively and operates a production function given by yt = kt↵ h1t ↵ , ↵ 2 (0, 1). The firm chooses in every period kt and ht to maximize profits. (d) Formulate the firm’s profit maximization problem and derive the first-order optimality conditions. Suppose now that in some period, the tax rate ? increases unexpectedly and permanently. (e) Derive an analytical expression for the response of equilibrium hours worked, in the initial period of the tax change, in terms of ? and the parameters of the model. (f ) Illustrate equilibrium in the labor market by drawing a supply-demand diagram, and explain the role of the parameters entering in the result derived under (e). Also, use the diagram to explain intuitively how the tax increase mentioned above a↵ects the equilibrium. 2 Question 3. Consumption (25 points) Consider the saving/consumption decision problem of an infinitely-lived household. Discounted utility at time zero is given by: 1 X t=0 1 t ct 1 1 , where ct denotes consumption in period t, 2 (0, 1) is the household’s subjective discount factor, and where > 0 is the coefficient of risk aversion. The household can invest in assets, denoted at+1 , which earn a constant net return r. The household’s budget constraint is given by: at+1 + ct = yt + (1 + r) at , t = 0, 1, 2, 3.. where (1 + r) at is wealth (at the beginning of the period) and yt is income, which evolves according to: yt+1 = ?yt where ? 2 (0, 1). The household chooses a time path {ct , at+1 }1 t=0 which maximizes discounted utility, subject to the two constraints and taking r, a0 > 0 and y0 > 0 as given. (a) Derive the first-order optimality conditions associated with the household’s decision problem and show that consumption growth is contant over time. (b) Derive a condition, in terms of r and , under which consumption declines over time. Explain the intuition behind this result. (c) Find an expression for ct as a linear function of at and yt . (d) Consider the e↵ect on consumption of an unanticipated, one-time change in either beginning-of-period wealth or current income. Which of the two triggers a larger consumption response? Relate your answer to the expression found under (c) and explain the intuition. Now suppose there is a large number of households, indexed by i = 1, 2, ....N, which behave as the household described above. The return r is still exogenously given. However, income and wealth levels may di↵er between households. (e) Consider the following statement: “In this model economy, only the average levels of wealth and income matter for aggregate consumption, not their distributions across households.” Do you agree? Explain your answer. 3 Question 4. Endogenous growth (5 points) (a) Consider the AK model as discussed in week 10. Discuss how the coefficient of risk aversion a↵ects the growth rate of output in this model. Explain the intuition.