Reference no: EM133142651

Question 1 - Consider a consumer choosing the level of consumption for three periods to maximise utility. The intertemporal utility function is

lnc₀ + βlnc₁ + β²lnc₂

c₀, c₁ and c₂ are the levels of consumption in period 0, 1, and 2 respectively. β is the discount factor. The consumer has income y₀, y₁ and y₂ in period 0, 1 and 2 respectively, and could borrow and lend with interest rate r. Assume that the consumer leaves no debt or savings by the end of period 2, the budget constraint of the consumer is c₀ + c₁/(1+r) + c₂/(1+r)² = y₀ + y₁/(1+r) + y₂/(1+r)² =.

(a) For each period, find the level of consumption that maximises the consumer's utility.

(b) What is the effect of an increase in interest rate on the level of consumption in period 0 and period 1? Discuss the economic intuition.

(c) Consider another model of consumer consumption for multiple periods, in which the consumption of the consumer in year t and t + 1 follow c_t+1 = β(1 + r)c_t for t = 0,1,2 ..., where c_t is the level of consumption in year t. It is assumed that the initial consumption is a constant c > 0, i.e. c₀ = c. Find the time path of consumption c_t. How does the time path of c_t depend on the interest rate r and discount factor β?

(d) Discuss the assumptions (including any underlying economic assumptions) for each of the two models. Are these reasonable assumptions, why or why not?

Question 2 - Consider a firm (e.g. a local movie theatre) that could supply services to consumers at both peak time (e.g. weekend and public holidays), and off-peak time (e.g. weekday). The firm offers peak and off-peak prices to the consumers depending on when they use the service - the price for peak service is p₁ and for off-peak service it is p₂. The quantity of services sold in peak and off-peak time is denoted as q₁ and q₂, respectively.

The firm starts with a capacity of Q^-, in which at both peak and off-peak times, the number of customers it serves (i.e. q₁ or q₂) cannot be greater than Q^-(e.g. Q^- is the number of total seats available in the movie theatre).

The firm faces a demand function for peak service as q₁ = (a₁/b₁) - (1/b₁)p₁, and a demand function for off-peak service as q₂ = (a₂/b₂) - (1/b₂)p₂, with a₁, a₂, b₁, b₂ > 0.

For each customer served, it costs the firm c to provide the services, regardless of whether it is a peak time or off-peak period customer. The firm cannot produce negative quantity in either of the peak and off-peak services.

(a) Express the total profit of the firm in terms of q₁, q₂, a₁, a₂, b₁, b₂ and c.

(b) If a₁ = 400, a₂ = 400, b₁ = 1, b₂ = 1 and c = 20, Q^- = 200, what is the profit maximizing quantity of peak and off-peak services for the firm? (Consider all constraints faced by the firm).

(c) If a₁ = 400, a₂, = 400, b₁ = 0.5, b₂ = 1 and c = 20, Q^- = 200, what is the profit maximizing quantity of peak and off-peak services for the firm?

(d) In the long run, the firm could choose the capacity (Q) by paying d for each unit of capacity, with d > 0. (e.g. the movie theatre could adjust the seating and it costs £d for each unit of capacity to be installed and maintained for service.) Assumes that the firm has no pre-existing capacity, so it has to pay for all units of capacity it chooses. What would be the quantity of services provided for peak and off-peak service, and the capacity, that the firm chooses to maximise profit? Assumes that a₁ > c + d and a₂ < c.

(e) For the demand functions given for the peak and off-peak services, what implicit economic assumption(s) did we make? Are they reasonable assumption(s), why or why not?

Question 3 - Let p(t) represent the price of one coin of a relatively new cryptocurrency, at a given time t. The change in the price of the cryptocurrency with respect to time, p(t) is given by 2e^-0.2t - 0.2p(t). One coin of this cryptocurrency has a price of $250.00 at time t = 2.

(a) Solve for the expression showing the cryptocurrency coin price as a function of time. What was the initial price of the cryptocurrency coin when first introduced, i.e. t = 0?

(b) Characterize the time path of price of this cryptocurrency coin.

(c) From your answers in a) and b) is investment in this cryptocurrency coin as a means for financial gain a good idea? Are there any potential economic reasons for this cryptocurrency to follow this particular time path of price? (Answer must be typed).

Question 4 - The labour market is described by the following labour demand and supply functions D(t) = a₀ + a₁W(t) + a₂W^·(t) + W^··(t) and S(t) = β₀ + β₁W(t). The market is in equilibrium if D(t) = S(t). Assume that a₀ = 16, a₁ = 70, a₂ = -16, β₀ = 400, β₁ = 6.

(a) What wage rate, W* is consistent with the labour market being in a long run steady-state, when labour supply equals demand?

(b) Find the time path of wages, when initially at time=0: W(0) = 10, dw/dt(0) = 4. Comment on your answers.

(c) With reference to the labour supply and demand functions you were given and your answers in (a), (b) and (c) is this labour market model economically plausible in the real world?

Question 5 - The aggregate change of consumption, C^·(t) and the aggregate change of the capital stock, K^·(t), are given by the following system of differential equations:

C^·(t) = (0.25K(t)^-0.7 - 0.04)C(t)

k^·(t) = 1.4k(t)^0.2 - 0.06K(t) - C(t)

(a) Find the steady-state of the capital stock and consumption?

(b) Write the above system of differential equations as a linear function of time around the steady-state?

(c) Find the time path of consumption and the capital stock for any level of initial conditions. What are the necessary mathematical conditions for the time paths for consumption and the capital stock to converge to the steady-state?

(d) Following from your answers in (a), (b) and (c), is such as model of the capital stock and consumption plausible for a real world economy?